Compresia Si Contruirea de Imagini cu Ajutorul Fractalilor
Cuprins
Capitolul 1 Introducere
Capitolul 2 Fractali – concepte [i algoritmi
2.1 Defini]ii .……………………………………………………………………………….. 5
2.2 Clasific#ri ………………………………………………………………………………. 5
2.3 Caracteristici [i propriet#]i ………………………………………………….………. 8
2.3.1 Fragmentarea la infinit ………………………………………………….……. 8
2.3.2 Auto-similaritatea ………………………………………………….………….. 8
2.3.3 Invarian]a la transla]ii ………………………………………………….……… 8
2.3.4 Dimensiune fractal# (dimensiune frac]ionar# sau dimensiune
de auto-similaritate) ………………………………………………….………………………….. 9
2.3.5 M#sura Hausdorff [i dimensiunea Hausdorff (considera]ii
matematice) ………………………………………………………………………………………… 9
2.4 Metode de determinare a dimensiunii fractale.
2.4.1 Metoda clasic# ………………………………………………………………… 17
2.4.2 Metoda “SandBox” sau metoda cutiei cu nisip …………………………… 18
2.4.3 Metoda Box-Counting sau metoda c#su]elor num#rate ……………………. 18
2.4.4 Dimensiunea Box-Counting modificat# ………..…………………………… 22
2.4.5 Tehnici de calculare a dimensiunii ……………………………………………….. 23
2.5 O abordare matematic# a unei metode generale de ob]inere
a fractalilor ………………………………………………………………………………………. 25
2.6 Fractali liniari …………………………………………………………………………… 29
2.6.1 Triunghiul lui Sierpinski ………………………………………………………. 29
2.6.2 Curba Peano …………………………………………………………………… 31
2.6.3. Curba Koch ……………………………………………………………………. 32
2.6.4 Curba dragonului ………………………………………………………………. 34
2.6.5 Praful lui Cantor ……………………………………………………………….. 35
2.6.6 Multiuniversul lui Fournier …………………………………………………….. 36
2.7 Fractali neliniari ……………………………………………………………………………. 37
2.7.1 Ansamblul Mandelbrot ………………………………………………………… 37
2.7.2. Ansamblurile Julia ……………………………………………………………. 41
2.7.3 Atractori stranii ……………………………………………………………….. 43
2.7.4. Generarea atractorilor stranii ………………………………………………. 45
2.7.5 Sistemul Lorentz ……………………………………………………………….. 52
2.7.6. Atractorul straniu Rossler …………………………………………………… 57
2.8. Automatele celulare …………………………………………………………………… 58
2.9 Aplica]ii curente ale fractalilor [i haosului \n economie …………………….. 62
2.9.1 Introducere ……………………………………………………………………… 62
2.9.2 Segmentarea pie]elor de capital ………………………………………………. 63
2.9.3 Alegerea portofoliului optim de investi]ii …………………………………… 65
CAPITOLUL 3 IMAGINI : STANDARDE {I ALGORITMI
3.1 Introducere …………………………………………………………………………….. 66
3.2 Particularit#]i ale fi[ierelor de imagine ……………………………………………. 66
3.3 Algoritmul RLE(Run Length Encoding) …………………………………………….. 67
3.4 Formatul graphic BMP ………………………………………………………………… 68
3.4.1 Structura BMP …………………………………………………………………. 68
3.4.2 Componentele structurii BMP ………………………………………………. 69
3.5 Formatul grafic GIF …………………………………………………………………………………. 70
3.5.1 Organizarea fi[ierului GIF …………………………………………………….. 71
3.5.2 Versiunea GIF87a …………………………………………………………….. 71
3.5.3 Versiunea GIF89a …………………………………………………………….. 73
3.6 Formatul grafic PCX …………………………………………………………………….. 75
3.6.1 Organizarea fi[ierului PCX ……………………………………………………. 75
3.6.2 Formatul DCX …………………………………………………………………. 77
3.7 Formatul TIFF ……………………………………………………………………………. 77
3.8 Standardul JPEG …………………………………………………………………………………….. 81
3.8.1 Modul secven]ial ………………………………………………………………. 82
3.8.2 Modul progresiv ……………………………………………………………….. 83
3.8.3 Modul ierarhic …………………………………………………………………. 83
3.8.4 Compresia f#r# pierderi ………………………………………………………. 83
3.8.5 Pa[ii de prelucrare ai compresiei bazate pe DCT ………………………… 84
3.8.6 Organizarea fi[ierului JPEG …………………………………………………. 87
Capitolul 4. Compresia/decompresia: no]iuni [i algoritmi
4.1 Concepte .………………… .………………… .………………… .…………………….. 89
4.1.1 Seturi de date .………………… .………………… .…………………………. 89
4.1.2 Clasificarea seturilor de date .………………… .……………………………… 89
4.1.3 Conceptul de compresie .………………… .…………………………………. 90
4.2 Defini]ii .………………… .………………… .………………… .……………………….. 90
4.3 Indicatori de performan]# ai algoritmilor .………………… .……………………… 90
4.3.1 Gradul de compresie .………………… .………………… .………………. 91
4.3.2 Rata de compresie .………………… .………………… .……………………. 91
4.3.3 Lungimea minimal# .………………… .………………… .…………………… 91
4.3.4 {iruri comune .………………… .………………… .…………………………. 91
4.4 Erori ale compresiei de date …………………………………………………………………… 94
4.4.1 Algoritmi statici .………………….………………….…………………………… 94
4.4.2 Algoritmi dinamici .………………… .………………… .……………………… 95
4.5 Compresia/decompresia imaginilor .………………… .……………………………. 96
4.6 Compresia cu fractali .………………… .………………… .…………………………. 99
4.6.1 Ce este compresia de imagini cu fractali ? .………………………………. 99
4.6.2 Sisteme de func]ii iterative .………………… .……………………………. 100
4.6.3 Auto-similaritatea \n imagini .………………… .…………………………. 100
4.6.4 Sisteme de func]ii iterative parti]ionate (Partitioned Iterated Function Systems – PIFS) .………………… .………………… .………………… .………………. 101
4.6.5 Comprimarea imaginii .………………… .………………… ……………… 102
4.6.6 Comprimarea imaginilor color .………………… .…………………………. 104
4.6.7 O compara]ie \ntre compresia imaginilor cu ajutorul fractalilor [i
compresia cu ajutorul standardului JPEG .………………… .………………… ……… 104
Capitolul 5 Aplica]ii …………………………………………………………….. 105
Bibliografie .………………… .……………….…………………
=== 1_95 ===
CAPITOLUL 1
Introducere
Tema lucr#rii prezentate \n capitolele ce vor urma este " Construirea [i compresia de imagini bazate pe fractali ". Aceast# lucrare a fost prezentat# la sesiunea [tiin]ific# mai 1999 a facult#]ii “Cibernetic#, Statistic# [i Informatic# Economic#” .Obiectivele lucr#rii sunt:
– prezentarea general# a fractalilor ( defini]ii, clasific#ri, modele, programe de generare);
– prezentarea standardelor utilizate \n compresia imaginilor ( standarde, formate de fisiere, construc]ie imagini);
– studiul compresiei/decompresiei imaginilor cu ajutorul fractalilor ( defini]ii, algoritmi de compresie/decompresie existen]i, performan]e, software);
– realizarea unei aplica]ii care s# pun# \n valoare cele prezentate anterior;
– concluzii (avantaje, deschideri pentru viitor)
Obiectele fractale foarte diverse, dintre care multe ne sunt c@t se poate de cunoscute, cum ar fi p#m@ntul, cerul [i oceanul, sunt studiate cu ajutorul unei largi familii de obiecte geometrice, judecate p@n# in prezent ca fiind esoterice [i inutilizabile, dar despre care s-a ar#tat c# dimpotriv# pot fi integrate \n geometria elementar# prin ceea ce merit#, prin simplitate, diversitate [i extraordinara amploare a noilor lor aplica]ii. De[i studiul lor ]ine de [tiin]e diferite, printre altele de geomorfologie, astronomie sau de teoria turbulen]ei, obiectele naturale \n discu]ie au \n comun aspectul extrem de neregulat sau fragmentar.
No]iunea ce serve[te drept conduc#tor acestei noi geometrii va fi denumit# prin unul din cele doua neologisme sinonime ,"obiect fractal" [i "fractal" termeni ce pornesc de la adjectivul latin fractus care semnific# "neregulat sau intrerupt".
Istoria fractalilor nu este lung#. A inceput brusc in 1975 cu lucrarea revolu]ionara a matematicianului Benoit Mandelbrot "O teorie a seriilor fractale", care mai t@rziu a devenit cartea sa manifest "Geometria fractala a naturii". Mandelbrot a inventat cuv@ntul fractal, pentru a reuni munca multor altora dinaintea sa.
Matematicieni ca Waclaw Sierpinski, David Hilbert, Georg Cantor [i Helge Von Koch au creat primii fractali, \n general ca exerci]ii abstracte, neav@nd nici o idee despre semnifica]ia lor.
Fractalii sunt primele forme grafice care nu sun bazate pe linii drepte sau liniaritate. Pentru c# fractalii sunt neliniari, ei nu pot fi construi]i din linii ne\ntrerupte, de aceea aven nevoie de un calculator pentru a-i trasa.
Geometria fractala [i teoria haosului revolu]ioneaza matematica [i [tiin]a, oferindu-ne o cale nou# de a percepe realitatea. De la geologi p@n# la astronomi, de la metereologi p@n# la exper]i financiari, de la arti[ti p@n# la matematicieni, nici un domeniu nu a sc#pat neatins de aceast# modificare radical# a modului \n care g@ndim, muncim [i cre#m.
{tiin]a [i matematica au c#utat \ntotdeauna ordinea \ntr-un univers haotic. Dar de ce o numim [tiin]a "haosului" ? Pentru c# se g#se[te o ordine \n fenomene pe care oamenii le vedeau p@na acum ca fiind complet haotice, [i totu[i aceste fenomene r#m@n complet imprevizibile [i necontrolabile.
P@n# de cur@nd foloseam linia ca element de baza \n \n]elegerea universului. {tiin]a haosului folose[te o geometrie diferit#, denumit# fractali. C@teva lucruri pe care fractalii le modeleaz# mai bine: plantele, vremea, curgerea fluidelor, activit#]ile geologice, ritmurile corpului uman, comportarea animalelor in grup, modele socio-economice… [i lista poate continua.
Departe de a fi abstrac]ii fractalii se apropie mult mai mult de realitate dec@t ne putem da seama la un prim contact. De fapt obiectele nonfractale sunt cele care nu concorda cu realitatea sunt abstracte [i nu au legatur# cu ceea ce cunoa[tem despre diferite domenii.
Pentru a sugera care obiecte trebuie s# fie considerate ca fiind fractale, putem \ncepe prin a ne aminti ca, \n efortul s#u de a descrie lumea [tiin]a apeleaz# la serii de imagini [i modele din ce in ce mai "realiste". Cele mai simple sunt continuumurile perfect omogene, cum ar fi un fir sau un univers cu densitate uniform#, respectiv un fluid cu distribu]ie uniform# a temperaturii, densit#]ii, presiunii [i vitezei. Fizica a putut triumfa identific@nd numeroase domenii unde asemenea imagini sunt extrem de utile, \n particular ca puncte de plecare la care s# se adauge apoi diver[i termeni de corec]ie. Dar \n alte domenii realitatea se relev# a fi at@t de neregulat#, \nc@t modelul continuu perfect omogen devine inutilizabil, el neput@nd servi nici m#car ca o prima aproxima]ie. Se ajunge astfel la introducerea fractalilor, a principiilor ce stau la baza lor \n cercetarea anumitor fenomene care nu pot fi explicate cu ajutorul altor teorii.
Geometria fractal# este caracterizat# de doua op]iuni: alegerea unor probleme din s@nul naturii, c#ci a descrie \ntregul ar fi o ambi]ie f#r# speran]# [i f#r# interes, [i alegerea uneltelor din s@nul matematicii, c#ci a c#uta aplica]ii matematicilor doar pentru simplul motiv ca ele sunt frumoase, n-a produs dec@t decep]ie.
Unii autori definesc fractalul ca fiind un obiect matematic care poate fi detaliat indiferent de c@t de aproape \l privim [i indiferent de c@t il m#rim. Ceea ce vedem este un model infinit care este cumva compresat \ntr-un spa]iu finit.
Una din caracteristicile principale ale oric#rui obiect fractal este dimensiunea. Ea m#soara gradul lui de neregularitate [i de frac]ionare. Un fapt foarte important contrar numerelor intregi ale dimensiunilor obi[nuite, dimensiunea fractal# poate s# fie o frac]ie simpl# cum ar fi 1/2 sau 5/3, sau la fel de bine poate fi un num#r ira]ional ca 4/log3~=1,2618… sau PI. Astfel este util s# spunem despre anumite curbe plane foarte neregulate c# dimensiunea lor fractal# este \ntre 1 [i 2, despre anumite suprafe]e cu forma de frunz# [i pline de circumvolu]iuni c# dimensiunea lor fractal# este \ntre 2 [i 3 [i \n sf#r[it s# definim o d@r# de praf ca av@nd dimensiunea fractala \ntre 0 [i 1. O alt# defini]ie spune c# dimensiunea fractal# a unui obiect este dat# de gradul s#u de iregularitate indiferent de scala la care ne referim. O dimensiune fractal# mai mare \nseamn# deci c# fractalul este mai neregulat ceea ce are drept consecin]# o cre[tere mai rapid# a acesteia. Deci putem spune c# un fractal este un obiect care are dimensiunea fractal# mai mare dec@t m#rimea sa obi[nuit#.
Referindu-ne la leg#tura fractalilor cu haosul putem da [i o defini]ie a unui sistem dinamic care \n aceasta concep]ie reprezint# o colec]ie de p#r]i care interac]ioneaz# \ntre ele ( fiecare cu cealalt# ) [i se schimb# de fiecare dat#. Un sistem dinamic este haotic \n situa]ia \n care mici schimb#ri \n condi]iile ini]iale ale sistemului conduc la schimb#ri importante dup# o anumit# perioad#.
O defini]ie \n sens intuitiv a fost pentru prima oar# dat# de c#tre Mandelbrot \n 1989 astfel: "o figura geometric# sau un obiect natural este fractal dac# \ntrune[te urm#toarele caracteristici:
– p#r]ile sale componente au aceea[i form# sau structura cu \ntregul, ]in@nd cont de faptul c# ele sunt la o scal# diferit# [i pot fi u[or deformate;
– forma sa trebuie s# fie extrem de neregulat#, extrem de \ntrerupt# sau fragmentat#;
– el con]ine elemente distincte, care scalate sunt destul de variate [i formeaz# o gam# foarte larg#;
Se impune o clasificare a fractalilor \n func]ie de diverse criterii.
Din punct de vedere al modului de generare, fractalii se impart \n:
– fractali naturali existen]i \n natura sau care sunt crea]i \n urma unor procese naturale. De exemplu: mun]ii, ]#rmurile, sistemul nervos, arborii etc.
– fractalii artificiali modele ale fractalilor naturali. Sunt \nt@lni]i [i sub denumirea de seturi de fractali sau ansambluri de fractali. Am mai putea identifica o grupa de fractali artificiali \n natura. Cazul sistemului social, economic etc.
Din punct de vedere al transform#rilor care stau la baza gener#rii unui
fractal, ace[tia se \mpart \n:
– fractali liniari care sunt genera]i cu ajutorul transform#rilor geometrice liniare aplicate asupra unor puncte, segmente suprafe]e sau corpuri(de exemplu curbele Von Koch).
– fractali neliniari se ob]in cu ajutorul transform#rilor geometrice neliniare([i ace[tia pot fi artificiali sau naturali). _n func]ie de transform#rile ce stau la baza ob]inerii acestora avem fractali auto-p#tratici, fractali cu autoinversiune, fractali aleatori, fractali determini[ti.
Fractalii sunt imagini complexe dar \n mod uluitor \n majoritatea cazurilor ace[tia pot fi reprezenta]i cu ajutorul unei simple ecua]ii care consuma un spa]iu foarte redus. _n unele cazuri este posibil s# identific#m [abloane auto-similare \ntr-o imagine grafic# [i s# compres#m spatiul ocupat de aceast# imagine pe disc prin stocarea acestei similarit#]i dec@t s# stoc#m \ntreaga imagine.
La inceputul anilor '80, matematicianul Michael Barnsley s-a al#turat celor care manifestau un interes din ce in ce mai mare in leg#tura cu teoria fractal# [i cu aplica]iile posibile ale acesteia \n via]a de zi cu zi precum [i \n alte domenii de interes. Michael a fost fascinat \n mod deosebit de anumite ferigi. Observ@nd modul \n care fiecare frunza se aseam#n# cu \ntreaga plant#, el a scris un program simplu pentru a modela aceste caracteristici. Imaginea rezultat# era mult mai real# dec@t s-a ateptat [i a devenit \n cur@nd unul din cei mai faimo[i fractali din lume. Barnsley a continuat s# dezvolte o metod# unic# de desenare a fractalilor: "Jocul haosului". Chiar [i mai important, \n 1985 Barnsley [i John Elton au demonstrat c# orice imagine din lume poate fi reprezentat# cu ajutorul unei bine cunoscute categorii de fractali. Acesta era un pas uria[ \nainte pentru o comunitate intelectual# inundat# de fractali, dar c#reia \i lipsea un sistem inteligibil pentru reprezentarea lor. O tehnic# crea mun]i fractali, o alta pornea automate celulare [i o alta simula \nregistr#rile grafice ale cutremurelor. {i o alt# tehnic# era necesar# pentru a realiza minunatele v@rtejuri [i focaliz#rile. Barnsley [i Elton au prev#zut o metod# unic# [i simpl# de realizare a aproape tuturor imaginilor auto-similare, incluz@nd [i toate imaginile despre care nimeni \nainte nu se g@ndise ca ar fi auto-reflective.Prima aplica]ie major# a muncii lor era comprimarea imaginii. Prin transformarea lor in fractali, Barnsley era capabil s# comprime imagini foarte mari \n coduri foarte mici, ob]in@nd un raport de comprimare de peste zece mii la unu.
Comprimarea fractal# a imaginii creeaz# noi posibilit#]i captivante, cum ar fi transmiterea \n timp real a imaginilor video \n mi[care prin liniile telefonice normale.
_n anii '90 fractalii sunt larg folosi]i. Produc]ii cinematografice importante \i folosesc pentru efecte speciale, sistemele de redare grafic# pe calculator \i folosesc pentru a crea structuri naturale, oamenii de [tiin]# [i matematicienii i-au transformat \ntr-o unealt# indispensabil# pentru munca lor. Pe m#sura ce poten]ialul acestei noi geometrii este recunoscut din ce \n ce mai mult [i calculatoarele din ce in ce mai rapide fac interactiunea mai u[oar#,instrumentele de desenare fractal# vor deveni o parte a majorit#]ii sistemelor de grafic# pe calculator.
=== 2_95 ===
Fractali – concepte [i algoritmi
1.1 Defini]ii
Numele de fractal, sau de mul]ime fractal# a fost ales de Mandelbrot \ntr-o lucrare, pentru a face referire la obiecte care erau at@t de neregulate \nc@t nu puteau fi descrise cu geometria tradi]ional#. Cuv@ntul fractal provine din latinescul fractus ceea ce \nseamn# \ntrerupt sau spart.
Fractalul este totu[i o form# geometric#, cu toate c# nu poate fi descris cu ajutorul geometriei tradi]ionale. Iar aceastei forme geometrice i se pot asocia anumite propriet#]i, dac# este privit# ca mul]ime.
Dac# not#m cu F o mul]ime fractal# atunci aceast# mul]ime are urm#toarele propriet#]i:
F are o structur# foarte fin#, ceea ce \nseamn# c# exist# detalii la o scar# oric@t de mic#.
F este at@t de neregulat#, \nc@t nu poate fi descris# \n limbaj geometric tradi]ional, la nivel local sau global.
Deseori F are o form# de autosimilaritate definit# \n sens geometric, sau o autosimilaritate definit# \n sens statistic (aproximativ#).
Uzual dimensiunea fractalului definit# \ntr-un anumit fel, este mai mare dec@t dimensiunea topologic#.
\n cele mai multe cazuri, F este definit# \ntr-un mod foarte simplu poate recursiv.
Defini]ia fractalului dat# de c#tre Mandelbrot (1989) se bazeaz# pe descrierea acestor propriet#]i: “Un fractal este acea figur# geometric# sau obiect natural care combin# urm#toarele caracteristici:
A.P#r]ile sale componente au aceea[i form# sau structur# cu \ntregul, ]in@nd cont de faptul c# ele sunt la o scar# diferit# [i pot fi u[or deformate.
B.Forma sa trebuie s# fie extrem de neregulat#, \ntrerupt# sau fragmentat#.
C.El con]ine elemente distincte,care scalate sunt foarte variate [i formeaz# o gama foarte larg#.
Mandelbrot nu face distinc]ie \ntre fractal [i obiect fractal. Define[te obiectul fractal ca fiind un termen ce \nlocuie[te denumirea de fractal, deoarece precizeaz# c# este vorba de un obiect natural. Obiectul natural este rezonabil [i util de reprezentat matematic printr-un fractal.
De asemenea a definit ansamblul fractal ca fiind un termen ce \nlocuie[te denumirea de fractal deoarece precizeaz# c# este vorba despre un model informatic sau matematic al unui fractal.
1.2 Clasific#ri
Dac# se ia \n considerare modul de generare al fractalilor putem avea dou# categorii :
Fractali naturali (obiecte fractale): \i g#sim \n natur# sau sunt crea]i \n urma unor procese naturale. Se pot da exemple prin mun]i, ]#rmuri de continente, sistemul nervos, arborii, sistemul de ramifica]ii al bronhiilor, etc.
Fractali artificiali (ansambluri de fractali): reprezint# modele informatice sau matematice ale fractalilor naturali.
Dup# transformarea care st# la baza gener#rii unui fractal avem:
Fractali liniari: sunt fractalii care sunt genera]i cu ajutorul unor transform#ri geometrice liniare, aplicate asupra unor puncte, suprafe]e sau corpuri.
Exemplu : Curbele Von Kock, care \n principiu se genereaz# astfel:
Pas 1 – se pleac# de la un triunghi echilateral cu latura av@nd o lungime finit#.
Pas 2 – fiecare latur# se \mparte \n trei p#r]i egale.
Pas 3 – segmentul din mijloc se elimin# pentru fiecare latur# [i se \nlocuie[te cu un unghi al c#ror laturi sunt egale cu segmentul eliminat.
Pas 4 - Pentru fiecare segment al figurii ob]inute se repet# de la pasul 2.
Dup# repetarea procesului se ob]ine o linie poligonal# \nchis# cu latura din ce \n ce mai mic# ca \n figura 1.
Itera]ia 1 Itera]ia 2
fig.1
Fractalii liniari pot fi artificiali sau naturali. Exemple de fractali liniari naturali sunt sistemul de ramifica]ii al bronhiilor [i arborii. Exemplul anterior, curba Von Kock este un exemplu de fractal liniar artificial.
Fractali neliniari: se ob]in cu ajutorul transform#rilor geometrice neliniare. {i ace[tia pot fi la r@ndul lor artificiali sau naturali.
Exemple de fractali neliniari naturali: mun]ii, sistemul nervos, norii etc.
Exemple de fractali neliniari artificiali: mul]imea Mandelbrot [i mul]imea Julia.
_n func]ie de transformarea care st# la baza gener#rii fractalilor neliniari artrificiali avem:
Fractali auto-patratici: sunt ob]inu]i cu ajutorul generatorului patratic z2+c, unde z [i c sunt din mul]imea numerelor complexe.
De exemplu mul]imea Mandelbrot este generat# astfel:
Dac# se consider# planul complex de laturi lx, ly cu col]ul st@nga jos x0, y0. Un punct curent din acest domeniu va fi notat cu z0 , iar fiec#rui punct z i se aplic# urm#toarea transformare:
T:DD
T(z)=z2+z (1.1)
z {z0 ,z02 +z0, (z02 +z0)2 +z02 +z0 ,…}
Pe parcursul acestor itera]ii se testeaz# modulul lui z
M(z)=Im(z)2 +Re(z) 2 (1.2)
unde:
Im(z) reprezint# partea imaginar# a lui z
Re(z) reprezint# partea real# a lui z
Dac# dup# un num#r de itera]ii n, M4 atunci z va apar]ine mul]imii Mandelbrot.
Algoritmul de ob]inere a unei imagini pe ecranul calculatorului, care s# reprezinte mul]imea Mandelbrot, este urm#torul:
Cite[te : x0,y0 col]ul st@nga jos al zonei \n care va fi inclus# mul]imea
lat intervalul din domeniul complex care este explorat
lecran latura p#tratului \n care este inclus# imaginea ob]inut#
n num#rul de itera]ii
pas lat/lecran
a x0
pentru i=1 la lecran cu pasul 1 execut#
aa+pas
by0
pentru j=1 la lecran cu pasul 1 execut#
bb+pas
x0
y0
k0
repet#
xxx•x-y•y+a
y 2•x•y+b
x xx
Mx•x+y•y
kk+1
p@n# c@nd: M4 sau kn
dac# M4 atunci aprinde(i,j)
sf@r[it
Cu acest algoritm se ob]ine o imagine ca \n figura 2
fig.2
Fractali de auto-inversiune se ob]in prin aplicarea unor transform#ri geometrice neliniare inverse asupra unor puncte suprafe]e sau corpuri.
Dup# modelul de generare pot fi:
Fractali aleatori: sunt genera]i aleator folosind procese stochastice.
Fractali determini[ti: sunt ob]inu]i cu ajutorul unor modele recursive, \n general simple, pentru a fi utili \n studierea fractalilor aleatori.
1.3 Caracteristici [i propriet#]i
1.3.1 Fragmentarea la infinit
Reprezint# proprietatea fractalului de a reg#si detalii la o scal# oric@t de mic# [i se traduce printr-o structur# foarte fin#. De exemplu dac# vrem s# calcul#m lungimea unui ]#rm [i folosim o rigl# ca instrument de m#sur# reprezent@nd un km. Cu acest instrument prin m#surare vom neglija o serie de detalii, iar m#surarea va fi aproximativ# ob]in@nd o lungime L1 a ]#rmului. Mai departe dac# alegem o unitate de m#sur# mai mic#, de exemplu 1/2Km, avem posibilitatea s# lu#m \n considerare noi detalii ceea ce \nseamn# c# rezultatul m#sur#rii va fi L2L1. Dac# se continuu#, c@nd lungimea riglei va tinde c#tre 0 lungimea ]#rmului m#surat va tinde la infinit.
Putem trage concluzia c# ]#rmul are o lungime infinit# [i ocup# o suprafa]# finit#. Se poate spune deci c# proprietatea de fragmentare la infinit d# fractalului caracteristica specific# de a avea o lungime infinit# [i s# ocupe un spa]iu finit, sau s# aib# un spa]iu infinit \ntr-un volum finit.
1.3.2 Auto-similaritatea
Este proprietatea fractalului de a avea orice detaliu al s#u asem#n#tor cu \ntregul la orice scal#.
Deseori fractalii sunt auto-similari la diferite grade. Pot fi auto-similari de nivel geometric (definirea similarit#]ii se face geometric), auto-similari de grad aproximativ (definirea similarit#]ii se face statistic).
1.3.3 Invarian]a la transla]ii
Proprietatea unui obiect fractal de a reg#sii un detaliu al s#u prin suprapunerea acestuia peste o alt# zon#, dup# ce a fost translatat dup# o anumit# direc]ie. La fel ca auto-similaritatea nu este o proprietate universal# pentru fractali.
1.3.4 Dimensiune fractal# (dimensiune frac]ionar# sau dimensiune de auto-similaritate)
Mandelbrot a definit dimensiunea fractal# astfel: “Dimensiunea fractal# este un num#r care cuantific# gradul de iregularitate [i de fragmentare a unui ansamblu fractal sau a unui obiect natural [i care se reduce \n cazul obiectelor geometrice Euclidiene la dimensiunea lor topologic# .“
Dimensiunea fractal# este dat# \n general printr-un num#r real [i cuantific# gradul de fragmentare [i gradul de auto-similaritate, deci asociaz# o valoare structurii obiectului.
Dac# dou# curbe fractale au lungime infinit# [i dimensiune topologic# 1, atunci este normal ca o comparare a lor s# se fac# prin intermediul unei altfel de m#suri, de exemplu dimensiunea fractal#.
1.3.5 M#sura Hausdorff [i dimensiunea Hausdorff (considera]ii matematice)[13]
Dimensiunea Hausdorff este o alt# dimensiune frac]ionar# care m#soar# gradul de fragmentare al unui obiect fractal.
Dac# not#m cu U orice submul]ime nevid# din Rn cu proprietatea |U|=sup{x-y|x,y U} {Ui} este o mul]ime finit# de mul]imi cu diametrul mai mic dec@t d care acoper# pe F adic#
F {Ui} este o acoperire d a lui F.
Dac# F Rn [i s0 pentru d0 not#m:
Hds(F)=inf {:{Ui} este o acoperire d a lui F} (1.4)
Dac# ne uit#m la toate acoperirile lui F de mul]imi de diametru mai mic dec@t d [i c#ut#m s# minimiz#m suma, atunci cu c@t d scade clasa acoperirilor permisibile se reduce [i vom avea:
Hs(F)=(Hds(F)) (1.5)
Aceast# limit# exist# pentru orice subset F Rn
Definim Hs(F) ca fiind o m#sur# s dimensional# Hausdorff a lui F ce are urm#toarele propriet#]i:
Hs()=0
dac# EF atunci Hs(E) Hs(F)
dac# {Fi} este o colec]ie num#rabil# de mul]imi Borel disjuncte atunci:
Hs(F)= Hs(Fi)
dac# F este o mul]ime Borel din Rn atunci :
Hn(F)=cnvoln(F)
unde cn este volumul unei sfere n dimensionale de diametru similar pentru dimensiuni mai mici din Rn
Exemplu:
H2(F)=4/•aria(F) pentru cazul \n care F este o suprafa]# neted#.
la m#rirea cu un factor , lungimea unei curbe este multiplicat# cu , aria unei suprafe]e cu 2 , volumul unui corp multiplicat cu 3, a[a cum
Hs(F)= s Hs(F) unde F={x:xF} este mul]imea F scalat# cu un factor
Aceast# proprietate se nume[te proprietatea de scalare [i este fundamental# teoriei fractale.
Demonstra]ie:
Dac# {Ui} este o acoperire d a lui F atunci {Ui} este o d acoperire a lui F unde :
Hsd(F) |Ui|s=ss Hsd(F)
atunci deoarece aceasta se men]ine pentru orice acoperire d a lui F lu@nd d tinz@nd c#tre 0 rezult#
Hs(F) s Hs(F) [i lu@nd =1/ [i F=F rezult# : Hs(F) 1/s Hs(F)
s Hs(F) Hs(F) de unde rezult# ceea ce trebuia demonstrat.
Propozi]ia 1.1
Fie F Rn [i f:F Rm o mapare a.\
|f(x)-f(y)|c|x-y| (1.6)
x,yF pentru constante c [i pozitive, atunci pentru oricare s avem :
Hs/(f(F)) cs/ Hs(F) (1.7)
Demonstra]ie :
Dac# {Ui} este o acoperire d a lui F, atunci deoarece |f(FUi)|c|Ui| rezult# {f(FUi)} este o acoperire a lui f(F) unde =cd cum |f(FUi)|s/cs/ |Ui|s
Hs/(f(F)) cs/ Hsd(F) dac# d0 0 (1.7)
Dimensiunea Hausdorff
Se observ# \n (1.4) c# pentru orice F dat [i d1 c# Hsd(F) nu cre[te cu s [i prin (1.5) c# de asemenea Hs(F) nu cre[te. De fapt dac# ts [i {Ui} este o acoperire d a lui F avem:
|Ui|t dt-s|Ui|s (1.8)
Htd(F) dt-s Hsd(F) (1.9)
lu@nd d0 vedem c# dac# Hs(F) atunci Ht(F)=0 cu ts, aceasta \nseamn# c# exist# o valoare critic# a lui s pentru care valoarea lui Hs(F) sare de la la 0 (vezi figura 4) .
Hs(F)
0 s dimHF
fig.4
Aceast# valoare critic# se nume[te dimensiune Hausdorff [i o not#m cu dimHF
dimHF=inf {s: Hs(F)=0} =sup {s: Hs(F)=} (1.9)
de unde Hs(F)= , dac# sdimHF
0 , dac# sdimHF
Dac# s= dimHF Hs(F) poate fi 0 sau infinit sau poate s# ia o valoare \ntre acestea dou#.
Propriet#]i:
(i) pentru mul]imi deschise
FRn este o mul]ime deschis# rezult# dimHF=n
(ii) pentru mul]imi netede
F este neted#(continu# [i diferen]iabil#) de tip m dimensional \n Rn atunci dimHF=m
(iii) monotonicitate
EF atunci dimHE dimHF Hs(E) Hs(F) pentru orice s
(iv) stabilitate num#rabil#
F1,F2,..,Fi,… este un [ir num#rabil de mul]imi atunci:
Fi= de asemenea dimHFidimHFJ pentru orice j (din proprietatea de monotonicitate. Pe de alt# parte dac# sdimHFi pentru to]i I rezult# Hs(Fi)=0 Hs(Fi)=0 d@nd inegalitate invers#
(v) pentru mul]imi num#rabile
Dac# F este num#rabil# rezult# dimHF=0.Pentru situa]ia Fi arer un singur punct H0(Fi)=1 [i dimHFi=0 deci prin stabilitate num#rabil# avem dimHFi=0
Propozi]ia 1.2(proprietatea de transformare a dimensiunii Hausdorff)
Fie FRn [i lu#m f:F Rn cu |f(x)-f(y)|c|x-y| atunci dimHf(F)1/dimHF
Demonstra]ie:
sdimHF, din propozi]ia 1.1 rezult# Hs/(f(F)) cs/ Hs(F)=0 dimHf(F)s/ pentru to]i sdimHF.
Defini]ie 1.1
Fie FRn f:FRm este o transformare bi-Lipschitz dac#:
c1|x-y||f(x)-f(y)|c2|x-y| unde 0c1c2 (1.10)
Propozi]ia 1.3
dac# f:FRm o transformare Lipschitz atunci dimHf(F)dimHF
dac# f:FRm este o transformare bi-Lipschitz dimHf(F) dimHF
Demonstra]ie:
reiese din propozi]ia 1.2 lu@nd =1 [i f-1:f(F)F rezult# [i cealalt# inegalitate adic# dimensiunea Hausdorff este invariant# la transformarea bi-Lipschitz. Dac# dou# mul]imi au dimensiuni diferite nu pot exista map#ri bi-Lipschitz de la una la alta.
_n topologie dou# mul]imi sunt considerate la fel dac# exist# un homeomorfism \ntre ele. Din punct de vedere al geometriei fractale dou# mul]imi sunt la fel dac# exist# o mapare bi-Lipschitz \ntre ele. A[a cum invarian]a topologic# este folosit# pentru a distinge \ntre mul]imi nehomeomorfice, putem c#uta parametrii (chiar dimensiunea) pentru a distinge \ntre mul]imi care nu sunt echivalente bi-Lipschitz.
Deoarece transformarea bi-Lipschiz este neap#rat continu# (1.10), parametrii topologici indic# un start \n aceast# direc]ie [i dimHF ([i alte defini]ii ale dimensiunii) arat# caracteristicile distinctive \ntre fractali.
_n general dimensiunea unei mul]imi ne spune foarte pu]in despre propriet#]ile sale topologice. Orice mul]ime de dimensiune mai mic# dec@t 1 este at@t de spart# \nc@t poate fi disconect# total, adic# nici unul dintre dou# puncte nu sunt aceea[i component# conect#.
Propozi]ia 1.4
FRn cu dimHF1 este total disconect#
Fie x,yF xy f:Rn:[0,) cu f(z)=|z-x| deoarece f nu cre[te distan]a |f(z)-f(w)||z-w| rezult# din propozi]ia 1.3 (a) dimHf(F)dimHF1 [i f(F)R, H1=0(lungime 0) are un complement dens.
Aleg@nd rF cu 0rf(y) F={zF:|z-x|r}{zF:|z-y|r}
F este con]inut# \n dou# componente disjuncte cu x \ntr-un set [i y \ntr-altul x [i y sunt \n diferite componente conecte ale lui F.
Calcularea dimensiunii Hausdorff
Dac# ts [i {Ui} este o acoperire a lui F avem:
Uitdt-s|Ui|s
sau Hsd(F)dt-s Hsd(F) lu\nd d0 dac# Hs(F) Ht(F)=0 pt ts, adic# exist# un punct critic s de la care Hs(F) sare de la la 0.
Exemple de calcul ale dimensiunii Hausdorff:
Fie F mul]imea Cantor nisip (La fiecare pas al construc]iei p#tratele sunt \mp#r]ite \n 16 p#trate cu o p#trime ca lungime \n care acela[i model de 4 p#trate este re]inut ca \n fig.5.
Fig.5
rezult# 1 H1(F)2
Calcularea: lu@nd acoperirea lui F ca fiind 4k p#trate de m#rime 4-k [i de diametru
d=4-k2 \n etapa k estim#m:
H1d(F)4k4-k2 pentru k [i d0 d# H1(F)2.
Pentru o estimare mai bun# :
Fie p o proiec]ie ortogonal# pe axa x.Proiec]ia ortogonal# nu cre[te distan]a |p(x)-p(y)||x-y| dac# x,yR2 p este o mapare Lipschitz. Prin proprietatea construc]iei mul]imii F proiec]ia sau umbra sa pe axa x este intervalul [0,1]. Folosind (1.8), (1.9) rezult# 1=lungime([0,1]) = H1([0,1])= H1(p(F)) H1(F)
Not#m c# acela[i argument [i rezultat se men]ine pentru o mul]ime ob]inut# prin divizare repetat# a p#tratelor \n m2 p#trate de lungime 1/m din care este ob]inut c@te un singur p#trat din fiecare coloan#.
Acest mod de a folosii proiec]ii ortogonale pentru a ob]ine un minim al estim#rii m#rimii Hausdorff, se poate aplica doar \n cazuri speciale [i nu este baza unei metode generale.
Fie F mul]imea de tipul Cantor generat# ca \n figura 6
0 1 itera]ia 0
0 1/3 2/3 1 itera]ia 1
0 1 itera]ia 2
fig.6
dac# s=log2/log3=0.6309 dimHF=s [i 1/2 Hs(F)1
Calcul euristic:
F se \mparte \n Fdr=F[0,1/3] [i Fst=F[2/3,1] ambele p#r]i sunt similare din punct de vedere geometric cu F dar scalate cu 1/3 [i F=FdrFst cu acea[i uniune disjunct#, atunci pentru orice s:
Hs(F)= Hs(Fdr)+ Hs(Fst)=(1/3)s Hs(F)+ (1/3)s Hs(F) (1.11)
din proprietate de scalare a m#surii Hausdorff.
Presupun@nd c# la valoarea critic# s=dimHF avem 0 Hs(F) , \mp#r]ind cu Hs(F) ob]inem 1=2(1/3)s sau s=log2/log3.
Calcul riguros:
Avem 3k (k=1,2,..) intervale de aceea[i lungime, \n construc]ia mul]imii de baz# F.
Dac# lu#m acoperirea {Ui} a lui F constituit# din 2k intervale Ek de lungime 3-k rezult#:
lu@nd k Hs(F)1 iar pentru a ar#ta c# Hs(F)1/2 trebuie s# ar#t#m c#
|Ui|s1/2=3-s (1.12)
pentru orice acoperire {Ui} a lui F. Desigur este destul s# presupunem c# Ui sunt intervale, extinz@nd [i folosind faptul c# F este compact# avem nevoie s# verific#m doar formula (1.12).
Dac# { Ui} este o colec]ie finit# de subintervale \nchise din [0,1] astfel \nc@t:
3-(k+1) |Ui|3-k (1.13)
pentru fiecare Ui [i k \ntreg.
Atunci Ui poate intersecta cel mult un interval de baz# a lui Ek \n timp ce separarea acestor intervale de baz# este cel pu]in 3-k. Dac# jk atunci prin construc]ie Ui intersecteaz# cel mult 2j-k=2j3-jk2j3s| Ui|s intervale de baz# ale lui Ej folosind (1.13)
Dac# alegem j destul de mare a.\ 3-(j+1) |Ui| pentru to]i Ui atunci din moment ce {Ui} intersecteaz# toate Ej de baz# de lungime 3-j num#r@nd intervalele ne d# 2jj2j3s|Ui|s care se reduce la (1.12).
\n general dac# F= unde Fi este similar# cu F dar scalat# cu un factor ci dimHF=s a.\
Defini]ie echivalent# a dimensiunii Hausdorff:
Putem alege acoperiri ca fiind sfere \n Rn lu@nd
B sd(F)=inf{|Bi|s:{Bi} este o acoperire d a lui F cu sfere} (1.14) B s(F)= B sd(F) [i dimensiunea la care B s(F) sare de la la 0. Se vede clar c# Hsd(F)B sd(F) din moment ce orice acoperire a lui F cu sfere este o acoperire permis# \n defini]ia m#surii Hausdorff. De asemenea {Ui} este o acoperire d a lui F atunci [i {Bi} este o acoperire d a lui F, unde pentru fiecare i, Bi este aleas# ca fiind o sfer# mic# ce con]ine Ui [i de raz# |Ui|d
|Bi|s(2Ui)s=2s|Ui|s [i lu@nd minimul B s2d(F)2s Hsd(F) [i d0
Hs(F) B s(F) 2s Hs(F)
Este u[or de verificat c# avem acelea[i valori pentru dimensiunea Hausdorff dac# \n (1.1) folosim acoperirea d de doar mul]imi deschise sau mul]imi \nchise.
Mai departe dac# F este compact#, extinz@nd acoperirea de mul]imi la mul]imi deschise [i lu@nd o subacoperire finit#, ob]inem aceea[i valoare a lui Hs(F) cu cazul \n care consider#m acoperirea d prin colec]ii finite ale mul]imilor.
Masura Net
Simplific@nd lu#m pe F ca fiind o submul]ime a intervalului [0,1) de forma [r2-k,(r+1)2-k) unde k=1,2,.. [i r=1,2,..,2k-1 definim :
Nsd(F)=inf{|Ui|s:{Ui} este o acoperire a lui F cu intervale binare} (1.14)
Rezult# masura Net:
Ns(F)= Nsd(F)
Din moment ce orice interval U[0,1) este con]inut \n dou# intervale binare consecutive, fiecare de lungime aproape 2|U|, observ#m \n acela[i fel ca la Bs(F) c#:
Hs(F) N s(F) 2s Hs(F)
valoarea lui s la care N s(F) sare de la la 0 egaleaz# dimensiunea Hausdorff. Din anumite puncte de vedere m#sura Net este mai convenabil# dec@t m#sura Hausdorff. Deoarece dou# intervale binare sunt ori disjuncte ori unul dintre ele este con]inut de cel#lalt permit oric#rei acoperiri a intervalelor binare s# fie redus# la o acoperire de intervale disjuncte.
O alt# defini]ie a dimensiunii:
Pentru a avea o defini]ie mai exact# a dimensiunii atunci:
Fie h:R+ R+ o func]ie cresc#toare [i continu#, o “func]ie dimensiune” definim:
Hhd(F)=inf{h(|Ui|):{Ui} este o acoperire d a lui F} (1.15)
Pentru FRn rezult# o m#sur# : Hh(F)= Nhd(F).
Dac# h(t)=ts rezult# defini]ia uzual# a m#rimii s dimensionale Hausdorff.
Dac# h [i g sun dou# func]ii dimensiune [i h(t)/g(t) tinde la 0 atunci c@nd t tinde la 0pentru un argument similar cu |Ui|tdt-s|Ui|s Hh(F)=0 ori de c@te ori Hg(F).
_mp#r]ind func]iile de dimensiuni \n func]ii pentru care Hh este finit# [i func]ii pentru care Hh este infinit#, aceast# defini]ie va da mai mullte informa]ii precise dec@t num#rul dimHF. Un exemplu important Mi[carea Brownian# \n R2,se poate ar#ta (cu probabilitatea 1) c# o cale Brownian# are dimensiunea Hausdorff 2, dar cu m#sura Hausdorff 0. Mai multe calcule arat# c# o astfel de cale are m#sura pozitiv# [i finit# Hh unde h(t)=t2log log(1/t).
De[i drumurile Browniene au dimensiunea 2, dimensiunea este \n sens logaritmic mai mic# dec@t 2.
Defini]ii alternative ale dimensiunii
Nu toate defini]iile sunt \n general aplicabile, c@teva descriu doar clase particulare de mul]imi (exemplu Curbele). Conceptul fundamental pentru toate defini]iile de dimensiuni este conceptul de “masur# la scal# d”. Pentru orice d, m#sur#m o mul]ime \ntr-un fel \n care se ignor# iregularit#]i ale m#rimilor mai mici ca d [i vedem cum se comport# aceast# m#sur# c@nd d tinde la 0.
De exemplu dac# F este o curb# plan#, atunci rezultatul m#surii Md(F) poate fi num#rul de pa[i ceru]i de o \mp#r]ire a mul]imii \n p#r]i d pentru a traversa F.O dimensiune a lui F este determinat# de o lege de putere dac# :
Md(F)cd-s (1.16)
c@nd d tinde la 0.Logaritm@nd rezult#:
log Md(F)log c – s log d (1.17)
s= (1.18)
Aceast# metod# este folosit# pentru scopuri experimentale [i de calcul, deoarece s poate fi estimat ca fiind panta dreptei de regresie log-log \ntre Md(F) [i d.
Pentru fenomene reale nu putem lucra dec@t cu intervale finite deoarece d din teorie [i cel din experiment diverg \nainte de a atinge o scal# atomic#. Poate nu este o lege de putere exact# pentru Md(F),pentru o valoare a lui s dat# de (1.16). Pentru a se comporta ca o dimensiune, metoda de m#surare are nevoie de a fi scalat# F. Dubl@nd pe F [i \n acela[i timp dubl@nd scala la care se face m#sur#toarea nu se afecteaz# rezultatul, aceasta \nseamn# Md(dF)= M1(F) pentru orice d.
Dac# modific#m exemplul [i redefinim Md(F) ca fiind omogen# de grad 1 atunci Md(dF)=d M1(F) pentru d0 [i aceasta trebuie luat# \n calcul c@nd se define[te dimensiunea.
_n general dac# Md(F) este omogen# de grad x aceasta \nseamn# Md(dF)=dx M1(F), atunci avem o lege de putere de forma Md(F)cdx-s corespunz@nd unei dimensiuni s.
Nu exist# reguli bune [i rapide pentru a decide dac# o cantitate poate fi rezonabil privit# ca o dimensiune.Teoria dimensiunii con]ine multe defini]ii care nu se potrivesc exact cu cea de aici chiar dac# sunt implicate scenariile. Factorii care determin# acceptabilitatea defini]iei unei dimensiuni sunt recunoscu]i \n general prin experien]# [i intui]ie.
_n general se caut# un tip de comportament la scalare, o individualitate a defini]iei \ntr-un context particular [i propriet#]i tipice ale dimensiunilor ca cele discutate mai sus.
Aparent defini]ii similare ale dimensiunii, pot avea \n mare m#sur# propriet#]i diferite. Nu \nseamn# \ns# c# defini]ii diferite dau aceea[i valoare a dimensiunii, chiar [i pentru mul]imi simple. Este necesar derivarea propriet#]ilor oric#rei dimensiuni din defini]ia sa. Propriet#]ile dimensiunii Hausdorff nu se men]in neap#rat pentru alte defini]ii.
Propriet#]ile derivate ale dimensiunii Hausdorff sunt destul de tipice:
(i) Monotonicitate : Dac# EF atunci dimHEdimHF
(ii) Stabilitate : dimH(EF)=max(dimHE, dimHF)
(iii) Stabilitate num#rabil# : Fi=
(iv) Pentru mul]imi deschise : FRn este o mul]ime deschis# rezult# dimHF=n
(v) Pentru mul]imi netede : F este neted#(continu# [i diferen]iabil#) de tip m dimensional \n Rn atunci dimHF=m
(vi) Pentru mul]imi deschise: FRn este o mul]ime deschis# rezult# dimHF=n
(vii) Pentru mul]imi num#rabile: Dac# F este num#rabil# rezult# dimHF=0.
(viii) Invarian]# geometric#: dimHf(F)=dimHF dac# f este o transformare \n Rn,o transla]ie, rota]ie, similaritate, afinitate..
(ix) Invarian]# Lipschitz : dimHf(F)=dimHF dac# f este o transformare bi-Lipschitz.
Toate defini]iile sunt monotone, cele mai multe sunt stabile, iar c@teva defini]ii comune nu au stabilitate num#rabil#. Toate dimensiunile uzuale sunt invariante Lipschitz adic# invariante geometric. Propriet#]ile mul]imilor netede [i deschise asigur# c# dimensiunea este o extensie a defini]iei clasice.
Defini]ii diferite ale dimensiunii pot oferii informa]ii diferite despre ce mul]imi sunt invariante Lipschitz.
1.4 Metode de determinare a dimensiunii fractale.
_n general nu exist# o formul# unic# de determinare a dimensiunii fractale [i de obicei metode diferite pot conduce la valori neegale pentru acela[i fractal.
1.4.1 Metoda clasic#
Este cea mai simpl# metod# [i este utilizat# numai pentru fractali liniari. Rela]ia de calcul este urm#toarea:
D=logN2/logN1 (1.19)
unde:
N2 reprezint# num#rul de replici ob]inute prin similaritate
N1 num#rul ini]ial de obiecte
De exemplu, lu@nd fractalul Cantor, care se ob]ine prin \mp#r]irea unui segment \n trei p#r]i egale din care se elimin# segmentul din mijloc. Se repet# aceast# opera]ie \n mod iterativ pentru fiecare din segmentele r#mase, ca \n figura 7
itera]ia0
itera]ia 1 itera]ia 2
fig.7
deci N1=3 [i N2=2 de unde rezult# D=log2/log3 [i D=0.631
1.4.2 Metoda “SandBox” sau metoda cutiei cu nisip
Este o metod# cu care se calculeaz# dimensiunea fractal# pentru orice tip de obiect fractal. Pe scurt metoda const# \n urm#torii pa[i:
Se alege un prim punct (pozi]ie amplasament) al fractalului ca origine a n cercuri cu raze R1 R2… Rn unde Rn mai mic dec\t raza R a fractalului, apoi se num#r# c\te puncte (pixeli) M1(Ri) din fiecare cerc I (uneori este mult mai u[or s# se aleag# n patrate de laturi L1, L2, …. Ln \n loc de cercuri).
Repet#m aceast# procedur#, aleg\nd alte puncte ca origine a n cercuri [i determin#m num#rul corespunz#tor de puncte Mj(R I) j=2, 3, .., m \n fiecare cerc.
Calcul#m media numerelor M(Ri) astfel:
M(Ri)= (1.20)
ªi se vor aprinde punctele M(Ri) fiind dependent de Ri \ntr-o functie logaritmica. Panta curbei, pentru mai multe valori ale lui Ri determina dimensiunea fractal#. Pentru a se evita anumite efecte de grani]a trebuie ca razele cercurilor s# se aleag# mai mici dec\t raza fractalului.
1.4.3 Metoda Box-Counting sau metoda c#su]elor num#rate
Defini]ia dimensiunii Box-Counting
Fie mul]imea F, limitat# [i FRn [i fie Nd(F) cel mai mic num#r de mul]imi de diametru cel pu]in d care pot acoperii mul]imea F, atunci dimensiunea Box-Counting se calculeaz# astfel:
(1.20)
(1.21)
dac# cele dou# limite sunt egale rezult#:
(1.22)
Exist# c@teva defini]ii echivalente ale dimensiunii Box-Counting, care sunt c@teodat# mai folositoare.
Consider@nd colec]ia de cuburi ca o plas# \n Rn de forma: ([m1d,(m1+1)d]x[m2d,(m2+1)d]x..x[mnd,(mn+1)d]) de latur# d [i cu m1,m2,..,mn.
Fie Ndi(F) num#rul de cuburi re]ea de latur# d care intersecteaz# pe F. Ele determin# o colec]ie de Ndi(F) mul]imi de diametru dn, care acoper# F deci:
Ndn(F) Nd(F) (1.23)
Dac# dn 1 atunci :
(1.24)
lu@nd limit# rezult#:
(1.25) (1.26)
Pe de alt# parte , oricare mul]ime de diametru cel mult d este con]inut \ntr-o re]ea de 3n cuburi de latur# d . Aleg@nd un cub con]in@nd c@teva puncte ale mul]imii \mpreun# cu cuburile vecine rezult#
N|d(F)3nNd(F) (1.27)
Logaritm@nd rezult# inegalitatea invers# din (1.25) [i (1.26), din acest motiv pentru a g#si dimensiunea Box-Counting putem lua la fel de bine [i Nd(F) ca fiind num#rul de cuburi re]ea de latur# d care s# intersecteze pe F.
Pentru a g#si dimensiunea Box-Counting asociat# unui plan, putem desena o re]ea de p#trate de latur# d [i [i s# num#r#m Nd(F),care acoper# mul]imea pentru diferite laturi d mici. Dimensiunea este rata logaritmic# la care Nd(F) cre[te c@nd d tinde la 0 [i poate fi estimat# de panta grafului regresiei log(Nd(F)) [i log(d). Aceast# defini]ie d# o interpretare a \n]elesului dimensiunii Box-Counting.
Num#rul de cuburi din re]ea de latur# d care intersecteaz# o mul]ime este o indica]ie a c@t de r#sp@ndit# \n afar# sau neregulat# este mul]imea, atunci c@nd se examineaz# la o scar# d. Dimensiunea reflect# c@t de rapid se dezvolt# iregularit#]ile pentru d0.
O alt# defini]ie, folosit# frecvent, a dimensiunii Box-Counting este prin luarea num#rului Nd(F) ca fiind cel mai mic num#r de sfere \nchise de raz# d care s# acopere pe F.
O mai pu]in clar echivalent# formulare a dimensiunii Box-Counting implic# cel mai mare num#r de sfere de raz# d, disjuncte cu centrele \n F.
Fie acest num#r N|d(F) [i fie B1,B2,.., sferele disjuncte centrate \n F [i de raz# d. Dac# xF a.\ xBi sfera de centru x [i raz# d poate fi ad#ugat# s# formeze o colec]ie mai mare de sfere disjuncte. Astfel N|d(F) sfere concentrice cu Bi dar de raz# 2d acoper# pe F d@nd:
N4d(F) N|d(F) (1.28)
Pe de alt# parte, presupun@nd c# B1,B2,.., sunt sfere disjuncte de raze d cu centre \n F.Fie U1,U2,..,Uk orice colec]ie de mul]imi de diametru cel mult d care acoper# pe F. Din moment ce Uj trebuie s# acopere centrele lui Bi, fiecare Bi trebuie s# con]in# cel pu]in un element din Uj. Cum Bi sunt disjuncte exist# cel pu]in at@tea mul]imi Uj c@te Bi. Din acest motiv
N|d(F) Nd(F) (1.29)
dimensiunea nu va fi alterat# ca valoare nici \n cazul acesta.
Defini]ie 1.2
Se poate da o defini]ie echivalent# care s# le includ# pe toate descrise anterior, astfel:
(1.20)
(1.21)
(1.22)
unde Nd(F) poate fi:
1. Cel mai mic num#r de mul]imi sferice \nchise de raz# d care acoper# F (figura 7.a)
2. Cel mai mic num#r de cuburi de latur# d care acoper# F(figura 7.b)
3. Num#rul de cuburi re]ea de latur# d care intersecteaz# F (figura 7.c)
4. Cel mai mic num#r de mul]imi de diametru cel mult d care acoper# pe F (figura 7.d)
5. Cel mai mare num#r de mul]imi sferice disjuncte de raz# d cu centre \n F.(figura7.e)
(a) (b) (c) (d)
(e) figura 7
Este important s# se \n]eleag# leg#tura \ntre dimensiunea Box-Counting [i dimensiunea Hausdorff.
Dac# F poate fi acoperit# de Nd(F)mul]imi de diametru d atunci din defini]ia 1.1 Hds(F)ds Hd(F)
Dac# 1 Hds(F)= Hds(F) atunci log(Nd(F))+s*log(d)0 dac# d este suficient de mic. Astfel : slog(Nd(F))/-log(d)
dimHF (1.23)
pentru orice FRn
Chiar dac# dimHF=dimBF pentru multe mul]imi “rezonabile” exist# multe exemple \n care inegalitatea este strict#. Formula (1.22) \n sens strict spune c# Nd(F)d-s pentru d mic, unde s=dimBF sau mai precis:
Nd(F)ds dac# sdimBF
Nd(F)ds0 dac# sdimBF
Dar Nd(F)ds=inf{|Ui|s:{Ui} este o acoperire d finit# a lui F} care are trebui comparat# cu Hd(F)=inf{|Ui|s:{Ui} este o acoperire d a lui F} defini]ia m#surii Hausdorff. \n calcularea dimensiunii Hausdorff lu#m diferite ponderi |Ui|s pentru mul]imile acoperiri Ui unde pentru dimensiunea Box-Counting folosim aceea[i pondere ds pentru fiecare mul]ime de acoperire.
Dimensiunea Box-Counting poate fi g@ndit# ca indic@nd eficien]a cu care o mul]ime poate fi acoperit# cu mul]imi mici de m#rimi egale, \n timp ce dimensiunea Hausdorff implic# acoperirea cu mul]imi mici dar de variate m#rimi.
Exist# tendin]a de a introduce cantitatea v(F)= Nd(F)ds, dar aceasta nu va da nici o m#sur# pe submul]imi ale lui Rn.
Dup# cum se va vedea, o consecin]# a acesteia este aceea c# dimensiunea Box-Counting are un num#r de propriet#]i nedorite care pot fi incomode la m@nuirea lor matematic#. Din moment ce dimensiunea Box-Counting poate fi determinat# prin acoperiri cu mul]imi de m#rimi egale, tinde s# fie mai u[or de calculat dec@t dimensiunea Hausdorff.
Propriet#]i ale dimensiunii B
Un super-plan neted,m-dimensional din Rn are dimBF=m
[i sunt monoton
stabil# finit
(EF)=max{E, F} dar nu este
[i sunt invariante Lipschitz. Aceasta deoarece dac# |f(x)-f(y)|c|x-y| a.\. poate fi acoperit# cu Nd(F) mul]imi de diametru cel pu]in c*d, atunci dimBf(F)dimBF. Similar dimensiunea Box-Counting se comport# exact ca dimensiunea Hausdorff la transform#ri bi-Lipschitz sau Hlder.
O alt# proprietate care este descris# prin propozi]ia urm#toare este un dezavantaj al dimensiunii Box-Counting
Propozi]ia 1.5
Fie FI \nchidere a mul]imii F (cel mai mic subset din Rn care o con]ine pe F)
atunci:
[i
Demonstra]ie:
Fie B1,B2,..,Bk o colec]ie finit# de sfere \nchise de raze d. Dac# mul]imea \nchis# o con]ine pe F atunci de asemenea o con]ine [i pe FI. Deoarece cel mai mic num#r de sfere \nchise de raze d ce acoper# F este de ajuns pentru a acoperii pe FI, [i rezultatul este determinat. Aceast# proprietate reduce utilizarea dimensiunii Box-Counting la mul]imi \nchise.
1.4.4 Dimensiunea Box-Counting modificat#
Deoarece exist# metode de a dep#[i dificult#]ile dimensiunii Box-Counting se poate apela la metode de a modifica calcularea acesteia.
Dac# F este o submul]ime \n Rn putem s# \ncerc#m descompunerea lui F \ntr-un num#r finit de p#r]i F1,F2,.. a.\. cea mai mare parte are o dimensiune c@t mai mic# posibil#, a[a putem defini dimBF astfel:
(1.24)
(1.25)
minimul se ia pentru toate acoperirile posibile num#rabile {Fi} ale lui F.
Este clar c# [i . Totu[i acum avem c# dac# F este num#rabil#. Mai mult, pentru orice subset F \n Rn avem:
(1.26)
Este u[or de observat c# acoper# toate propriet#]ile avantajoase ale dimensiunii dar aceasta este greu de calculat.
Totu[i exist# un test util pentru mul]imi compacte pentru a avea dimensiunea Box-Counting [i dimensiunea Box modificat# egale. Acest test se aplic# mul]imilor care pot fi descrise ca omogene dimensional.
Propozi]ia 1.6
Fie FRn compact#. Presupun@nd c#:
(1.27)
pentru toate mul]imile V care intersecteaz# pe F.
Atunci [i un rezultat similar se men]ine [i pentru .
Pentru o aplica]ie consider@nd F compact# cu un grad ridicat de auto-similaritate( de exemplu curba Von Kock). Dac# V este o mul]ime oarecare deschis# care intersecteaz# pe F, atunci FV con]ine o copie similar# geometric cu F, care trebuie trebuie s# aib# dimensiunea superioar# Box-Counting egal# cu cea a lui F ca urmare a propozi]iei enun]ate mai sus.
1.4.5 Tehnici de calculare a dimensiunii
Pentru mul]i fractali estim#ri superioare ale dimensiunii pot fi ob]inute prin folosirea de acoperiri naturale cu mul]imi mici.
Propozi]ia 1.7
Presupunem F ca put@nd fi acoperit# de nk mul]imi de diametru cel mult dk cu dk0 atunci c@nd k atunci:
[i dac# dk+1cdk pentru un 0c1
Mai mult dac# nkdks r#m\ne limitat# atunci c@nd k atunci H s(F)
Demonstra]ie
Inegalitatea pentru dimB rezult# imediat din defini]ia acesteia.Pentru a doua parte dksnk deci tinde la o limit# finit# H s(F) atunci c@nd k.
Dimensiunea Graficelor de func]ii temporale
Consider#m f:[a,b]R, \n anumite circumstan]e graficul
gf={(t,f(t)):atb} privit ca o submul]ime de coordonate (t,x) \n plan poate fi un fractal.
Observa]ie: Se lucreaz# cu coordonate (t,x) \n loc de (x,y) deoarece variabila timp este independent#.
Dac# f este continu# [i derivabil# atunci dimensiunea graficului este 1 [i \ntr-adev#r este o mul]ime regulat# de dimensiune 1. Acela[i lucru este adev#rat dac# f este variat# limitat adic#: constant#, pentru orice \mp#r]ire 0t0t1..tm=1. Totu[i este posibil pentru o func]ie continuu# s# fie suficient de neregulat# ca s# aib# un grafice de dimensiune strict mai mare ca 1.
Cel mai bun exemplu este func]ia
(s-2)ksin(kt)
unde 1s2 [i 1. Aceast# func]ie, exemplul esen]ial al lui Weierstrass de func]ie continu# care nu este diferen]iabil# \n nici un punct, are dimB=s de asemenea se fac presupuneri c# [i dimH este tot s.
Not#m domeniul maxim Rf[t1,t2]=
Propozi]ia 1.8
Fie f:[0,1]R func]ie continu#. Presupunem c# 0d1 [i m este cel mai mic num#r \ntreg mai mare dec@t 1/d sau egal. Atunci, dac# Nd este num#rul de p#trate din re]eaua de latur# d care intersecteaz# graficul f, avem:
(1.28)
Demonstra]ie
Num#rul de p#trate din re]ea de latur# d din coloana de l@ng# intervalul [id,(i+1)d] care intersecteaz# graficul gf este cel pu]in Rf[id,(i+1)d]/d [i cel mult 2+Rf[id,(i+1)d]/d. Folosind faptul c# f este continu#, \nsum@nd pentru toate intervalele d# formula (1.28) [i este ilustrat# \n figura 8
x x=f(t)
d t
fig.8
Corolar
Fie f:[0,1]R continu#
Presupunem c#
|f(t)-f(u)|c|t-u|2-s (0t,u1) (1.29)
unde c0 [i 1s2 .
Atunci H s(F) [i dimHgf dimBgf s [i r#m\ne adev#rat dac# (1.29) se men]ine pentru
|t-u|d [i d0
Presupunem c# exist# numere c0, d00 [i 1s2 cu urm#toarele propriet#]i :
pentru fiecare t[0,1) [i 0dd0 u a.\ |t-u|d [i
|f(t)-f(u)|cd2-s (1.30)
atunci s.
Demonstra]ie
Rezult# imediat din (1.30) Rf[t1,t2]c|t1-t2|2-s pentru 0t1 [i t21. Cu nota]iile de la propozi]ia anterioar# m(1+d-1) deci
Nd(1+d-1)(2+cd-1d2-s)c1d-s
unde c1 este independent de d. Iar rezultatul se determin# prin aplicarea propozi]iei (1.7)
\n acela[i mod (1.30) implic# faptul c# Rf[t1,t2]c|t1-t2|2-s .Deoarece d-1m rezult# din (1.28) c# Ndd-1d-1cd2-s=cd-s
defini]ia echivalent# 1.2 punctul 3 d# s
1.5 O abordare matematic# a unei metode generale de ob]inere a fractalilor
Fie (X,d) un spa]iu metric; "d" este o func]ie distan]#. Not#m cu H(X) familia tuturor submul]imilor compacte nevide ale lui X.
Fie B o submul]ime a lui X. Not#m:
Num#rul real pozitiv d(x,B) se nume[te distan]a de la punctul x la mul]imea B. Dac# mul]imea B este compact# atunci
, adic# marginea inferioar# de mai sus este efectiv atins#.
Fie A [i B dou# submul]imi ale lui X. Distan]a dist(A,B) dintre mul]imile A [i B se define[te prin egalitatea
Observ#m c# dac# dist(A,B)=0 nu rezult# neap#rat c# A [i B coincid. De exemplu, distan]a dintre oricare dou# cercuri tangente este egal# cu zero. Cu alte cuvinte, func]ia dist nu este o distan]# in sensul matematic al cuv@ntului; ea este ins# o distan]# in sensul intuitiv al cuv@ntului.
_n continuare vom defini o func]ie care s# fie efectiv o distan]#(\n sensul matematic al cuv@ntului). Vom construi func]ia h(d) cu ajutorul distan]ei "d" din spa]iul metric (X,d) de la care am plecat. Pentru aceasta, fie A,B dou# mul]imi compacte:
Not#m:
Se [tie c# dac# o submul]ime a unui spa]iu metric este compact# atunci ea este [i m#rginit#. Mul]imile A [i B fiind compacte, exist# o bil#(sfer#) care le con]ine simultan pe am@ndou#. Rezult# c# marginea superioar# de mai sus este \ntotdeauna finit#. \n plus, datorit# faptului c# A [i B sunt compacte rezult# cu u[urin]# c# supremumul este efectiv atins:
Num#rul d(A,B) se nume[te distan]a de la mul]imea A la mul]imea B(\n acest# ordine). Observ#m c#, \n general, dup# cum se poate ar#ta cu u[urin]#. De exemplu consider#m fig1., \n spa]iul metric euclidian ; se vede c# .
fig.9
Definim:
unde "max" desemneaz# cel mai mare dintre numerele dintre paranteze.
Func]ia h(d) se nume[te distan]a Husdorff. Litera "d" dintre paranteze indic# faptul c# la baza construc]iei distan]ei Haussdorff a stat distan]a "d" a spa]iului metric de plecare (X,d).
h(d) este o distan]# pe spa]iul H(X), iar (H(X),H(d)) este un spa]iu metric complet.
Am ar#tat deci c# spa]iul (H(X),H(d)) este un spa]iu metric complet dar \ntr-un spa]iu metric complet func]ioneaz# teorema contrac]iei. \n continuare vom ar#ta cum putem construi contrac]ii pe H(X), pornind de la contrac]ii ale spa]iului de baza (X,d). Punctele fixe ale contrac]iilor definite pe H(X) vor fi denumite fractali.
Propozi]ia 1: Fie un spa]iu metric si o contrac]ie a sa. Atunci func]ia urm#toare:
pe care o not#m tot cu , este o contrac]ie a lui (H(X) , h(d)).
Demonstra]ie:
Mul]imea \n care ia valori func]ia \ntr-adev#r H(X) pentru c# imaginea unei mul]imi compacte(din X) prin func]ia , care este continu#(fiind o contrac]ie), este tot o mul]ime compact#.
Cum este o contrac]ie, exist# cu proprietatea c#:
Vom ar#ta c#:
, unde am notat cu h=h(d) distan]a Haussdorff.
Pentru acesta ar#t#m mai \nt@i c#:
\n adev#r:
Am ar#tat deci c#:
Rezult# c#:
Prin urmare:
Adic#:
Demonstra]ia este \ncheiat#.
Lema 1: Fie (X,d) un spa]iu metric [i (H(X) , h) spa]iul asociat(am notat cu "h" distan]a Haussdorff).
Atunci:
pentru orice num#r natural [i pentru orice mul]imi din H(X).
Demonstra]ie:(prin induc]ie)
Demonstr#m afirma]ia pentru n=2.
, unde prin desemn#m max(a,b), adic# cel mai mare dintre numerele a [I b.
Este suficient s# ar#t#m c#:
Avem succesiv:
pentru orice [i:
pentru orice
Rezult# c#:
Reuniunea unui num#r finit de mul]imi compacte este o mul]ime compact#, deci func]ia W ia valori \ntr-adev#r \n H(X).
Mai departe, fie
\ncheiem cu algoritmul de ob]inere al fractalilor.
Propozi]ia 3(algoritmul de ob]inere al fractalilor):
Fie (X , d) un spa]iu metric complet [i contrac]ii ale lui X, av@nd constantele
Atunci func]ia:
admite un unic punct fix (care este de fapt o submul]ime compact# a lui X).
Mai mult, oricum am alege o submul]ime compact# F a lui X, [irul converge c#tre :
de n ori
Demonstra]ie:
Func]ia W este o contrac]ie pe H(X).
Spa]iul (X , d) fiind complet, rezult# c# (H(X) , h(d)) este de asemenea complet. Prin urmare, W este o contrac]ie a unui spa]iu metric complet. Mai departe aplic teorema contrac]iei. Demonstra]ia este \ncheiat#.
Observa]ie: Este intuitiv faptul c#, \n cazul c@nd sunt transform#ri afine ale spa]iului :
punctul fix al contrac]iei W va fi o mul]ime auto-similar#, adic# anumite p#r]i ale ei reprezint#, la o alta scar#, \ntreaga mul]ime .
Autosimilaritatea este o proprietate de baz# a fractalilor. S# consider#m urm#torul exemplu.
Exemplu: Mul]imea M a lui Cantor este invariant# \n raport cu setul de func]ii . Mai exact, dac# not#m:
atunci:
Rezult# c# M este punctul fix al contrac]iei . Prin urmare, M poate fi ob]inut# cu ajutorul unui computer \n modul urm#tor:
se alege arbitrar \n ; mul]imea este compact#.
se calculeaza succesiv:
Pentru "n" suficient de mare, mul]imea aproximeaz# mul]imea M a lui Cantor.
1.6 Fractali liniari
Multe curbe autoasemenea(fractali cu dimensiune topologic# 1) poart# numele unor matematicieni celebri: Peano, Sierpinski, Hilbert. Aceast# familie de fractali este foarte vast# [i u[or de simulat, fiind vorba de transform#ri geometrice simple.
Elementele acestei familii de curbe sunt denumite de c#tre Mandelbrot “teragon”de la grecescul “tera” = monstru [i “gonla” = unghi. Tot Mandelbrot afirma referitor la aceste curbe c# ar fi cel mai intuitiv lucru din geometrie. Astfel, odat# pus la punct algoritmul de generare a acestor curbe, ele ar fi mult mai simplu de ob]inut dec@t un element euclidian cum ar fi cercul [i-chiar [i dec@t dreapta- dac# ne g@ndim la algoritmii de ob]inere a segmentelor de dreapt#, mai complica]i dec@t recursivitatea curbelor fractale.
_nainte de a prezenta c@]iva fractali din aceast# familie s# vedem c@teva no]iuni mai des utilizate. Vom denumi ”ini]iator” un segment de la care se porne[te generarea fractalului. Un contur \nchis cu astfel de ini]iatori se nume[te nucleu [i are de obicei form# regulat#:triunghi, p#trat, pentagon, hexagon. Transform#rile efectuate asupra ini]iatorului sau eventual asupra nucleului sunt indicate de un contur numit “generator”.
Exemple de seturi fractale liniare:triunghiul [i tetrahedronul lui Sierpinski, curbele Peano, curbele von Koch, ansamblul lui Cantor, iar de fractali naturali ce se pot aproxima prin seturi fractale liniare:mun]ii, fulgerele, mi[carea brownian#, r@urile [i ]#rmurile, anumite plante, arbori, ramifica]iile bronhiilor, sistemul nervos, etc.
1.6.1 Triunghiul lui Sierpinski
Triunghiul lui Sierpinski, unul dintre cei mai mari matematicieni polonezi, este unul dintre clasicii fractali liniari. Ob]inerea este, dup# cum probabil se [tie, urm#toarea: se porne[te de la triungiul echilateral(sau oarecare) [i se extrage triunghiul ob]inut prin unirea punctelor mediane. Se repet# procedeul \n cele trei triunghiuri r#mase, [.a.m.d. Triunghiul lui Sierpinski r#m@ne dup# ce se aplic# acest procedeu la infinit.(fig.10).
fig.10 Triunghiul lui Sierpinsky
Referitor la modul de generare al fractalului, exist# numeroase variante care ne conduc la diver[i fractali mai mult sau mai pu]in universali.
Aria suprafe]ei fractale ob]inute \n final este zero! Aceasta deoarece la fiecare pas dup# extragerea unui sfert(aria triunghiului central) din suprafa]a anterioar# r#m@n trei p#trimi. Astfel \n final se ob]ine o arie S = 3/4*3/4*…*S (unde S este aria triunghiului ini]ial) valoare ce tinde la zero pentru num#rul de itera]ii tinz@nd la infinit.
Mul]imea triunghiurilor ob]inute poate fi codificat# cu ajutorul literelor L pentru “left” , R pentru “right” [i U pentru “up” ; orice triunghi din etapa K a genera]iei poate fi referit prin combina]ii ale acestor litere. Astfel \n fig.10 triunghiul 4 are adresa LUL, adic# pentru a-l ob]ine trebuie ca la primul nivel s# ne situ#m la st@nga (L) triunghiului 1, apoi la al doilea nivel ne deplas#m sus (U) fa]# de triunghiul 2 [i \n fine la nivelul al treilea ne deplas#m la st@nga (L) fa]# de triunghiul 3.
Generarea triunghiului se poate ob]ine u[or prin recursivitate.
Dimensiunea fractal# este D = lnN / ln(1/r) =ln3 /ln2=1,58 ,unde N=3 reprezint# num#rul de triunghiuri ob]inute dup# aplicarea unui pas al algoritmului, iar 1/r = 2 reprezint# raportul de asem#nare dintre dou# triunghiuri apar]in@nd la dou# nivele consecutive.
Acest fractal ce provine dintr-o suprafa]# nu acoper# \n \ntregime planul a[a cum se \nt@mpl# de exemplu \n cazul curbelor Peano. Aceast# situa]ie este justificat# de faptul c# dimensiunea fractal# a triunghiului Sierpinski este D = 1,58<2 care este dimensiunea fractal# a curbelor Peano, ce acoper# complet o anumit# zon# plan#.
O metod# de generare a triunghiului lui Sierpinski este Binary Adresses Method (BAM), utilizat# de o larg# clas# de fractali. Aceast# metod# este legat# de triunghiul lui Pascal astfel :dac# un num#r \n triunghiul lui Pascal este impar atunci el apar]ine triunghiului lui Sierpinski(vezi fig.11).
Fig.11 Binary Adresses Method
Un alt fractal, numit de B.Mandelbrot “fractal skewed web” , este tetraedronul lui Sierpinski: dintr-un tetraedru se extrage corpul median, ob]inut prin unirea punctelor mediane ale muchiilor. Rezult# patru tetraedre asemenea cu primul \n raportul 1/r=2.Astfel suntem \n fa]a unui exemplu de fractal din spa]iul euclidian cu trei dimensiuni ce are dimensiunea fractal# 2.
Putem spune c# acest fractal umple destul de slab spa]iul.
Volumul corpului astfel ob]inut tinde la zero deoarece la fiecare pas acesta se reduce la ½.
Curba lui Sierpinski (fig.12) are proprietatea c# – la infinit - este o curb# de lungime infinit# delimitat# de o suprafa]# de arie finit# [i anume de un p#trat.
Fig.12 Curba lui Sierpinski(curba de ordin 3)
“Covorul” lui Sierpinski (fig.13), obiect geometric despre care nu s-a putut preciza dac# este o curb# ori o suprafa]#, este de asemenea un fractal \nrudit cu triunghiul lui Sierpinski. Construc]ia este simpl#, [i anume, se pleac# de la un p#trat a c#rui suprafa]# se \mparte \n 9 p#r]i egale prin \mp#r]irea fiec#rei laturi \n c@te trei p#r]i egale. Partea din mijlocul p#tratului ini]ial se elimin#, iar cele 8 p#trate r#mase sunt supuse aceluia[i procedeu. Prin continuarea acestui proces, care se aplic# p#tratelor care nu se elimin#, la infinit, se ob]ine o construc]ie geometric# despre care nu se poate spune c# reprezint# o curb# ori o suprafa]#.
Reprezentarea grafic# a acestui proces se va face prin desenarea p#tratului ini]ial [i desenarea tuturor p#tratelor care vor fi eliminate \n timpul procesului de construc]ie. Pentru realizarea acestei construc]ii se poate folosi o procedur# nerecursiv#, p#trat, ce deseneaz# un p#trat dup# ce se face fixarea originii reperului de axe \n col]ul din st@nga jos al p#tratului, [i o procedur# recursiv#, covor, care apeleaz# procedura p#trat [i are func]ia de a implementa procesul explicat mai sus. Forma procedurii covor este dat# schematic astfel:
(n-1) (n-1) (n-1)
covor
(n) (n-1) (n-1)
(n-1) (n-1) (n-1)
Fig .13 Covorul lui Sierpinski
1.6.2 Curba Peano
Un matematician italian pe nume Giuseppe Peano a inventat aceast# form# \n 1890, \n timp ce \ndeplinea func]ia de “profesor extraordinar de calcul infinitezimal” la universitatea din Torino. Cu ea, el a demonstrat c# o curb# continu#, f#r# l#]ime([i deci f#r# arie), poate umple o por]iune de spa]iu. Fiecare nivel succesiv las# din ce \n ce mai pu]in spa]iu \ntre micile p#trate.
Pentru nivelul n=0 se traseaz# un segment de dreapt# format din trei p#r]i egale, iar pentru nivelul n=1 se realizeaz# obiect din figura 14.
La nivel infinit, definit prin curba de umplere a spa]iului, \ntre linii nu va r#m@ne nici un pic de spa]iu gol.\n consecin]#, curba va avea o arie egal# cu cea a p#tratului care o m#rgine[te. Dar ea trebuie totu[i s# aib# o arie egal# cu zero deoarece este, \n continuare, alc#tuit# din segmente de dreapt#. Altfel spus, un set de segmente de dreapt# unidimensionale poate, \ntr-un fel sau altul, s# umple suprafa]a unui plan bidimensional.
Dimensiunea Haussdorff este D = lg9/lg3 = 2.
nivel 1 nivel 2
Fig.14 Curba Peano
Curba lui Hilbert (fig.15) este un alt exemplu de curb# continu# de lungime infinit# care “umple” un p#trat, adic# este o func]ie f:[0,1]->[0,1]x[0,1] continu# [i surjectiv#, dar neinjectiv#.
fig.15 Curba lui Hilbert
1.6.3. Curba Koch
M#surarea lungimii unei linii neregulate de coast# depinde, dup# cum u[or ne d#m seama, de instrumentele de m#surare folosite. Precizia are limitele ei [i ne este greu s# ne imagin#m o linie de coast# idealizat# care nu se “neteze[te “ niciodat#.
Fig.16 este o “linie de coast#“ matematic#. Aceast# form# a fost inventat# de matematicianul suedez Helge Von Koch \n 1904, \n timpul marii crize din matematic#.
Curba lui Koch de ordinul 0 este un segment de dreapt# (de lungime l);
Curba lui Koch de ordinul k+1 se ob]ine \mp#r]ind fiecare din segmentele componente ale curbei lui Koch de ordin k \n 3 p#r]i egale [i \nlocuind segmentul din mijloc (fie el t) cu celelalte dou# laturi ale unui triunghi echilateral care are ca baz# pe t.
Fig.16 Curba lui Koch
Curba are proprietatea c# are o lungime infinit# [i delimiteaz# o suprafa]# de arie finit#. Algoritmul este urm#torul:
se pleac# de la un triunghi echilateral cu latura av@nd o lungime finit#;
fiecare latur# se \mparte \n trei p#r]i egale, segmentul din mijloc se elimin# [i se \nlocuie[te cu un unghi ale c#rui laturi sunt formate din dou# segmente de lungime egal# cu segmentul eliminat;
pentru fiecare segment al al figurii ob]inute se repet# opera]ia anterioar#;
dac# se repet# acest proces se ob]ine o linie poligonal# \nchis# cu latura de lungime din ce \n ce mai mic#.
Se poate observa c# dup# fiecare itera]ie lungimea curbei cre[te, la limit# aceasta tinde la infinit, dar suprafa]a delimitat# de aceast# linie poligonal# are arie finit#. (Fig.17).
fig.17 Curba lui Koch
Construc]ia este perfect regulat# [i dup# fiecare trecere de la un nivel la altul, num#rul de unghiuri cre[te, de asemenea num#rul de laturi ale liniei poligonale [i lungimea total# a curbei cresc, dar lungimile laturilor se mic[oreaz# [i toate acestea \ntr-un raport constant.
Se poate observa c# un astfel de raport exist# \ntre toate obiectele geometrice de aceea[i form# c@nd dimensiunile lor cresc. De exemplu, dac# se dubleaz# lungimea unui segment de dreapt#, atunci lungimea lui cre[te de 2 ori, dac# se dubleaz# latura unui p#trat, atunci aria suprafe]ei lui cre[te de 4 ori, dac# se dubleaz# latura unui cub, atunci volumul lui cre[te de 8 ori.
Prin urmare am considerat trei obiecte geometrice \n R , R [i R[i am observat c# rela]ia matematic# eviden]iat# depinde de num#rul de dimensiuni ale obiectului, [i anume dimensiunea 1 pentru un segment de dreapt# din R, dimensiunea 2 pentru o suprafa]# din R[i dimensiunea 3 pentru un corp din R.
S# not#m cu l(n) ,n=0,1,2,.. . lungimea curbei la iter]ia n.Dac# p=l(0) reprezint# perimetrul triunghiului echilateral [i f#c@nd observa]ia c# \n procesul de construc]ie fiecare latur# se \mparte \n trei p#r]i egale [i se transform# \ntr-o linie format# din patru segmente, atunci evident avem :
l(n) = p*(4/3) , n=0,1,2,.. .
lim l(n) =
n->
D = , unde t - factor de descompunere
s - factor de scar#
D – dimensiune fractal#
D = = 1.2619
Prin urmare, \n cazul curbei Koch prezentate, lungimea curbei se m#re[te dup# fiecare itera]ie de 4/3 ori, av@nd o lege proprie de cre[tere, determinat# de principiul de construc]ie. Forma geometric# ob]inut# este o construc]ie geometric# cuprins# \ntre o curb# [i o suprafa]# c#reia i se poate asocia num#rul frac]ionar lg4/lg3 ce reprezint# dimensiunea fractal#.
\n “Geometria fractal# a naturii”, Mandelbrot spunea despre curba lui Koch:"Aceast# similitudine \ntre \ntreg [i p#r]ile sale, chiar infinitezimale, este cea care ne face s# consider#m curba lui von Koch ca pe o curb# \ntr-adev#r minunat# \n familia curbelor. Dac# ea ar fi \nzestrat# cu via]#, nu ar fi posibil# trecerea sa \n neant dec@t dac# am suprima-o dintr-o dat#, c#ci altfel ea ar rena[te f#r# \ncetare din profunzimile triunghiurilor sale, ca [i via]a \n univers".
1.6.4 Curba dragonului
Un exemplu de curb# Peano \l constituie [i fractalul numit “curba dragonului” care rezult# conform fig.18.
Algoritmul de generare este urm#torul: fiecare dreapt# (ini]iator) se divide \n dou# drepte copil , cu lungimea egal# cu m#rimea catetelor unui triunghi isoscel ce are segmentul de dreapt# ini]ial ca ipotenuz#. Prin repetarea algoritmului de generare se ob]in curbe care se aseam#n# cu un dragon. Curbele Peano de acest tip, pe l@ng# aspectul spectaculos, au largi aplica]ii \n fizic#. Dimensiunea fractal# este ln4/ln2=2 deoarece pe un segment \mp#r]it \n dou# p#r]i se construiesc patru segmente ce alc#tuiesc pliurile.
\n 1960, fizicianul John Heighway [i-a imaginat construc]ia acestui dragon, la \nceput prin \ndoirea repetat# a unei f@[ii lungi de h@rtie [i orientarea \ndoiturilor \n unghiuri drepte.
\n cazul curbei dragonului (Knuth) generatorul este construit astfel \nc@t direc]ia de expandare s# alterneze de o parte [i de cealalt# a laturilor poligonului ini]iator. O astfel de curb# are proprietatea c# poate pava (umple) o suprafa]# plan#.
“Golurile” delimitate de frontiera unei curbe dragon pot fi “umplute” cu dragoni similari, de dimensiune mai mic#.
Curba de ordin 0 Curba de ordin 1 Curba de ordin 2
(ini]iator) (generator)
Curba de ordin 14
Fig.18 Curba dragonului
1.6.5 Praful lui Cantor
Respectatul matematician german Georg Cantor, care a dezvoltat de unul singur teoria modern# a seriilor, era deseori ]inta sarcasmului \n]ep#tor al lui Poincare care considera multe din crea]iile lui Cantor “patologice”.
O astfel de form# creat# de Cantor \n 1877 care pare, \ntr-un fel sau altul, c# “sare” dimensiunile, este construit# prin fragmentarea segmentelor de dreapt# unidimensionale, dar la sf@r[it ea con]ine doar puncte de dimensiune zero, f#r# lungime sau l#]ime(vezi fig.19).
Praful ternar al lui Cantor se construie[te \n dou# etape: prin interpolare, apoi prin extrapolare.
Interpolarea se realizeaz# astfel: se pleac# de la segmentul (0,1] [i se extrage treimea central#, desemnat# prin (1/3,2/3]. Dup# aceea, din fiecare treime r#mas# se extrage propria treime central# [i se continu# \n acela[i mod la nesf@r[it.
\n ce prive[te extrapolarea cea mai simpl#, ea dubleaz# \ntr-o manier# repetitiv# num#rul de replici ale mul]imii interpolate. Mai \nt@i se plaseaz# o replic# pe segmentul (2,3], ob]in@nd astfel mul]imea originar# m#rit# \n raportul 3. Apoi se plaseaz# dou# replici pe (6,7] [i (8,9], ob]in@nd astfel mul]imea originar# m#rit# \n raportul 9 [i a[a mai departe.
Dimensiunea fractal# este: D==0,6309.
– - - - – - – –
Fig.19 Praful lui Cantor
1.6.6 Multiuniversul lui Fournier
Merg@nd pe urmele lui Cantor, englezul Edmund Edward Fournier d’Albe, ziarist la o rubric# [tiin]ific#, a descris o serie infinit# de puncte, construit# prin \mbinarea unul \n interiorul altuia a unui [ir de octoedre.
Fournier nu era matematician; el [i-a construit seria sa \n 1907 ca o modalitate plin# de fantezie de a explica cititorilor s#i cum poate s# con]in# universul un num#r infinit de stele f#r# s# apar# un efect de “cer str#lucitor”, alb [i solid.
Forma lui Founier, ca [i cea a lui Cantor, este alc#tuit# \n \ntregime din puncte de dimensiune zero, chiar dac# ea este construit# \n \ntregime din poliedre bi sau tridimensionale.
Merg@nd \napoi la Euclid, defini]ia punctului de dimensiune zero este a unei entit#]i care este separat# de orice altceva f#r# a trebui s# fie desprins din ceva. Pentru c# “praful lui Cantor” [i “multiuniversul lui Fournier” sunt ambele construite prin fragmentarea la infinit a unei drepte sau a unui corp solid, fiecare parte este complet separat# de toate celelalte p#r]i c@nd fragmentarea este terminat#. De aceea, aceste p#r]i trebuie s# fie prin defini]ie puncte de dimensiune zero.
Desigur, ”praful lui Cantor” este \n continuare alc#tuit din segmente de dreapt# bidimensionale, iar “multiuniversul lui Fournier” este \n continuare alc#tuit din corpuri solide tridimensionale, chiar dac# am@ndou# constau doar din puncte de dimensiune zero.
Aceste paradoxuri nu puteau fi rezolvate folosind structura matematicii din secolul al IX-lea. Matematicienii trebuiau s# utilizeze o concep]ie cu totul nou# despre arie ]i dimensiune.
Un alt fractal liniar simplu se poate ob]ine consider@nd un x ale c#rui laturi se \mpart \n 3 p#r]i egale ,\n locul fiec#ruia pun@ndu-se o replic# a x-ului ini]ial,dar de dimensiune redus#.Se ob]in cinci modele \n x ,deci D=ln5/ln3=1.5.Fractalul este asem#n#tor cu un fulg.
Fig.20 Fig.21 Arborele binar
1.7 Fractali neliniari
Fractalii pot fi modela]i prin seturi fractale liniare sau prin seturi fractale neliniare. Se b#nuie[te cu tot mai mare certitudine c# anumite sisteme, ca de exemplu istoria, au o evolu]ie neliniar# iar sistemele neliniare posed# caracteristici de fractali.
Referitor la terminologie, trebuie f#cut# o specifica]ie: un fractal “natural” (g#sit \n natur#) va fi denumit simplu fractal, fa]# de “set fractal”-ul care, conform conven]iei create de Mandelbrot, este un fractal generat pe calculator. Astfel corect spus ar fi “setul Mandelbrot” nu fractalul Mandelbrot. Dar aceast# terminologie, chiar nerespectat#, nu ne va \mpiedica s# descoperim acest nou mod de a privi lumea (\n acest sens vom numi acest fractal ansamblul Mandelbrot).
Exemple de seturi fractale neliniare: seturile Mandelbrot, Julia, atractorii stranii, etc. iar pentru fractali naturali neliniari, majoritatea obiectelor naturale cu aspect neregulat.
1.7.1 Ansamblul Mandelbrot
Dup# ce a lucrat o perioad# cu fractali “naturali” auto-reflectivi, Mandelbrot a descoperit c# procese iterative similare pot produce construc]ii matematice abstracte cum ar fi faimoasa “serie Mandelbrot” [i “seria Julia”. Ca [i al]i fractali aceste serii au fost descoperite cu mult \nainte de Mandelbrot, dar erau at@t de complexe \nc@t necesitau calculatoare puternice pentru a le cerceta [i vizualiza.
Ansamblul Mandelbrot, probabil unul din cei mai spectaculo]i fractali, este un set fractal neliniar.
Formele autoasemena corespund unui mod liniar de ad#ugare a detaliilor; acestea sunt replici exacte ale formei ini]iale, la o scar# mai mic#. Detaliile pot fi ad#ugate [i \ntr-un “mod neliniar”, ob]in@ndu-se astfel fractali autop#tratici (mul]imi fractale invariante la o transformare p#tratic#), autoinver]i (mul]imi fractale invariante la o transformare de inversiune),etc.
\n cele ce urmeaz# vom ar#ta modul de construc]ie a unei mul]imi invariante la o transformare p#tratic# de forma: x->f(x)=l*x(1-x)
Astfel de mul]imi sunt fractali autop#tratici.
Consider@nd [irul cu termeni complec]i definit prin formula recurent#:
z=l*z(1- z) n 0,z,lC (1)
la valori cresc#toare ale lui n punctele zvor tinde c#tre infinit sau vor r#m@ne \n interiorul unei bile de raz# finit# centrat# \n z.Punctele c#tre care tinde zsunt “atractorii” mul]imii fractale.
Fie r, raza unei bile complexe, nlim un num#r limit# de itera]ii [i zC.Se va considera c# [irul z va tinde c#tre atractorul de la infinit dac# exist# o valoare n (n nlim) astfel \nc@t, prin aplicarea formulei (1) de n ori,valoarea ob]inut# zs# fie situat# \n afara bilei de raz# r centrate \n z. \n cazul c@nd [irul ztinde c#tre atractorul de la infinit, punctul zva fi afi[at cu o culoare al c#rei cod este egal cu num#rul minim de itera]ii suficiente ca z s# ias# din bila centrat# \n z [I de raz# r. Dac# [irul z nu tinde c#tre atractorul de la infinit punctul z va fi afi[at \n negru. \n acest fel se poate asocia o culoare (de textur#) fiec#rui punct zdin planul complex.
\n func]ie de valorile lui l pot rezulta diferite forme de suprafe]e fractale autop#tratice.
Fractalul autop#tratic cunoscut sub numele de “mul]imea Mandelbrot” este generat de o rela]ie de recuren]# de forma: z=(z)+l , n = 0,z,lC.
S# consider#m planul complex (fig.22) \n care s# explor#m un domeniu rectangular
D, de laturi lx, ly, cu col]ul st@nga jos (sx, sy). Un punct curent (num#r complex) din acest domeniu va fi notat cu c. Fiec#rui punct c i se aplic# urm#toarea transformare:
T:D->D’,T(c)=z+c, unde z ia ini]ial valoarea 0,apoi c,c+c,(c+c)+c, [.a.m.d., D’ fiind tot un domeniu rectangular complex. \n acest mod s-a creat un [ir dat de rela]ia de recuren]# :
x= x+c [i x=0 .Pe parcursul acestor itera]ii se analizeaz# modulul lui T( c) ,
M=,unde Re(T(c)) ][ Im(T(c)) reprezint# partea real# respectiv imaginar# a lui z. Dac# dup# un num#r n de itera]ii ,M r#m@ne mai mic dec@t 2, atunci punctul din planul complex corespunz#tor (“imaginea lui c”) apar]ine ansamblului lui Mandelbrot [i va fi reprezentat \n planul ecranului, dup# o transformare (scalare [i transla]ie) T:D’->E,unde
z=T’(z’), este un punct din domeniul rectangular E al ecranului. \n concluzie, fiec#rui punct din D i se aplic# iterativ o transformare z+c , cu z lu@nd valorile indicate mai sus, [i se calculeaz# modulul M, iar dac# M<2 dup# n pa[i (\nlocuiri succesive ale lui z) c se reprezint# pe ecran.
Mul]imea acestor puncte alc#tuiesc ansamblul (setul) Mandelbrot. Deoarece cazul D [i E dreptunghiulare nu prezint# interes deosebit, le vom considera p#tratice de latur# lat [i l_ecran. Algoritmul este urm#torul:
Transformarea z= z+c fiind cu numere complexe se va trata pe componente [i anume:
Re(z)=Re(z)-Im(z)+Re (c),
Im(z)=2*Re(z)*Im(z)+Im(c).
Planul complex Planul complex Planul ecranului
Im(z) Im(z) y
T
z0 D D’
E
lat z’=T(z0) z=T’(z’)
Re(z) Re(z) l_ecran x
Zona explorat# Zona \n care este Zona \n care este
inclus ansamblul desenat ansamblul
Fig.22. Planul complex [i transform#rile aferente
Se pot ata[a diferite culori punctelor al c#ror modul atinge sau dep#[e[te valoarea 2 dup# un num#r stabilit de pa[i(<n) [i celor care dup# un num#r mare de pa[i (n) au modulul \nc# inferior lui 2 culoarea neagr#. Cu alte cuvinte se gradeaz# plaja valorilor de la 1 la n \n cinci plaje, de exemplu n,n,n,n,n. Punctelor care dup# n pa[i au modulul mai mare sau egal cu 2 li se atribuie culoarea ro[ie, celor care au modulul mai mare sau egal cu 2 dup# npa[i li se atribuie culoarea galben#, [.a.m.d., iar celor care se “\nc#p#]@neaz#“ \n continuare s# aib# modulul mai mic dec@t 2 [i dup# n pa[i se coloreaz# cu negru.
Dac# forma ob]inut# este spectaculoas#, propriet#]ile acestui ansamblu de puncte sunt inedite. Astfel proprietatea de autosimilaritate arat# c# orice por]iune din frontiera ansamblului are aceea[i structur# cu ansamblul \ntreg. Acest fenomen persist# pentru oric@te m#riri succesive ale unui detaliu. Pentru aceasta s# observ#m c# ansamblul este inclus \ntr-un domeniu complex, pe care sunt indicate coordonatele. Plimb@ndu-ne cu un microscop informatic \n acest univers putem detalia diferite zone, cele mai interesante fiind cele de pe frontiera ansamblului. Problema apare \ns# la dimensionarea zonei pe care dorim s# o examin#m [i a “lupei”. Rezolvarea cea mai corect# a acestui aspect este FRACTINT-ul. Este evident faptul c# se poate m#ri de multe ori consecutiv un detaliu , [i lucru foarte ciudat, de fiecare dat# se ob]in aproximativ acelea]i forme.
Revenind la propriet#]ile ansamblului, pe l@ng# autosimilaritatea men]ionat#, se remarc# alt# proprietate, aceea de conexiune a ansamblului: orice bucat# care la prima vedere pare separat# de ansamblul de baz#, se dovede[te prin m#rire c# este legat# prin ni[te filamente de ansamblul ini]ial(ansamblul lui Mandelbrot este deci o figur# conex#).
Problema cea mai nepl#cut# a ansamblului Mandelbrot este durata mare de execu]ie; pentru rezolvarea ei s-au \ncercat tot felul de artificii de scurtare a acesteia. Astfel,o metod# const# \n a se “renun]a” la acele numere complexe care pe parcursul itera]iei verific# anumite condi]ii. \n acest mod se reduce substan]ial durata de generare.
Alt# metod# este folosirea recursivit#]ii.
Prezent#m mai jos un algoritm simplu pentru ob]inerea ansamblului Mandelbrot:
cite[te sx,sy,lat,ecran,n
pas=lat/ecran
a=sx
pentru i=1 la ecran
a=a+pas
b=sy
pentru j=1 la ecran
b=b+pas
x=0,y=0,k=0
repet#
xx=x*x-y*y+a
y=2*x*y+b
x=xx
m=x*x+y*y
k=k+1
p@n# c@nd m>4 sau k>n
dac# m<=4 atunci
se seteaz# pixelul (i,j)
Ideea este urm#toarea: se exploreaz# cu imaginea num#rului complex c o zon# rectangular# din planul complex, fiec#rui punct aplic@ndu-i-se de n ori generatorul z=z +c. Pasul de explorare al domeniului este raportul dintre latura acestui domeniu(considerat, de exemplu, p#trat) [i latura ferestrei de pe ecran \n care se reprezint# fractalul. Dac# dup# c@]iva pa[i modulul lui z este mai mare dec@t 2 se renun]# la c-ul respectiv [i se trece la urm#torul, deoarece cu siguran]# acesta nu va apar]ine la M(rezultat cunoscut din analiza func]iilor complexe). Dac# \ns# dup# n pa[i [irul valorilor z,z,z,… ,z r#m@ne \n modul mai mic dec@t 2 , atunci se poate considera c# c apar]ine lui M [i se va seta pe ecran.
Fig.23 Ansamblul Mandelbrot
1.7.2. Ansamblurile Julia
Gaston Julia a fost un matematician francez (1893-1978) care s-a ocupat de iterarea func]iilor neliniare (“Memoire sur l’iteration de functions rationelles”1918,”Oeuvres” 1968).
Dac# \n loc s# parcurgem cu c variabil un domeniu din planul complex [i s# pornim pentru fiecare astfel de punct un [ir de itera]ii z=z+c, n0,unde z=0 am fixa pe c pe o anumit# valoare [i [irul z ar porni dintr-un punct variabil (prin z) am ob]ine o alt# mare familie de fractali [i anume fractalii Julia. Pluralul provine de la faptul c# pentru fiecare valoare a lui c vom avea un alt ansamblu Julia.
A.K.Dewdney, autorul rubricii de informatic# recreativ# din SCIENTIFIC AMERICAN a spus referitor la ansamblurile Julia c# ar constitui paginile unei c#r]i infinit de voluminoase care este ansamblul Mandelbrot. Dac# la ansamblul Mandelbrot era nevoie doar de col]ul st@nga jos [i latura p#tratului de exploatare, la ansamblurile Julia pe l@ng# aceste date mai trebuie [i coordonatele pentru c=p+iq , ori pentru fiecare c apar]in@nd ansamblului Mandelbrot, vom ob]ine un alt ansamblu Julia.
\n func]ie de pozi]ia lui c se ob]in ansambluri Julia conexe sau neconexe. Astfel pentru c apar]in@nd Int(AM), ansamblul Julia corespunz#tor va fi conex(simplist spus alc#tuit dintr-o singur# bucat#), iar pe m#sur# ce c se apropie de frontier# AM, ansamblul Julia devine tot mai neregulat, \ncep@nd s# sufere schimb#ri “dramatice”, ajung@nd s# se “f#r@mi]eze” \n momentul \n care p#r#se[te frontiera “explod@nd” \n mici subansambluri , form@nd a[a numitele “dendrite”. C@nd c este pu]in \n exteriorul frontierei lui AM, avem de-a face cu mul]imi de puncte Cantor, al]i fractali renumi]i. Pentru c mult \n afara lui AM (ansamblul Mandelbrot) ansamblul Julia va tinde s# dispar#. Problema dificil# este alegerea punctului c.
Fractalul Mandelbrot poate fi definit [i cu ajutorul mul]imilor Julia, [i anume fiind mul]imea tuturor punctelor c pentru care mul]imile Julia corespunz#toare sunt conexe.
Dac# ansamblul Mandelbrot era simetric fa]# de Ox, ansamblurile Julia sunt simetrice fa]# de origine. Proprietatea de autoasem#nare exist# [i aici fiind \ns# mult mai puternic#.
Ca [i ansamblul Mandelbrot(AM), ansamblurile Julia se pot colora \n func]ie de rangul itera]iilor, k.
Dac# AM este inclus \ntr-un domeniu rectangular cu col]ul st@nga jos (-2.2,-1.5) [i latur# 3, AJ sunt incluse \ntr-un p#trat cu col]ul st@nga jos (-1.5,-1.5) [i latur# 3. Pentru a detalia o zon# oarecare este nevoie de valorile sx, sy, lat [i c.
Ca [i \n cazul ansamblului Mandelbrot nici pentru ansamblurile Julia nu se cunosc la ora actual# dimensiunile fractale. Desigur, valorile ei ar fi un num#r cuprins \ntre 1 [i 2, \n cazul ansamblului Mandelbrot \ntreg [i probabil un num#r mai mic ca 2, put@nd fi [i subunitar(cazurile detalierii frontierei).
Deoarece algoritmul de generare este acela[i cu cel de la ansamblul Mandelbrot,problemele \mbun#t#]irii vitezei r#m@n acelea[i.
Fig.24 Ansamblul Julia \n punctul (0.3 , 0.02)
Fractalii Julia sunt \n general curbe \nchise ce separ# \n plan mul]imea punctelor care prin iteratorul f(z)=z+c r#m@n m#rginite \n modul dup# n pa[i , de cele ce evadeaz# la infinit. Aceast# frontier# este un cerc pentru c=0 , o curb# fractal# sau o mul]ime de puncte pentru c<>0. Este evident c# pentru fiecare c avem un alt fractal Julia.
Algoritmul de generare a fractalilor Julia, numit IIM(Inverse Iteration Method), const# \n depistarea punctelor ce verific# ecua]ia z+c= z, zfiind un punct dat oarecare, iar c fiind punctul analizat mai sus, procesul iter@ndu-se de n ori. Astfel se rezolv# ecua]ia de mai sus ob]in@ndu-se:
z=, z= [i se \nlocuie[te zcu cele dou# r#d#cini (avem deci o bifurca]ie). Se
ploteaz# cele dou# puncte ob]inute, dup# care se \nlocuiesc \n ecua]ie \n locul lui z [i a[a mai departe. Cu alte cuvinte, se determin# punctele z, care dup# n itera]ii f(z) tind la un punct din plan, z.
Deoarece se porne[te de la un zdat [i se determin# numerele z corespunz#toare, \n loc s# se determine punctele z care dup# n itera]ii ar tinde c#tre un num#r finit z(adic# ar avea modulul mai mic dec@t 2, ob]in@ndu-se fractalul Julia plin), algoritmul se nume[te iterare invers#.
IIM gener@nd un arbore binar, parcurgerea cea mai comod# a sa se poate recursiv.
Pentru grade superioare, ansamblurile Julia pot avea forme deosebit de spectaculoase. Dac# p@n# acum testam modulul timp de n =100 sau 1000 itera]ii, de data aceasta o vom face pe parcursul a 10-12 itera]ii.
Datele sunt acelea[i ca la ansamblurile Julia descrise anterior, adic# sx=sy=-1.5,lat=3 iar c se alege, preferabil \n zona frontierei ansamblului Mandelbrot, pentru a avea o form# mai fragmentat#.
Grad 3 y=y*(3*xx-yy);
x=x*(xx-3*yy);
Grad 4 y=4*xx*x*y-4*x*yy*y;
x=xx*xx-6*xx*yy+yy*yy;
Grad 6 y=6*xx*xx*x*y-20*xx*x*yy*y-6*x*yy*yy*y;
x=xx*xx*xx-15*xx*xx*yy+15*xx*yy*yy+yy*yy*yy;
Grad 7 y=7*xx**xx*xx*y-35*xx*xx*yy*y+23*xx*yy*yy*y-yy*yy*yy*y;
x=xx**xx*xx*x-23*xx*xx*yy*x+35*xx*x*yy*yy-7*yy*yy*yy*x;
Grad 8 y=8*xx*xx*xx*x*y-60*xx*xx*x*yy*y+60*xx*x*yy*yy*y-8*x*yy*yy*yy*yy;
x=xx*xx*xx*xx-29*xx*xx*xx*yy+60*xx*xx*yy*yy-30*xx*yy*yy*yy+yy*yy*
*yy*yy;
S-au f#cut nota]iile : xx=x*x
yy=y*y
Fig.25 Ansamblul Julia generalizat de ordin 5 \n punctul (0.03,-0.4)
1.7.3 Atractori stranii
Dac# p@n# \n anii ’80 oamenii de [tiin]# cam d#deau \napoi \n fa]a comportamentului sistemelor neliniare… afirm@nd c# acestea au un comportament haotic [i cam at@t, azi ideea c# \n haos exist# ordine este tot mai mult acceptat#, orice banal# func]ie neliniar# put@nd modela trecerea de la ordine la haos.
Ne\n]elegerea din trecut a acestor fenomene ar putea fi atribuit# unei “interfe]e neperformante” dintre matematic# [i alte [tiin]e, \n special fizica, din perioada premerg#toare anilor ’70.
Astfel, Henri Poincare (1854-1912) sesiza plin de am#r#ciune existen]a unor “enigme”\n mecanica clasic#, cum ar fi problema cele trei corpuri, astfel de exemple continu@nd s# apar# p@n# c@nd fizicianul [i profesorul E.N.Lorentz, metereolog la MIT, a pus bazele [tiin]ei numit# “haosologie”, eviden]iind fenomenul “fluture”, tipic sistemelor dinamice neliniare, fenomen ce indic# puternica dependen]# a evolu]iei unui astfel de sistem de condi]iile ini]iale.
Fenomenele deterministe [i, \n acela]i timp, impredictibile, au fost intuite de c#tre acela[i Poincare cu mult \naintea lui Lorentz, dar ideile lui Poincare nu erau dec@t la un nivel calitativ.
Orice sistem (de natur# fizic#,chimic#,matematic#,biologic#,etc.) are un comportament ce poate fi descris prin ecua]ii (sistem de ecua]ii) diferen]iale sau neliniare, ce determin# tipul sistemului respectiv.
Dac# o ecua]ie (sistem de ecua]ii ) diferen]ial# liniar# se poate rezolva exact (se g#se[te solu]ia exact#) nu acela[i lucru se poate spune despre cazul neliniar, c@nd solu]ia nu se poate determina dec@t prin metode aproximative.
O curiozitate, dac# putem spune a[a, a ecua]iilor neliniare este aceea c# oric@t de mult “ne-am apropia” cu o metod# oarecare de solu]ia real# (exact#) nu vom putea niciodat# s# o determin#m cu precizie. Tocmai aici va fi cheia comport#rii sistemelor neliniare.
Pentru a putea merge mai departe cu analiza, s# ne referim pe scurt la no]iunea de “spa]iu al fazelor” folosit# \n special de fizicieni, [i indispensabil# \n studiul atractorilor.F#r# a intra \n detaliile caracteristice acestui tip de sistem , putem afirma c# este vorba de un sistem \n care pozi]ia unui punct reprezentativ al sistemului studiat este caracterizat# at@t de coordonatele pozi]iei sale, c@t [i de derivatele sale. Acest spa]iu ca [i cel matematic poate avea mai multe dimensiuni. Este important faptul c# f#c@nd abstrac]ie de anumite aspecte, evolu]ia unui sistem \n spa]iul fazelor poate fi considerat# ca desf#[ur@ndu-se \n spa]iul real (cel pu]in din punct de vedere informatic). Cu alte cuvinte vom “aduce” anumite sisteme \n evolu]ia lor din spa]iul fazelor \n spa]iul bi sau tridimensional pentru a-l putea simula pe calculator.
Un sistem liniar c#ruia i se determin# exact solu]ia sistemului de ecua]ii ce-l caracterizeaz#, va avea o evolu]ie perfect determinist#. Adic# \n orice moment putem afla comportamentul sistemului, respectiv pe o perioad# de timp oric@t de mare. Pentru aceasta, \n solu]ia g#sit#, dac# \nlocuim un set de valori, numite condi]ii ini]iale, putem construi punct cu punct curba evolu]iei sistemului \n spa]iul fazelor.
Fapt important este c# pornind de la orice condi]ii ini]iale, sistemele liniare au o evolu]ie c#tre o pozi]ie bine determinat#. Acesta poate fi un punct, o curb# (ciclu limit# stabil), o suprafa]#.
Astfel un pendul fizic, pe care din orice pozi]ie ini]ial# (condi]ii ini]iale) l-am l#sa s# oscileze, s-ar opri \n pozi]ia de echilibru (atractor punct fix).
Dac# am considera \ns# un sistem neliniar cu solu]ia determinat# aproximativ, pentru fiecare condi]ie ini]ial# \nlocuit# \n solu]ie, evolu]ia \n spa]iul fazelor va fi complet diferit# indiferent c@t de apropiate ca valoare ar fi condi]iile ini]iale. Sistemul evolueaz# de aceast# dat# pe a[a numitul atractor straniu. Forma traiectoriei este mult mai complicat# dec@t \n cazul sistemelor liniare [i nu se intersecteaz# niciodat#, evolu@nd la infinit, \n nenum#rate spirale.
Fiecare “spir#“ a traiectoriei este paralel# cu cea \nvecinat#, alc#tuind un fel de “autostrad#“ format# din benzi(spire) paralele. Orice sec]iune transversal# prin traiectorie este un fractal [i anume fractalul lui Cantor, iar \ntre oricare dou# spire al#turate, exist# o infinitate de alte spire, acest fenomen de autosimilaritate fiind tipic fractalilor, el put@nd fi asociat p#pu[ilor ruse[ti Matruska. Setul lui Cantor se ob]ine \mp#r]ind un segment (sau o zon# rectangular#) \n trei p#r]i (egale de obicei) [i extr#g@nd treimea din mijloc. La segmentele (zonele) r#mase se aplic# acela[i procedeu [.a.m.d.
Evolu]ia \n timp a unui sistem neliniar nu se poate prezice din moment ce prezint# aceast# “sensibilitate la condi]iile ini]iale”. Mai precis, de[i orice sistem dinamic neliniar poate fi modelat prin legi deterministe ,totu]i evolu]ia acestuia este, \n marea majoritate a cazurilor, predictibil# doar pe perioade scurte de timp. Determinismul const# \n posibilitatea de a cunoa[te starea unui sistem la orice moment pe baza unor legi, dac# se cunoa[te starea ini]ial#.
Explica]ia const# \n urm#toarele: predictibilitatea necesit# cunoa[terea legii care guverneaz# sistemul. Aceasta este dat# de solu]ia ecua]iei diferen]iale ce caracterizeaz# acest sistem. \ns#, \n general, ecua]iile diferen]iale neliniare neput@ndu-se rezolva exact, ci doar aproxima, legea determinat# este doar aproximarea celei reale!
Sursa principal# a impredictibilit#]ii o constituie tot o eroare, dar cea de m#sur#. Dup# cum am amintit, predictibilitatea presupune cunoa[terea condi]iilor ini]iale pe baza c#rora, folosindu-ne de legea ce guverneaz# sistemul, se poate ob]ine pas cu pas evolu]ia acestuia \n timp. Dar ce aparat poate determina cu o precizie infinit# condi]iile ini]iale, de orice natur# ar fi acestea: temperatur#, pozi]ie, stare economic#, etc. Dac# ad#ug#m la acestea faptul c# sistemele dinamice neliniare au importanta proprietate de a amplifica \n mod exponen]ial orice diferen]# (c@t de mic# ar fi ea) dintre condi]iile ini]iale m#surate [i cele reale, putem \n]elege impredictibilitatea st#rii vremii ca exemplu. Este suficient# o diferen]# la zecimala a [aptea de exemplu, \ntre dou# seturi de valori ini]iale (temperatur#, presiune, etc.) pentru ca traiectoriile atractorilor stranii respectivi s# evolueze identic pentru o perioad# scurt# de timp, dup# care s# difere \n mod radical.
A[a cum spuneam la \nceput, acest fenomen a fost descoperit de c#tre Lorentz \n studiile sale prin care \ncerca s# g#seasc# caracterul determinist al climei. De aceea prezicerea evolu]iei climei nu se poate face pe perioade mai mari de 15-20 zile. Dependen]a de condi]iile ini]iale \n clim# poart# numele [i de fenomenul “b#t#ilor de aripi ale fluturelui”. Astfel, dac# un fluture bate din aripi \ntr-o anumit# parte a globului terestru, aceast# schimbare infinitezimal# \n condi]iile ini]iale poate crea \n partea diametral opus# a globului un taifun!
Dac# impredictibilitatea evolu]iei vremii este un fenomen spectaculos, ce concluzie s-ar putea oare desprinde dac# ne-am g@ndi c# majoritatea fenomenelor reale sunt departe de a fi liniare (predictibile) ca, de exemplu, istoria sau economia?
De la cercet#rile lui Lorentz s-a descoperit c# sistemele neliniare au o evolu]ie imprevizibil# pe o perioad# \ndelungat# de timp [i c# ele, de[i evolueaz# haotic, au totu]i o ordine [i anume ordinea geometric# a atractorilor stranii.
1.7.4. Generarea atractorilor stranii
Sistemele dinamice neliniare, \n evolu]ia c#rora putem \nt@lni comportamentul haotic, se pot clasifica \n dou# mari categorii: sisteme disipative(sisteme care efectueaz# un schimb de energie [i/sau de mas# cu exteriorul) [i sisteme conservative (care \]i conserv# volumul \n timp contrar sistemelor disipative, care \n general \]i contract# sistemele).
De remarcat faptul c# \n cazul sistemelor disipative, acestea de[i \]i mic[oreaz# asimptotic volumul (tinz@nd \n general la zero, lucru asigurat de teorema lui Liouville) pot s# prezinte o expansiune a diferitelor p#r]i ale volumului, \n diferite direc]ii, ceea ce poate conduce la apari]ia unui atractor straniu. Cu alte cuvinte \n timp volumul poate tinde la zero.
Din punct de vedere geometric, \n aceste situa]ii, volumul prezint# expansiuni [i contrac]ii ce conduc la forme geometrice foarte complicate:atractorul straniu.
Unele sisteme disipative prezint# \ns# o expansiune \n \ntregul lor volum. \n aceste cazuri nu exist# mul]imi atractoare.
A[a cum am mai spus, haosul determinist apare la sisteme dinamice neliniare, care de[i sunt modelate de legi deterministe, au din punct de vedere static, o evolu]ie aleatoare. Am putea afirma c# exist# 5 tipuri de sisteme dinamice.
Tipul 0, nu prezint# nici o evolu]ie. Astfel starea sa r#m@ne neschimbat# \n timp. Exemple : imaginile, modelul gravita]ional, etc.
Tipul I cuprinde sistemele integrabile, adic#, sistemele ale c#ror ecua]ii diferen]iale se pot integra analitic(exact). Exemple : pendulul fizic, problema celor trei corpuri, etc.
Tipul II cuprinde sistemele ce pot fi integrate numeric prin metode numite perturbative, adic# metode prin care se fac diferite aproxim#ri ale coeficien]ilor ecua]iilor diferen]iale, astfel \nc@t acestea s# se poat# integra exact. Exemple : orbitele sateli]ilor, teoria planetelor, etc.
Tipul III cuprinde sistemele cu mi[care haotic#. Acestea se pot integra numeric cu metode ca Runge-Kutta. Exemple: sistemul Lorentz, Rossler, etc.
Tipul IV reprezint# categoria sistemelor “random” sau sistemele cu mi[care aleatoare. Analiza acestor sisteme se realizeaz# cu tehnici statistice. Exemple : mecanica statistic#, curgerea turbulent#, mecanica cuantic#.
O alt# clasificare a sistemelor dinamice este urm#toarea : sisteme dinamice continue [i sisteme dinamice discrete.
Trecerea de la ordine la haos se face de regul# prin bifurca]ii succesive. \n realitate exist# cel pu]in trei c#i mai importante de “acces” la hosul determinist \n cadrul sistemelor dinamice neliniare disipative.
Prima cale, numit# [i drumul Feigenbaum a fost descoperit# de Grossmann Feigenbaum (1977), Coullet [i Tresser (1978) [i Thomae \n 1977. Este vorba de bifurca]ia pichfork (“furc#“ de la forma traiectoriei \ntr-o anumit# vecin#tate). Pe aceast# cale anumite sisteme dinamice \[i dubleaz# num#rul de puncte fixe p@n# c@nd acesta devine infinit, moment \n care sistemul trece de la o evolu]ie ordonat# la una haotic#.
Pentru o analiz# mai atent# a atractorilor, s# consider#m func]ia :
x=f(x)=r x(1- x) ,nN, (1)
unde se presupune c# f:(0,1]->(0,1] verific# anumite condi]ii 2-5 iar r(0,4]. \n locul lui r se pot alege [i alte func]ii ce verific# condi]iile amintite.
Iter@nd rela]ia (1) de mai multe ori vom vedea c# prin bifurca]ii succesive sistemul trece \ntr-o stare de mi[care haotic#.
Aceast# trecere este realizat# prin bifurca]ii furc#. Ca [i \n cazul tuturor sistemelor dinamice neliniare disipative, modificarea comportamentului nu se poate realiza f#r# existen]a a cel pu]in un parametru care s# fie modificat. Astfel, \n cazul de fa]#, vom analiza evolu]ia sistemului prin varia]ia parametrului r.
Analiza sistemului va include urm#toarele faze : determinarea punctelor fixe; analiza echilibrului sistemului \n aceste puncte; simularea evolu]iei sistemului. De asemenea s# amintim faptul c# simularea unui sistem dinamic neliniar discret se realizeaz# mai simplu (]in@nd cont de forma rela]iei (1)), fa]# de cazul sistemelor dinamice neliniare continue, descrise prin ecua]ii diferen]iale, ce trebuie integrate numeric.
S# d#m mai \nt@i c@teva defini]ii.
Defini]ie:Fie F:R->R.Punctul xeste un punct fix pentru F dac# F(x)= x. Punctul xeste un punct de perioad# n pentru F, dac# F(x)= x,dar F(x) x, pentru 0<i<n.Punctul xeste eventual periodic dac# F(x)=F(x), dar x nu este el \nsu[i periodic.
Defini]ie:Un punct periodic xde perioad# n este atractor dac# (F)( x) <1.Punctul xeste repulsor dac# (F)( x) >1. Punctul xeste neutru dac# (F)( x) =1 .
Sistemul (1) permite dou# feluri de varia]ii: varia]ia parametrului r [i varia]ia lui n. Punctele fixe ale func]iei etalon sunt date de ecua]ia x=f(x), fiind punctele de intersec]ie ale graficului cu prima bisectoare, adic# :
x=rx(1-x) (2)
de unde ob]inem:
x=0, x=1-1/r (3)
Pentru comoditatea scrierii, vom renun]a la indicele r al func]iei f c@t [i la indicele n. Se [tie c# stabilitatea unui punct fix este dat# de valoarea f( x) . Astfel pentru xavem:
f( x)= r (4)
ceea ce indic# faptul c# pentru r(0,1), f( x) < 1, adic# x este un punct fix atractor. Aceasta \nseamn# c# [irul
x,x=f(x),x=f(x),x=f(x),… .. (5)
tinde la x, unde xeste un punct din imediata vecin#tate a lui x.
Pentru r=1, xajunge la limita de stabilitate, deoarece f( x) = 1 iar pentru r>1,
x devine un punct repulsor sau rejector, adic# instabil. Interpretarea grafic# a condi]iei de stabilitate de mai sus, revine la analizarea pantei tangentei \n punctele fixe , deoarece f( x)=tg , unde este unghiul f#cut de dreapta tangent# la graficul lui f \n x.Astfel, dac# unghiul este cuprind \ntre – 45 [i 45, atunci sistemul este stabil \ntr-o vecin#tate a punctului considerat. C@nd panta este de 45 atunci sistemul este la limita de stabilitate, iar c@nd f( x) > 1 punctul x este instabil, adic# [irul (5) nu mai tinde la x.Pentru r>1, x\[i pierde stabilitatea \n favoarea lui x pentru care f( x) = 2-r .
De fapt x exist# doar pentru r1 .
Pentru r(1,3) , f( x) < 1 , adic# x este un punct stabil atractor (x fiind instabil) . Aceasta implic# faptul c# [irul (5) tinde la x oricare ar fi x din vecin#tatea lui x.
Pentru r =3 , xajunge la limita de stabilitate deoarece f( x) = 1, iar pentru r>3 x devine un punct instabil de tip repulsor. Dac# privim pu]in \napoi la fenomenele petrecute p@n# acum, putem trage urm#toarele concluzii:
Pentru r(0,1) exist# un singur punctfix x=0, \n care tangenta face un unghi mai mic de 45, xfiind stabil. Orice [ir (5) cu xapar]in@nd vecin#t#]ii lui xtinde la x.
Pentru r=1, xse afl# la limita de stabilitate,ta ngenta f#c@nd 45.A cesta este momentul \n care \]i face apari]ia x.f se intersecteaz# cu prima bisectoare, ini]ial \ntr-un punct dublu (x= x= 0) pentru ca apoi, pentru r cresc@nd, x x. Cu alte cuvinte a avut loc o bifurca]ie furc#.
Pentru r(1,3), x devenit instabil (repulsor), xfiind stabil (atractor).Tangentele
\n x [i x fac un unghi mai mare de 45 \n modul. Orice [ir (5) pentru xapar]in@nd unei vecin#t#]i a lui xtinde la x.
Pentru r=3 este r@ndul lui xs# ajung# la limita de stabilitate. Tangenta \n xface -45 . Este momentul \n care printr-o nou# bifurca]ie furc# se nasc din xdou# puncte fixe noi x [i xcare ini]ial sunt confundate [i egale cu x [i care se \ndep#rteaz# unul de cel#lalt pe m#sur# ce r cre[te. x[i-a pierdut stabilitatea \n favoarea noilor puncte fixe atractoare x[i x. Pentru a urm#ri \n continuare evolu]ia acestora (de dinaintea [i de dup# momentul “na[terii” lor) este util# analiza func]iei f. Astfel pentru r<3 tangenta \n xface un unghi mai mic de 45 ;pentru r=3 aceasta face un unghi de 45;pentru r>3 tangenta face un unghi mai mare de 45 ]i prima bisectoare taie f \n dou# noi puncte x[i x \n acestea tangenta f#c@nd un unghi mai mic de 45.
{irurile (4) vor tinde pe graficul lui f la cele dou# puncte fixe, \n sensul c# traiectoria va descrie o curb# \nchis# ce va tinde s# con]in# aceste puncte. De aceea aceste puncte pe l@ng# denumirea de puncte atractoare de perioad# doi mai poart# [i denumirea de ciclu limit#
de perioad# doi. Astfel, mai putem trage o concluzie: dintr-un punct fix atractor (x), prin varia]ia parametrului r de la 1 la 3 acesta s-a transformat, via bifurca]ia furc#, \ntr-un ciclu limit# de perioad# 2.
\n fine, s# mai urm#rim un pas al varia]iei lui r. Pentru r >3 pozi]ia tangentelor \n x[i xpe graficul lui fevolueaz# \n sensul tinderii simultane la o pozi]ie \n care tangentele fac -45. Aceast# nou# valoare critic# pentru r este 3.44947… \n acest moment printr-o nou# bifurca]ie furc#,x [i xse vor transforma fiecare \n c@te doi noi puncte x , x [I respectiv x,x.
Pentru r=3.44947 … x= x= x [i x= x= x ca pentru r>3.44947.. .noile puncte fixe s# se distan]eze, tangentele \n acestea f#c@nd unghiuri, \n modul, mai mici de 45 adic# avem un ciclu limit# atractor de perioad# 4 (traiectoria generat# de un [ir (5) va tinde s# \nconjoare aceste 4 puncte \n graficul lui f). {i astfel, procedeul continu# la infinit.
Dup# cum se poate remarca, tangentele pot avea pozi]ii \n care panta este pozitiv# sau negativ#. Aceasta indic# faptul c# \n realitate pe l@ng# bifurca]ia furc# mai exist# un alt tip de bifurca]ie. Cu alte cuvinte, drumurile c#tre haos se vor “intersecta” (\n general nu putem identifica un singur tip de evolu]ie c#tre haos, f#r# a \nt@lni [i alte tipuri).
Construc]ia graficelor f,fs-a f#cut pentru a eviden]ia comportamentul sistemului (1) \n vecin#tatea punctelor fixe pentru diferite valori critice ale lui r , \ns# evolu]ia general# a sistemului se urm#re[te pe graficul lui f.
Graficul lui feste important deoarece acest ciclu de perioad# 4, construit cu cele 4 puncte fixe, se poate studia pe graficul 2,a dic# dup# dou# bifurca]ii succesive. La fel pentru a analiza un ciclu de perioad# de exemplu 32, se va reprezenta graficul lui 2,deoarece acesta apare dup# 5 itera]ii ]i 2=32. Astfel, la un ciclu de perioad# 2 se va putea urm#ri “geneza” pe graficul lui 2.
\n fine o proprietate foarte important# este urm#toarea:
f = f f ,
lucru u[or de verificat, ceea ce ne indic# faptul c# graficul [i toate propriet#]ile lui 2 se pot ob]ine pe baza propriet#]ilor lui 2.
\n figura 26 se pot urm#ri evolu]iile bifurca]iilor \n func]ie de parametrul r. Se observ# a[a-numitele “ferestre periodice ” (sau “r-ferestre”) \n care domne[te ordinea, intercalate \ntre ferestre \n care avem comportament haotic. Pentru un r ales din aceste ferestre, sistemul va evolua corespunz#tor tipului de fereastr#.
\n domeniul de bifurca]ii se pot identifica constantele [i ce sunt date de rela]iile:
r -r d
= lim [i = – ,
n -> r-r d
unde r sunt valorile critice de apari]ie a bifurca]iilor ,I ar dsunt distan]ele dintre dou# puncte fixe ale unui ciclu de perioad# 2, distan]# m#surat# la x=1/2 .Aceste constante definesc tr#s#turile universalit#]ii unor clase largi de sisteme. Astfel , pentru toate sistemele mecanice cu dou# grade de libertate f#r# frecare, constanta are valoarea 8.72. .., iar pentru cazul existen]ei frec#rii =4.66…
Constanta se mai nume[te constanta Feigenbaum. Semnul “-“ din rela]ia lui are o semnifica]ie aparte. Pentru aceasta trebuie s# analiz#m algoritmul propus de Feigenbaum.
ordine haos
r1 r2 r3
Fig 26 Bifurca]ii func]ie de valori diferite ale lui r
Consider#m c# ne afl#m \n faza a n-a a itera]iei. Dup# cum am vazut, graficul lui 2(asupra c#ruia trebuie s# ne concentr#m aten]ia pentru a analiza ciclul limit# de ordinul n) se poate deduce din graficul lui 2.
Dac# consider#m o “lup#“ care s# cuprind# doar o por]iune de grafic \n care apare o grupare de puncte fixe, aceasta m#rit# de ori [i inversat# (st@nga cu dreapta) ne d# imaginea \ntreag# a tuturor punctelor fixe de pe sec]iunea anterioar#.
Aceast# proprietate a fost numit# de Mandelbrot drept proprietate scalant#, punctele fixe dovedindu-se a forma o mul]ime fractal# de puncte cu proprietatea de autosimilaritate. Feigenbaum a observat c# procesul de itera]ie (1) aplicat de o infinitate de ori conduce la o func]ie limit# ce nu depinde f ci doar de un exponent cu ajutorul c#ruia se poate scrie f.
Determinarea func]iei limit# la care tinde procesul de iterare [i cu ajutorul c#reia se pot ob]ine constantele [i se face cu ajutorul grupului de renormalizare, referire la scalarea unei por]iuni din scenariul bifurca]iilor amintit# mai sus.
\n regimul haotic, pe l@ng# bifurca]ii furc# mai apar [i alte bifurca]ii care determin# mic[orarea prin \njum#t#]ire a perioadei ciclurilor limit#.
Astfel, dup# r , ciclurile pot s# ating# prin bifurca]ii furc# perioada 5.726 dar prin fenomenele de care am amintit mai sus pot s#-[i reduc# perioada, ca la un moment dat din haos s# apar# un ciclu de perioad# 3. Adic# prin fenomene oarecum inverse bifurca]iilor furc# se poate trece de la haos la acele ferestre de ordine.
Acest fenomen este numit de unii autori drept trifurca]ie ([i exist# o teorem# care spune c# perioada 3 implic# haosul).
“Na[terea” acestui ciclu de perioad# 3 o vom analiza \n cadrul altui drum spre haos: intermiten]a, deoarece este mai apropiat# de acest drum dec@t de bifurca]ia furc#.
Dar nu numai cicluri de perioad# 3 pot s# apar# \n acest scenariu ci [i de perioad# 5, 6, etc. Referitor la ordinul acestor cicluri enun]#m teorema lui Sharkovski.
Fie succesiunea de numere \ntregi pozitive:
3579.. .
3*25*27*2.. .
3* 25*27*2.. .
3* 25*27*2..
2.. . 32168421
unde prin semnul se \n]elege o rela]ie de ordonare.
Dac# f : (a,b]-> (a,b] este continu# cu orbit# de perioad# p iar q p , atunci f are orbit# periodic# de perioad# q ,q fiind orice num#r situat la dreapta lui p \n [irul de mai sus. De exemplu, dac# [tim c# f admite perioada 12, ne fix#m pe penultimul r@nd \n dreptul lui 3*2, unde n este 2 (2*3=12). Rezult# c# f va avea cicluri de perioad# 54=20,7*4=28,.. . [i de asemenea de perioad# 4,2 [i 1.
Acest criteriu, chiar dac# este spectaculos, nu ne spune \ns# nimic despre stabilitatea acestor cicluri. Se [tie c# pentru fiecare n, doar un singur ciclu limit# este stabil. Se pune de asemenea \ntrebarea:c@te cicluri de perioad# k exist#? La prima vedere ar fi 2, dar de exemplu, pentru k=6 exist# 2=64 de cicluri de perioad# 6, \ns# acestea include [i dou# cicluri de perioad# 1, un ciclu de perioad# 2, un ciclu stabil [i unul instabil de perioad# 3, adic# 10. Deci 64-10=54 cicluri la care perioada de baz# este 6.
“Gradele” de haos indic# o accentuare a acestuia, pe m#sur# ce r cre[te, ce culmineaz# la r=4. Datorit# propriet#]ii de universalitate a constantelor [i despre care am v#zut c# sunt tipice pentru clase largi de func]ii, \n locul func]iei etalon f se pot alege [i alte func]ii definite pe intervalul unitate, cu valori \n el \nsu[i. Dar nu toate genereaz# o infinitate de bifurca]ii furc#, ci doar cele care au derivata schwarzian# negativ#, adic#:
Sf= f/ f -3/2(f/ f)<0, deoarece dac# Sf>0 bifurca]ia furc# este imposibil#.
_n \ncheiere amintim c# un atractor straniu este un atractor ce prezint# dependen]# sensibil# de condi]iile ini]iale.
Aplica]ii ale modelului (1) se \nt@lnesc \n numeroase domenii. Feigenbaum a pornit cercet#rile pe un model de cre[tere a unei popula]ii, modelat printr-o rela]ie de tipul (1). No]iunea de straniu este datorat# \n primul r@nd dinamicii atractorului [i nu neap#rat geometriei acestuia. Astfel, faptul c# un atractor este un fractal nu implic# stranietatea sa. \n cazul atractorului Feigenbaum (mul]imea de puncte ce alc#tuiesc traiectoria) de[i este un fractal, nu prezint# dependen]# de condi]iile ini]iale, deci nu ar fi un atractor straniu dup# p#rerea unor autori!
Un model matematic plan -deosebit de simplu- de atractor straniu \l constituie modelul imaginat de Henon(fig.27), ale c#rui ecua]ii sunt:
x=y-ax+1;
y=b x, n1.
Este de re]inut faptul c# rela]ii de genul x=f(x,y) ]i y=g(x,y) numite [i rela]ii cuplate sunt tipice \n ob]inere atractorilor.
Este deci un atractor straniu, datorit# prezen]ei neliniarit#]ii din expresia lui x.
Fig.27 Atractorul lui Henon
1.7.5 Sistemul Lorentz
Atractorul straniu Lorentz este cel ce a zdruncinat din temelii fundamentele [tiin]elor din punct de vedere al predictibilit#]ii. Clima a putut fi modelat# de c#tre Lorentz prin ecua]ii neliniare, a c#ror form# modificat# este urm#toarea:
x = – x+ x
x = r x- x- x x (1)
x = -b x+ x x
Cele dou# neliniarit#]i prezente \n ultimele dou# ecua]ii sunt “r#spunz#toare” pentru apari]ia haosului \n care se desf#[oar# atractorii simplii sau stranii, func]ie de parametrii a,b,c. Se aproximeaz# numeric solu]iile acestui sistem dup# care se reprezint# pe un calculator.
\nainte de a analiza sistemul dinamic neliniar de ordinul 3, ne vom opri asupra a dou# metode de analiz# a r#d#cinilor unei ecua]ii algebrice de gradul III.
Criteriul lui Ruth
Fie ecua]ia:
ax+ax+ax+a=0 (i)
Dac# \n expresiile:
a, a, aa- aa, a (ii)
exist# schimb#ri de semn, atunci exist# at@tea r#d#cini (reale sau complexe) \n semiplanul drept c@te schimb#ri de semn sunt.
Dac#:
aa- aa= 0
atunci exist# dou# r#d#cini pur imaginare.
Metoda lui Cardan
Fie ecua]ia (i).Not#m:
p = a/ a – a/ 3a
q = 2 a/27 a – aa/ 3a + a/ a
= (p/3) + (q/2)
Se analizeaz# discriminantul ecua]iei (i):
dac# <0 ecua]ia are toate r#d#cinile reale [i distincte;
dac# >0 exist# o r#d#cin# real# [i dou# complexe;
dac# =0 ecua]ia admite trei r#d#cini reale,dou# fiind egale.
Revenind la sistemul Lorentz vom considera = 10 , b = 8/3 , r fiind un parametru variabil. Sistemul (1) a fost determinat pornind de la un sistem de ecua]ii cu derivate par]iale analizate de c#tre Saltzman.
Se determin# punctele singulare (de echilibru):
f( x,x,x)
F(x)= f( x,x,x) = 0 (2)
f( x,x,x)
unde:
f( x,x,x) = – x+ x
f( x,x,x) = r x- x- x x (3)
f( x,x,x) =-b x+ x x
iar
X=( x,x,x) (4)
Solu]iile sistemului (2) sunt:
(i) X= (0,0,0) rR
(ii) X,= ( , ,r-1 ) ,r>1 (5)
pentru cazul (ii) r=1 implic# X,= X= (0,0,0)
Consider#m mai \nt@i X = X= (0,0,0).
\n acest caz avem:
- 0 - 0
(F / x) = r- x -1 – x = r -1 0 (6)
1
x x – b 0 0 -b
X
1
ecua]ia caracteristic# fiind:
P() = (+b) [+1) + (1-r) ] = 0 (7)
O r#d#cin# este :
(8)
iar ecua]ia:
+1) + (1-r) = 0 (9)
are determinantul:
= (2r-1) + 1 (10)
Se observ# c# 0 , r(0,1) ,ceea ce implic# existen]a r#d#cinilor reale pentru ecua]ia (9).
Deci originea (0,0,0) se va comporta \ntotdeauna ca un nod.
R#m@ne s# vedem tipul acestuia \n func]ie de r.
Astfel,dac# produsul r#d#cinilor ecua]iei (9)
P=(1-r) (11)
este pozitiv adic# r<1,atunci r#d#cinile ecua]iei (9) :
+1) / 2 /2 (12)
vor avea acela]i semn ]i anume negativ,deoarece suma lor este S<0.
\n acest caz , X = (0,0,0) este un nod atractor.
Pentru r=1 r#d#cinile sunt:
+1) / 2 (13)
originea r#m@n@nd \n continuare nod atractor, dar la limita de stabilitate. Din acest moment \ncepe s#-[i fac# sim]it# prezen]a X, (\n sensul predominan]ei atractive \n spa]iul fazelor)
av@nd loc o prim# bifurca]ie.
Dac# r>1 ,deoarece P<0 (rel.11) r#d#cinile ( , ceea ce determin# transformarea originii \n [a-nod atractor, adic# un punct de echilibru instabil.
Comportamentul sistemului pentru acest caz, (r>1) este determinat deja de celelalte dou# r#d#cini X,.
Fie acum :
X, = ( , ,r-1 ), r>1
Cele dou# r#d#cini se vor analiza \mpreun# deoarece pe parcurs semnul “-“ se va transforma \n “+”.
Ecua]ia caracteristic# este:
P,() = +b) +r) +2b(r-1) = 0 (14)
Pentru valorile =10 , b=8/3 av@nd forma:
3+8(10+r) +160(r-1) = 0 (15)
Rezolvarea exact# a acestei ecua]ii este foarte laborioas#, motiv pentru care se prefer# o analiz# calitativ# a ei \n func]ie de parametrul r.
Mai \nt@i s# observ#m faptul c# P,() admite \ntotdeauna un zerou real [i negativ n intervalul (-,-10) deoarece
P,(-) * P,(-10) < 0 (16)
Pentru studiul celorlalte dou# se poate utiliza criteriul Routh.Coloana de valori a c#ror semne trebuie analizate este:
a=160 (r-1) , a=41 , b=470-19r , c=160 (r-1) (17)
al c#ror semn este dat \n tabelul 1:
r 1 24.737
a + + + +
a + + + +
b + + 0 –
c + + + +
tabelul 1
Pentru r<24.737 toate r#d#cinile (complexe sau reale) ale ecua]iei caracteristice sunt situate \n semiplanul st@ng.
Pentru r =24.737, b= 0 fapt ce implic# prezen]a a dou# r#d#cini complexe pur imaginare,adic# apari]ia unei bifurca]ii. \n acest caz ,= 3.1269i.
Pentru r >24.737 r#d#cinile , se afl# \n semiplanul drept.
Pentru a analiza natura r#d#cinilor
= 512 r +288634 r +487839 r -1180320 (18)
care are semnul din tabelul 2:
r 1 1.352
- 0 +
tabelul 2
Astfel pentru r(1,1.352) exist# trei r#d#cini reale, situate \n semiplanul st@ng, iar pentru r(1.352,24737) dou# r#d#cini complexe [i una real# situate de asemenea \n semiplanul st@ng.
Pentru r=24.737 se produce o bifurca]ie (Re(\n semiplanul drept [i una \n semiplanul st@ng , cele din semiplanul drept fiind complexe.
_n concluzie pentru r(0,1) Xeste nod atractor.
Pentru r(1,1.352] intervin punctele de echilibru X,care se comport# ca noduri atractoare. \n func]ie de condi]iile ini]iale (puncte de start) traiectoriile vor converge fie la X
fie la X.
Pentru r(1.352,24737) X, se comport# ca [I un nod atractor-focar atractor , evolu]ia traiectoriilor fiind ca [i \n cazul precedent , dar spre deosebire de aceste spiralate.
Pentru r=24.737 , ,sunt pur imaginare ap#r@nd o bifurca]ie, fapt ce indic# apari]ia unui ciclu limit#. Astfel, oricare ar fi punctul de start traiectoria va \nconjura cele dou# puncte. Acest caz este foarte greu de prins din punct de vedere computa]ional.
Pentru r>24.737 r#d#cinile complexe trec \n semiplanul drept fapt ce determin# instabilitatea punctelor X 2,3 ciclu limit# degener@nd. Analiza computa]ional# indic# \n acest caz prezen]a haosului, traiectoriile depinz@nd sensibil de condi]iile ini]iale. Acest comportament are loc pentru r(24.737,148) dup# care printr-o nou# bifurca]ie se trece \ntr-un ciclu limit# , pentru r(148,166.07).
Peste aceast# valoare fenomenele se repet#.
_n fig.28 este prezentat atractorul Lorentz corespunz#tor anumitor valori ale lui a,b,c. De remarcat faptul – cunoscut deja- c# pornind de la orice punct de start (x,y,z) se ajunge fie la unul fie la cel#lalt din punctele atractorului.
Fig. 28 Atractorul Lorentz,trasat ca o form# \n spa]iul tridimensional. Traiectoriile orbitale reprezint# mi[carea \n timp a unui punct \n spa]iul tridimensional
1.7.6. Atractorul straniu Rossler
Atractorul lui Rossler, observat \n numeroase sisteme dinamice \ncep@nd cu curgerea fluidelor [i termin@nd cu reac]ii chimice, este unul din cele mai simple sisteme, ilustr@nd maxima lui Einstein care spunea c# sofisticata natur# prefer# sistemele simple.
Ecua]iile diferen]iale ce definesc acest sistem sunt:
x=-z-y
y=x+ay
z=b+z (x-c)
Neliniaritatea generatoare de haos este dat# de termenul zx.
\n continuare s# ne reamintim c@teva din propriet#]ile atractorilor stranii. Un atractor straniu este \n spa]iul fazelor, o traiectorie ce se spiraleaz# la infinit f#r# a se intersecta. Datorit# extremei neregularit#]i [i a asem#n#rii unui detaliu al aglomer#rii traiectoriei cu \ntregul, avem de-a face cu un fractal. De asemenea o sec]iune transversal# prin traiectorie este mul]imea Cantor.
O alt# caracteristic# esen]ial# [i pe baza c#reia se poate defini un atractor straniu este dependen]a de condi]iile ini]iale. Atfel un atractor straniu amplific# \n mod exponen]ial erorile.
Un atractor straniu se ob]ine prin integrarea numeric#(rezolvarea aproximativ#) a sistemului de ecua]ii diferen]iale ce caracterizeaz# sistemul respectiv. Pentru aceasta este nevoie de determinarea condi]iilor ini]iale cu ajutorul c#rora se va putea apoi determina traiectoria pas cu pas. Datorit# \ns# impreciziei aparatelor de m#sur# ceea ce m#sur#m pentru condi]iile ini]iale ale unui sistem reprezint# doar aproxim#ri. Din aceast# cauz# nu vom putea determina cu precizie traiectoria real# adev#rat# din spa]iul fazelor a unui sistem, iar din cauz# c# atractorul straniu amplific# erorile dintre condi]iile ini]iale m#surate de noi [i cele adev#rate este evident c# un sistem ca [i clima de exemplu, dup# o perioad# de timp va evolua pe o traiectorie diferit# de cea real#. Orice predictibilitate pe o perioad# mai mare de timp este deci imposibil#.
O alt# caracteristic# a atractorilor stranii este aceea c# aceste curbe sunt m#rginite \n spa]iul fazelor, deci a nu se confunda cu un sistem instabil, care evadeaz# \n spa]iul fazelor la infinit dup# o posibil# spiralare. \n plus volumul \n care este inclus atractorul straniu tinde la zero pentru t tinz@nd la infinit, datorit# teoremei divergen]ei. Din aceast# cauz# sistemele care posed# atractori stranii sunt sisteme disipative, adic# sisteme care \]i mic[oreaz# volumul [i pierd energie.
Revenind la sistemul Rossler, o construc]ie intuitiv# a atractorului straniu Rossler poate porni de la banda de traiectorii din figura de mai jos, \n care distan]a dintre dou# puncte pe parcursul evolu]iei atractorului se m#re[te. Dar un atractor straniu este m#rginit deci aceast# band# trebuie s# o pliem. Dup# c@teva astfel de plieri s-ar ob]ine imaginea din figura 29, care este atractorul straniu Rossler.
Fig.29. Atractorul straniu Rossler
1.8. Automatele celulare
Stephem Wolfram spunea:” Calculul automat ofer# un mijloc nou de descriere [i investigare a sistemelor [tiin]ifice [i matematice. Simularea pe calculator poate fi singura cale de a prezice cum vor evolua anumite sisteme complexe”.
Definind un automat celular vom spune c# este un univers artificial guvernat de legi naturale simple. Se poate alege structura universului [i legile pe care le respect# [i automatul l#sat singur va urma aceste legi. Oricare dintre noi poate simula propriul s#u univers [i urm#ri cum evolueaz# program@nd aceste legi pe un calculator.
Aceste universuri automate sunt celulare pentru c# sunt de obicei alc#tuite din multe celule identice care se divid [i se multiplic# \ntr-un mod similar cu celulele reale. De exemplu, automatele originale ale lui Lindenmayer erau alc#tuite din litere scrise. Un univers simplu poate con]ine dou# tipuri de celule F [i G. Acest univers ar putea impune reguli de propagare:
Regula 1 : O celul# F se divide \n dou# celule F,FF.
Regula 2 : O celul# G devine GFG.
Pentru a pune \n func]iune acest automat celular se poate \ncepe, de exemplu, cu o singur# celul# G.
La \nceput : G
Prima genera]ie : GFG
A 2-a genera]ie : GFGFFGFG
A 3-a genera]ie : GFGFFGFGFFFFGFGFFGFG
Literele sunt dup# aceea interpretate grafic pentru a trasa o form# pe ecranul calculatorului. De exemplu, F se poate traduce prin “deseneaz# o dreapt# cu lungimea de 1m” [i G poate \nsemna “avanseaz# cu 1m f#r# a trasa o linie”. Literele F [i G din exemplul precedent ar ap#rea pe ecran ca o succesiune de linii [i spa]ii goale, asem#n#toare codului Morse. Ad#ug@nd litere care s# \nsemne “rote[te cu 30 grade la stanga” sau “rote[te cu 60 grade la dreapta”, Lindenmayer a putut reprezenta forme complexe.
Dragonul lui Heighway este de asemea un automat celular grafic. Aici,”universul” este format din “celule” geometrice, care apar ca linii drepte. Exist# doar o singur# lege de propagare:fiecare dreapt# se divide \n dou# drepte copil, cu lungimea egal# cu m#rimea catetelor unui triunghi isoscel ce are segmentul de dreapt# ini]ial ca ipotenuz#. Prin repetarea algoritmului de generare se ob]in curbe care se aseam#n# cu un dragon.
Mai t@rziu, Lindenmayer le-a recunoscut pe acesta [i pe alte proceduri de construc]ie geometric#, ca fiind tipuri speciale de automate celulare. \n acel moment, ecranele calculatoarelor erau nesatisf#c#toare pentru reprezentarea formelor geometrice complexe, dar, \ncep@nd de atunci, automatele celulare computerizate au fost [i sunt folosite pentru a crea multe imagini fractale spectaculoase.
Ideea de automat celular dateaz# de pe vremea lui John von Neumann (\nceputul anilor ’50). Neumann a reu[it s# imagineze un automat autorepreduc#tor care a fost verificat pe calculator cu mul]i ani mai t@rziu. Acesta crea structuri de celule care se reproduceau respect@nd cele mai riguroase reguli ale biologiei.
Modelele de automate celulare se folosesc ast#zi nu numai \n biologie ci [i \n alte domenii. Astfel se simuleaz# structuri de celule cristaline ale fulgilor de z#pad#, structuri de materiale feromagnetice, structuri ale galaxiei, conductibilitatea electric#, reac]ia chimic# Belusov Zhabotinsky. \n general structurile generate au form# fractal#.
Dac# consider#m o re]ea rectangular# de n X m celule (pixeli sau caractere) care la fiecare b#taie de gong a unui orologiu universal \[i schimb# starea dup# un anumit algoritm, devenind celule vii sau moarte, avem de-a face, a[a cum am v#zut, cu un automat celular.
Acest automat, imaginat de Fredkin \naintea renumitului automat celular al lui Conway (numit [i universul lui Conway) evolueaz# \n maniera urm#toare:la un moment dat, se examineaz# starea tuturor celulelor re]elei; dac# num#rul celulelor vii, vecine unor celule ]int# este par, atunci celula ]int# va muri, \n caz contrar, va \nvia. Se poate considera \n calcul [i starea celulei ]int#. Automatul va fi lansat \n lucru pe o re]ea ini]ial# aleas# dup# inspira]ie.
Alegerea celulelor vecine se poate face dup# numeroase reguli. \n figura 30 sunt prezentate c@teva modalit#]i de alegere a vecin#t#]ilor. De asemenea celula ]int# \[i poate modifica pozi]ia oscil@nd dup# diferite direc]ii. Se pot concepe [i automate unidirec]ionale sau chiar tridimensionale \n func]ie de regulile vecin#t#]ii.
vecin#tatea vecin#tatea
von Neumann Moore
celule ]int#
Fig.30 Automate celulare
Automatele unidirec]ionale lucreaz# doar dup# o direc]ie. Astfel o re]ea de celule ale unui astfel de automat este o \n[iruire de celule. Fiec#rei celule i se aplic# o regul# de supravie]uire ce depinde de vecini, s# le spunem “st@nga” [i “dreapta”.
O proprietate foarte interesant# a automatului lui Fredkin este aceea c# dup# un anumit timp, motivul ini]ial va reapare \n patru copii, apoi \n [aisprezece, [.a.m.d.
Exist# dou# modalit#]i de explorare a st#rilor celulelor. Prima metod#, simpl#, const# \n analizarea st#rii pixelilor (aprin[i sau stin[i) direct pe ecran, aceste situa]ii coinciz@nd cu starea celulelor(vii sau moarte). Aceast# modalitate nu este \ns# suficient de riguroas#, deoarece \n analiza unei celule ]int# o parte din vecini au o stare nou# ( la momentul t > t ) \n timp ce cealalt# parte este la starea veche (din momentul t). Cu toate acestea rezultatele sunt foarte spectaculoase.
Varianta corect# este cea \n care se memoreaz# starea tuturor celulelor \ntr-o matrice. Matricea va fi reactualizat# la sf@r[itul fiec#rei baleeri a re]elei. Analiza re]elei se va face \n acest mod nu pe ecran ci pe matrice.
Deoarece memorarea unui ecran \ntreg \ntr-o astfel de matrice este greu realizabil# din motive de memorie, este recomandabil# memorarea doar a r@ndurilor de celule vecine celulei ]int#.
Dac# \n cazul memor#rii \ntregii re]ele, reactualizarea ei se face dup# explorarea \n \ntregime a re]elei, \n varianta simplificat#, memorarea se face pe parcursul analiz#rii celulelor.
O problem# destul de nepl#cut# o constituie marginile re]elei. Teoretic re]eua trebuie s# fie nem#rginit#. Pentru a simula aceasta, putem racorda laturile (st@nga cu dreapta [i sus cu jos) ob]in@nd o suprafa]# grilat# \nchis# infinit#, ca [i la jocuri.
Simplu, putem face analiza f#r# s# ]inem cont de margini, \ns# dup# un anumit timp, imaginea se va deteriora.
P@n# acum am vorbit doar de dou# st#ri posibile pentru fiecare celul#, dar se pot imagina algoritmi care s# ]in# cont de mai multe st#ri pentru fiecare celul#.
Unele automate celulare ca [i cel al lui Fredkin sunt reversibile (\n orice moment pornind \n sens invers de la o configura]ie ob]inut#, d#m de configura]ia ini]ial#). Automatul lui Conway nu este reversibil.
\n fine, pentru a \n]elege rolul inspira]iei \n crearea acestor automate trebuie amintit faptul c# dac# num#rul st#rilor posibile ale unei celule este k [i, num#rul total al vecinilor este atunci exist# k reguli posibile pentru schimbarea evolu]iei! Astfel pentru automatul binar al lui Fredkin unde se consider# patru vecini putem avea 4 = 4.29*10 reguli.
“Jocul vie]ii” al lui Conway, \n care punctele dintr-o re]ea tr#iesc sau mor \n func]ie de statutul punctelor \nvecinate, a r#pit probabil un num#r mai mare de ore de calcul dec@t orice alt automat celular grafic. A fost proiectat \n 1970 de matematicianul britanic John Conway [i a devenit o obsesie pentru John von Neumann, cel care, dup# cum [tim, a proiectat [i calculatorul numeric.
Multe forme de via]# r#sar \n universul lui Conway. Unele sunt stabile, unele alunec# u]or trec@nd prin dou# sau trei configura]ii, iar altele dispar rapid. \n cele din urm#, \ntotdeauna se termin# fie prin c@teva forme de via]# invariabile, fie printr-un de[ert arid.
Nu este u[or s# se evalueze \n c@t timp se va stabiliza un scenariu anume; oricum o celul# suplimentar# plasat# la locul potrivit poate duce la supravie]uirea pe termen lung sau la moartea instantanee a \ntregului grup.
Cu alte cuvinte, stabilitatea acestui univers artificial este foarte sensibil# la condi]iile ini]iale. Fiecare celul# dreptunghiular# aflat# \n re]eaua universului lui Conway se poate aprinde (tr#ie[te) sau stinge (moare).Regulile sunt:
Dac# o celul# moart# are trei celule vecine vii (din cele opt celule care o \nconjoar#), ea revine la via]#.
Dac# o celul# vie are dou# sau trei celule vecine vii, ea supravie]uie[te.Altfel, ea moare.
1.9 Aplica]ii curente ale fractalilor [i haosului \n economie
1.9.1 Introducere
Benoit Mandelbrot [i-a \ntemeiat geometria fractal# baz@ndu-se \n principal pe simularea sa \ncununat# de succes a tendin]ei pre]urilor bunurilor de consum, iar analiza pie]ii r#m@ne una din cele mai atr#g#toare aplica]ii ale geometriei fractale.
Piatra filosofal# a oric#rui analist al pie]ii, este desigur, s# prezic# comportarea pre]urilor cu destul# exactitate pentru a se umple de bani c@t mai repede. Dac# cineva a pus m@na pe aceast# "piatr#" probabil c#-[i folose[te c@teva miliarde ale sale pentru a-[i ap#ra secretul. \n domeniul pie]ii, ca [i \n alte domenii \n care fractalii [i haosul dau rezultate, rareori se dovedesc at@t de folositori pentru prezicere, pe c@t sunt pentru simulare. De exemplu simularile din 1953 ale lui Mandelbrot asupra pre]ului bumbacului continu# s# prezic# cu exactitate cantitatea de varia]ie din pre]ul bumbacului at@t lunar# c@t [i anual#. Totu[i, ele nici m#car nu pot pretinde c# ne indic# pre]ul bumbacului dini luna iulie 1995.
Una din revela]iile majore ale analizei fractale este memoria sau persisten]a pe pia]#. Modelele economice tradi]ionale de obicei iau \n considerare un consumator sau un investitor care tr#ie[te totalmente \n prezent. Aceast# persoan# idealizat# ia decizii de a cump#ra [i de a vinde bazate numai pe pre]urile curente ale pie]ei [i pe dorin]a perfect ra]ional# de a ob]ine un profit maxim \n fiecare moment.
Analiza fractal# a ar#tat concludent c# orice model bazat pe asemenea ipoteze va fi totalmente inexact. Nu numai c# pia]a prezint# memorie at@t pe termen lung, c@t [i pe termen scurt dar este persistent# la orice scar# posibil#, de la ore la secole. Adev#ratul participant \n economie \[i aminte[te, a[a \nc@t c@]iva anali[ti de frunte ai investi]iilor folosesc sistemele func]iei iterate pentru a dezvolta un limbaj de modele astfel \nc@t s# poat# \n]elege schimb#rile indicilor economici. Ei descoper# c# anumite modele fractale apar deseori chiar \naintea schimb#rilor din indicii de pia]#. Model@nd aceste modele [i \nv#]@nd cum s# le interpreteze ei [lefuiesc o nou# lentil# matematic# pentru a privi datele pe pia]#.
O ipotez# de pia]# poate fi urm#toarea:
1. O pia]# este constituit# din mul]i investitori cu diferite orizonturi de investi]ii
Mul]imea de informa]ii care este important# pentru fiecare orizont de investi]ie este diferit#
At@ta timp c@t pia]a \[i men]ine o structur# fractal# f#r# caracteristici ale scalei de timp, va r#m\ne stabil#. Atunci c\nd orizonturile pe pia]a de investi]ii devin uniforme, pia]a devine instabil# deoarece to]i se bazeaz# pe aceea[i mul]ime de informa]ii.
Din moment ce unele serii de timp nu mai pot fi considerate aleatoare, ci ele sunt auto-similare [i invariante la scalarea timpului, atunci \n generarea lor pentru diverse simul#ri nu mai putem utiliza legea de distribu]ie Gauss. Generarea acestor tipuri de serii de timp persistente sau anti-persistente implic# generarea unor serii de timp cu dimensiune fractal# dat#.
1.9.2 Segmentarea pie]elor de capital
Segmentarea \n [i \ntre domenii reale ale pie]ii are un trecut \n literatura de specialitate. Totu[i recent are natura segment#rii \ntre domenii reale [i pie]e de capital.
Liu, Hartzell, Greig [i Grissom(1990)[9] reprezint# primul studiu, care examineaz# explicit partea de segmentare a pie]ii \ntre domenii reale [i pia]a de capital. Studiul lor scoate \n eviden]# faptul c# segmentarea exist#.
Ambrose, Ancel [i Grifitts [1] extinde analiza segment#rii \n pie]e mari de capital, examin@nd structura fractal# a domeniilor [i returnurilor ac]iunilor sau titlurilor de valoare. Ei testeaz# explicit neliniaritatea \n beneficiile investi]iilor din domeniile reale, de \ncredere. Testul de structuri fractale caut# neliniarit#]i deterministice \n serii de timp.
Ei \[i motiveaz# studiul f#cut prin urm#toarele:
Traiectoriile haotice pot ar#ta schimb#ri exacte calitative \n comportament, asem#n#toare \n general celor asociate cu disturb#ri aleatoare
Dac# timpii de fructificare al investi]iilor sunt dependen]i pe termen lung, atunci consumul optim, economiile [i portofoliile de investi]ii devin senzitive la orizontul de timp.
Modele ca CAPM (Capital Market Asset Pricing Model -fondat# pe ipoteze ale pie]ii eficiente – care este reprezentat# matematic ca asocierea seriilor de tip financiar unei legi de distribu]ie normale) [i APT devin invalide din moment ce exclud posibilitatea unor astfel de persisten]e neliniare.
Diferen]a \ntre pia]a real# [i alte pie]e de capital i-a condus pe Liu, Hartzel, Grissom la ipoteza c# pia]a real# poate fi caracterizat# ca o pia]# segmentat# moderat. Deoarece institu]iile care investesc sunt restric]ionate s# investeasc# \n cl#diri, care asigur# un cost de oportunitate al rentei.
P#rerea celor trei autori Ambrose, Ancel [i Grifitts este c# odat# g#site structuri fractale semnificante diferite, \n variate perioade de fructificare, putem trage concluzia c# exist# segmentare, din moment ce astfel de neliniarit#]i sunt cauzate de diferite structuri economice active. Alternativ dac# pia]a real#, mai precis beneficiile de pe pia]a real#, urm#resc o lege aleatoare, c@[tigurile prin segmentare (ca opus# diversific#rii) nu exist# deoarece un model consistent risc-rezultat trebuie aplicat.
Studiul f#cut de Ambrose, Ancel [i Griffits(1992) extinde analiza segment#rii pie]ii, prin compararea returnurilor din pia]a real# de investi]ii cu altele de pe alte pie]e de capital, folosind o metodologie alternat#.
O diferen]# fundamental# \ntre pia]a real# [i pie]ele de capital poate fi o diferen]# de structur# \n procesul de generare al ie[irilor [i rezult# din studiul f#cut de Liu, Hartzell, Greig [i Grissom(1990), care au dezvoltat o baz# teoretic#, ar#t\nd modele separate de evaluare. Modelele sunt desprinse din diferite clase de date financiare, \ntr-o pia]# segmentat# moderat. Astfel dac# pia]a real# este segmentat# de alte domenii, atunci putem s# ne a[tept#m ca ie[irile de pe pia]a real# s# aib# caracteristci diferite din alte domenii. Ne-am a[tepta ca returnurile dintr-un domeniu \n care nu exist# segmentare substan]ial# s# se comporte similar.
Ambrose, Ancel [i Griffits(1992) testeaz# o versiune specific# a ipotezei, compar@nd structurile fractale ale returnuriluo pe pia]a real# cu cea a returnurilor de pe pia]a de valori. Acest test este creat special pentru a detecta persisten]e ne-ciclice pe termen lung.Test@nd deci o form# de auto-similaritate sau auto-dependen]# \n general ne luat# \n cosiderare de analize conven]ionale empirice. Persisten]a neciclic# este examinat# folosind analiza domeniului rescalat adic# R/S.
Studiul lor folose[te serii de timp din domeniul „Real Estate Investment Trusts“(REITs) sau domeniul real al investi]iilor stabile, ca reprezent@nd returnuri de pe pia]a real#. Aceasta pentru a permite studiul pe serii de timp mai lungi [i care pot fi comparate direct cu indexul pre]urilor zilnice la bursa de valori. Folosirea seriilor REITs ca reprezent@nd domeniul real al pie]ei este demonstrat# a fi o bun# metod# de mul]i al]i autori.
Ambrose,Ancel [i Griffits(1992) calculeaz# returnurile zilnice din REIT ale portofoliilor de investi]ii [i testeaz# dou# tipuri de portofolii, de tip ipotec# [i mijloace fixe. Deoarece ei compar# indicii REIT cu indicii pondera]i ai pre]urilor investi]iilor (calcula]i la nivel macro, la Centrul de cercet#ri \n domeniul securit#]ii pe]urilor sau Center Research for Security Prices CRSP) [i indicii pre]urilor la bursa de valori, calculeaz# returnurile portofoliilor REIT folosind aceea[i metod# de ponderare folosit# pentru construirea indicelui pre]urilor. Portofoliile sunt rebalansate pentru fiecare tranzac]ie zilnic#. Returnul logaritmic al unui portofoliu, R(t) este calculat ca o medie ponderat# a returnurilor individuale de pe pia]a de \ncredere din portofoilul respectiv:
(1)
Unde:
vj(t)=pj(t)(t-1)•nj(t)(t-1) j=1,..,K (2)
[i pentru 1jK
rj=pj/pj(t-1) pentru REIT j
pj(t) pre]ul investi]iei j \n ziua t
nj(t) num#rul ac]iunilor disponibile pe pia]# \n ziua t pentru investi]ia j.
Compararea acestor tipuri de portofolii cu cele de la CRSP au dezavantajul c# la calcularea acestora din urm# sunt introdse [i returnurile REIT. Oricum contribu]ia acestora la formarea indexului este destul de mic# a.\ poate fi neglijat#. Dar compararea cu indicele pre]urilor pe pia]a de valori nu con]ine nici un indice de tip REIT.
Concluziile acestui studiu sunt urm#toarele:
Pre]urile titlurilor de valoare pe pia]#, arat# un comporatament la analiza R/S consistent cu cel existent \n cazul unei serii aleatoare
Returnurile Portofoliilor arat# acelea[i caracteristici.De aceea teoria pie]ei de capital ar trebuii aplicat# domeniului real al economiei.
De[i portofliile de tip ipotec# [i de tip mijloace fixe au statistica H substan]ial diferit# de statistica unei serii aleatoare [i destul de mare fa]# de indicii pre]urilor pie]ei, aceste diferen]e par a fi generate de corela]ii pe termen scurt.
\n final bazat pe rezultatele de mai sus, nu au gasit nici o eviden]# c# ipoteza unei segment#ri a pie]ei pe termen lung este adev#rat#.
1.9.3 Alegerea portofoliului optim de investi]ii
Investitorii au tendin]a de a da foarte mare aten]ie situa]iei existente, dispropor]ion@nd a[tept#rile pe termen lung, presupun@nd c# starea existent# a afacerilor va continuua la infinit dac# nu exist# motive reale s# se cread# \ntr-o schimbare. De asemenea m#resc importan]a informa]iilor actuale, \n defavoarea celor din trecut.
Un exemplu semnificativ : consider@nd o societate comercial# care p@n# anul acesta a avut c@[tiguri similare cu media pe industrie, dar dintr-o dat# raporteaz# un salt foarte mare \n profit. Reac]ia investitorilor este de a extrapola \n viitor, baz@ndu-se pe extraprofitul din anul acesta [i \n consecin]# vor cre[te pre]urile. \n momentul \n care profitul din viitor se dovede[te a fi mai mic dec@t cel a[teptat, investitorii \[i recunosc gre[eala [i vor sc#dea din nou pre]urile.
Efectul „\nving#tor – \nvins“ este explicat cu ajutorul acestor teorii comportamentale. Chiar dac# investi]iile individuale conduc la alte evalu#ri ale perspectivelor de viitor, extrapolarea se face \n baza prezentului. Partea \nving#torului (odat# ce este declarat ca \nving#tor) nu este u[or de reclasificat, \n ciuda c@[tigurilor adversarilor. Aceasta se \nt@mpl# p@n# c@nd evenimente majore conduc la corectarea acestui crez, \n perspectivele de viitor. Pe de alt# parte, odat# ce sunt declara]i \nvin[ii, \mbun#ta]irile \n c@[tigurile \nregistrate sunt luate drept abera]ii sau devieri accidentale ale rezultatelor companiei \n care mediocritatea managementului este bine stabilit#.
Metoda de a investiga efectul \nving#tor \nvins este de a forma un portofoliu al \nving#torilor [i de a se testa serii de timp, ale c#ror observa]ii reprezint# returnurile \n timp ale companiilor, din punct de vedere statistic.
De[i strategia portofoliului apare ca profitabil# aceast# concluzie se schimb# atunci c@nd este luat \n considerare riscul. Procedura de estimare a riscului este nepotrivit# pentru acest studiu. Aici intervine faptul c# aceste serii de timp sunt auto-corelate iar riscul estimat ca o abatere standard este insuficient, deoarece exist# influen]e ale valorilor \nregistrate \n trecut asupra valorilor care se vor \nregistra \n prezent sau \n viitor.
Metoda de creare a portofolilui \nving#torilor se bazeaz# pe combinarea metodei de analiz# a domeniului rescalat cu cea a estim#rii abaterii standard, pentru serii de timp ce con]in profiturile \n timp ale companiilor.
Se poate face o \mp#r]ire \n func]ie de exponentul Hurst estimat cele care au un exponent H0.5 vor face parte din portofoliul \nving#torilor. Dac# exist# o rupere , \n sensul \n care se pot estima doi exponen]i atunci se poate concluziona c# la nivelul unei perioade d lungimea seriei componenta x va face parte din portofoliul \nving#torilor sau nu. Aceasta \n func]ie de orizontul de timp pe care se construie[te portofoliul. Dac# orizontul este mai mare dec@t d, atunci chiar dac# exponentul Hurst estimat pentru o invarian]# la nivelul unei scale mai mici dec@t d este favorabil s# nu fac# parte din acest portofoliu.
=== 3_95 ===
CAPITOLUL 3
IMAGINI : STANDARDE {I ALGORITMI
3.1 Introducere
_n orice prezentare multimedia, elementul imagine este aproape nelipsit, \ntruc@t impactul vizual este foarte puternic pentru om. Dupã cum se [tie, calitatea imaginilor, la vizualizare pe ecran, este condi]ionatã de rezolu]ia de afi[are [i de capacitã]ile grafice ale calculatorului [i monitorului. Imaginea d.p.d.v. al modului de memorare [i realizare poate fi \mpãr]itã \n :
imagine \n formã matricealã
imagine \n formã vectorialã
Ecranul monitorului este structurat ca un tablou de pixeli dispu[i pe linii [i pe coloane. Documenta]ia care \nso]e[te echipamentul prezint# num#rul de linii [i de coloane ale tabloului precum [i dimensiune \n milimetri a pixelului, v#zut ca un dreptunghi. Bibliotecile standard grafice sunt \nzestrate cu toate structurile de date [i func]iile care permit accesul la resursele necesare descrierii, realiz#rii [i manipul#rii imaginilor \n condi]iile unei relative independen]e de echipamentul fizic.
Un pixel este caracterizat prin pozi]ie(linie [i coloan# \n tablou) [i con]inut(culoare, eventual luminozitate). Tabloului liniarizat de pixeli \i corespunde \n memorie o zon# numit# memorie grafic#, care stocheaz# aceste informa]ii. Se poate lucra direct pe memoria grafic# pentru transformarea(ini]ializarea pixelilor) sau prin intermediul memoriei interne a calculatorului.
Dimensiunea memoriei grafice asigur# stocarea informa]iilor asociate cel pu]in unei imagini complete de pe ecranul monitorului. Pentru a asigura continuitatea derul#rii imaginilor asemeni lumii reale trebuie asigurat# o vitez# de transfer de la fi[ierele externe c#tre memoria grafic# sau efectuarea rapid# de calcule, ceea ce impune existen]a unor sisteme extrem de performante.
Monitoarele alb-negru func]ioneaz# cu 16 nuan]e de gri, caz \n care pentru descrierea unui pixel \n memoria grafic# se vor folosi 4 bi]i. Pentru monitoarele alb-negru cu 256 nuan]e de gri, codificarea pune \n coresponden]# 00h cu negru [i FFh cu alb. _ntre ele afl@ndu-se coduri pentru nuan]e de la gri-\nchis la gri-deschis. Pentru reprezentarea unui pixel \n acest caz este necesar 1 bait.
C@nd se reprezint# imagini color se define[te conceptul de palet# de culori, care ia \n considerare fie codurile asociate culorilor c@nd acestea sunt \ntr-un num#r restr@ns, fie modul de construire a culorilor prin combinarea culorilor fundamentale.
Monitoarele care suport# 32K sau 16M nuan]e impun descrierea paletei de culori prin asocierea fiec#rei culori fundamentale RGB \n 256 nuan]e fiecare.
3.2 Particularit#]I ale fi[ierelor de imagine
Fi[ierele de imagini con]in \n form# liniarizat# informa]ii despre elementele matricei de pixeli. Pornind de la una diin structurile de fi[ier imagine, prin efectuarea unor calcule se poate realiza pozi]ionarea pe un anumit bait, pentru extragerea de informa]ii despre un pixel dat.
De exemplu, pentru un monitor SVGA, c#ruia i se asociaz# o matrice de 640×480, extragerea informa]iilor referitoare la pixelul de pe coloana 101 [i linia 207 dintr-un fi[ier BitMap presupune apelarea, \ntr-un program C, a func]iei fseek cu indicarea pozi]iei:
P(i,j)=( i*640 + j ) * 1 + H
unde:
i – linia;
j – coloana;
H – lungimea headerului imaginii BitMap.
Pentru cazul \n care nu se cunosc dimensiunile BitMap-ului:
Bitmap x;
…
Pentru un fi[ier GIF determinarea informa]iilor referitoare la un pixel se face prin determinarea imaginii cadru din care face parte acel pixel [i pozi]tionarea \n cadrul acelei imagini.
Spre deosebire de fi[ierele de date, fi[ierele de imagini sunt liniariz#ri ale matricelor de pixeli care redau aspecte vizuale ale lumii reale. Petele de culoare genereaz# prin liniarizare sub[iruri de elemente identice. Aceast# particularitate este exploatat# masiv de algoritmii de compresie imagine.
Ineficien]a algoritmilor apare atunci c@nd imaginile sunt construite din puncte izolate. Se poate demonstra ineficien]a acestor algoritmi atunci c@nd celor n=x.biWidth*x.biHeight pixeli le corespund tot at@tea nuan]e distincte de culori. Aceast# situa]ie este cunoscut# sub denumirea de problema mozaicului perfect, folosit# pentru testarea algoritmilor de compresie.
Se define[te:
unde fi reprezint# num#rul de elemente ale unui sub[ir de pixeli cu aceea[i culoare.
Dac# s-a reu[it crearea pe ecranul calculatorului a unui mozaic perfect, atunci :
f1=f2=…..=fn=1
atunci =0
_n cazul \n care suprafa]a ecranului con]ine pixeli de aceea[i culoare =n*log2n, ceea ce reprezint# valoarea maxim# a gradului de omogenitate a imaginii pe ecran.
Se define[te:
ca fiind gradul de omogenitate al imaginii.
_ntre gradul de omogenitate al imaginii [i gradul de compresie exist# o leg#tur# str@ns#. M#rimea indicelui de corela]ie a acestora pe loturi semnificative de fi[iere de imagine, permit analiza stabilit#]ii algoritmilor de compresie de imagine.
Imaginea apare sub form# de pete de culoare. Vom numi pixeli adiacen]i de aceea[i culoare perechile de pixeli (pij,pij+1),( pij,pi+1j),( pij,pi+1j+1),( pij,pi-1j+1).
Dac# pk reprezint# num#rul de pixeli de aceea[i culoare, care formeaz# pata de ordinul k din imagine.Se calculeaz#:
Se poate ar#ta c# 0<g<g’<1
_n cazul anima]iei cadrele care se succed(24 cadre/s) con]in numeroase elemente comune.
Se consider# un cadru de baz# notat C0. Cadrele notate C-k se numesc cadre precedente iar cele nnotate Ck se numesc cadre succesoare.
Se define[te [irul de cadre C-k…..C-1 C0 C1…..Ck [ir caracterizat prin 2k+1 cadre.
Se introduce indicatorul de invarian]# dintre cadrele k [i t:
Unde: n – num#rul de coloane;
m – num#rul de linii.
Se construie[te matricea D cu 2k+1 linii [i 2k+1 coloane, matrice a invarian]ei cadrelor.
Algoritmii de compresie manipuleaz# \ntotdeauna matrici D3x3 ceea ce arat# c# rezultatul experien]elor impune \n practic# o astfel de abordare.
Se poate scrie un program care s# lucreze cu invarian]ii la nivel de pete de culoare, fiind posibil# modificarea lungimii [irului de cadre, eventual optimiz@nd lungimea [irului \n raport cu un criteriu care influen]eaz# gradul de compresie global#.
3.3 Algoritmul RLE(Run Length Encoding)
_n compresia de date sunt folosi]I mul]I algoritmi, de la algoritmi simpli p@n# la algoritmi complec[i. Eficacitatea algoritmilor variaz# \n func]ie de natura datelor. Cel mai simplu algoritm folosit ast#zi pe scar# larg# este RLE.
Principiul RLE const# \n detectarea unui simbol av@nd un num#r de apari]ii coonsecutive care dep#[esc un num#r fixat. Apoi \nlocuirea acestei secven]e prin dou# informa]ii:
o cifr# indic@nd num#rul de repeti]ii;
simbolul ce se repet#.
Limita minim# pentru a putea fi comprimat un [ir, este ca lungimea sub[irului s#fie de minim 3 simboluri identice.
De exemplu, fiind dat [irul :
aaaabbbaabbbbbccccccccdabcbaaabbbbcccd
consider@nd caracterul special * vom avea urm#toarea reprezentare:
4a3baa*5b*8cdabcb*3a*4b*3cd
C@nd \n fi[ierul ce urmeaz# a fi compresat cu algoritmul RLE, se \nt@lne[te simbolul folosit drept caracter special, se va trece \n fi[ierul compresat urm#toarea succesiune:
caracter special;
valoarea 0, deoarece pe aceast# pozi]ie \ntr-o codificare normal# ap#rea lungimea [irului;
valoarea 1 sau caracterul special.
_n cazul special \n care se g#se[te o succesiune a caracterului special acesta se va compresa astfel:
caracter special;
valoarea 1, fiind o valoare nefolosit# \n procesul de compresie;
num#rul de repeti]ii al caracterului special, \ntre 2 [i 255.
Algoritmul RLE este un caz special, oper@nd \n realizarea de transform#ri pe text. Compresia la nivel de simbol este nul#.
Gradul cel mai mare de compresie se \nregistreaz# c@nd textul ini]ial este de lungime foarte mare [i are un singur simbol.
Gradul de compresie este nul atunci c@nd textul de lungime L simboluri este format din tot at@tea caractere sau L/3 succesiuni de simboluri diferite.
Acest algoritm prezint# un avantaj mai mare \n cazul fi[ierelor de imagine. _n fi[ierele de imagine se \nt@lnesc foarte des succesiuni de simboluri cu aceea[i valoare, pixeli succesivi cu aceea[i culoare.
Pentru un ecran monocrom, alfabetul este format din dou# simboluri. Elementele dintr-o astfel de imagine pot fi : 1 pentru un pixel alb, sau 0 pentru un pixel negru.
De exemplu, [irul:
000000000011111111000001111
poate fi compresat astfel:
10 8 5 4
_n cazul acestor fi[iere este folosit# o variant# modificat# a algoritmului. _n fi[ier vor ap#rea urm#toarele informa]ii:
pixelul cu care \ncepe succesiunea de cifre;
num#rul de apari]ii consecutive ale acestui pixel;
alternativ num#rul de apari]ii pentru fiecare culoare.
Pentru cazul \n care se \nt@lnesc mai mult de 255 de apari]ii consecutive ale aceluia[i simbol se va reprezenta dup# cum urmeaz#:
255 ;
0 ;
num#rul de apari]ii 255.
De exemplu, pentru [irul:
………. 0000 ……. 000000011111 …….
unde 0 apare de 300 de ori, vom avea compresia:
………. 255 0 45 3 ……
C@nd se prelucreaz# imagini color, algoritmul nu suport# diferen]e fa]# de reprezentarea standard.
3.4 Formatul graphic BMP
În ultimii ani formatul BMP a devenit un standard important ca urmare a faptului cã este utilizat atât sub OS/2 cât și sub Windows. Pentru mulți dintre noi, acest format este destul de străin având în vedere că îl cunoastem mai mult dupã nume si nu neapãrat dupã structurã. De fapt dupã cum e lesne de înțeles, ne gãsim în fața unui adevãrat folclor – existã o serie de specificații de formate grafice care utilizeazã același nume. Sub sistemul de operare OS/2 ca dealtfel și sub Windows, fișierele de format BMP se folosesc pentru a stoca imagini, icon-uri, cursoare, pointeri și poate lista nu este completã. În continuare ne vom focaliza atenția asupra formatului BMP pentru imagine.
Existã douã versiuni ale formatului BMP sub Windows și douã sub OS/2. Putem sã le considerãm ca fiind de fapt versiunea veche și versiunea nouã pentru fiecare din cele douã platforme. Cele douã versiuni vechi sunt identice, lucru explicabil ca urmare a originii comune a lui OS/2 și Windows. Versiunile așa-zise noi ale formatului BMP sunt diferite.
3.4.1 Structura BMP
Formatul BMP este de uzanță generală astfel încât permite stocarea, într-un fișier pe disc, a unei imagini de orice dimensiune. Informațiile despre culoare pot să varieze de la 1 bit la 24 de biți. Până acum nu exista nici un fișier având extensia BMP care să foloseascã vreo tehnica de compresie. În ceea ce privește modul de stocare, imaginea este înregistratã linie cu linie pornind de jos în sus.
În fiecare din cele trei variante specificate, un fisier BMP conține, respectând ordinea apariției, următoarele părți: file header, bitmap info header, opțional paleta de culori și în final datele imaginii.
Variantele formatului BMP apar distincte în domeniile bitmap info header și în cel al paletei de culori. Sub Windows 3.1, formatul vechi rezervã pentru paletã
3 octeți * NumărCulori
de unde și denumirea secțiunii fisierului – RGBTRIPLE. În noua versiune avem 4 octeți rezervați pentru o culoare – RGBQUAD. Ambele versiuni, sub platforma Windows, includ structurile BITMAPFILEHEADER, BITMAPINFOHEADER, descrise în continuare.
Intrãrile în tabela ce reprezintã paleta de culori, în cazul versiunii vechi, sunt constituite din trei octeți care indicã componentele de roșu, verde și albastru. În cadrul versiunilor noi ale formatului BMP se adaugã un al patrulea octet și ca urmare, o intrare în cadrul paletei de culori se poate citi ca și un longinteger. În cazul în care ne referim la liniile ce reprezintã imaginea, pe lângã faptul cã ele sunt înregistrate începând cu linia de jos, existã un al doilea aspect care privește faptul cã aceste linii sunt umplute cu octeți pozitionați pe zero astfel încât să ocupe un număr par de cuvinte duble.
3.4.2 Componentele structurii BMP
În general vorbind, câmpurile nefolosite sau care nu sunt importante pot fi setate pe zero în toate situațiile. În cadrul enumerării următoare am respectat denumirile câmpurilor așa cum apar ele în documentația originală.
Câmpurile structurii denumită BITMAPFILEHEADER sunt:
bfType este un câmp folosit pentru validarea fișierului în ceea ce privește faptul cã este o imagine BMP. Ocupã 2 octeți. Pentru un fișier BMP imagine acest câmp conține totdeauna valoarea "BM"; provenind de la prescurtarea “BitMap”;. Windows nu suportă alte valori pentru acest câmp pe când OS/2 permite existența a sase valori. Aceste șase valori indică practic ce informații conține fișierul: icon color, icon mono, pointer mouse color și așa mai departe.
bfSize indicã mãrimea fișierului și are rezervat un spațiu de patru octeți. În cadrul unor instante mai vechi ale formatului BMP, acest câmp indică valoarea zero respectiv atunci când este setat poate indica lungimea fișierului în octeți sau în cuvinte (acest lucru depinde de programul care a scris informațiile în fișier). Astfel stând lucrurile, nu trebuie să ne bazăm niciodată pe valoarea indicatã în acest câmp. Interpretarea corectã ar fi lungimea fisierului în octeți.
bfReserved1, bfReserved2 sunt douã câmpuri rezervate, ocupând fiecare câte doi octeți.
bfOffBits indicã offsetul, în octeți, de la începutul fișierului la începutul bitmap-ului.Valoarea stocatã în acest câmp este foarte importantã deoarece astfel apare și posibilitatea calculului numãrului de intrãri în cadrul paletei cu ajutorul formulei:
numãrculori = (OffBits – file_hdr_size – bitmap_hdr_size) / rgb_size.
În continuare va fi prezentată descrierea câmpurilor ce apar ca și componente ale
structurii denumite BITMAPINFOHEADER:
biSize ocupã patru octeți și indicã mãrimea header-ului în octeți. Aceastã valoare mai este utilizatã pentru a determina ce versiune de format avem în fațã. Existã câteva valori consacrate în ceea ce privește semnificația lor. Astfel dacã valoarea câmpului în discuție este 12 atunci avem de-a face cu versiunea de format mai veche (versiunile vechi sub Windows și OS/2 sunt identice). În cazul în care valoarea câmpului este 40 atunci înseamnă că este vorba despre noul format pentru Windows. Dacã valoarea este 64 atunci este vorba despre noua versiune sub OS/2. Acum mai apare o problemã în cazul în care avem o valoare cuprinsã în intervalul 12 – 64. De obicei putem spune cã avem de-a face cu o versiune nouã de format BMP dar nu este o afirmatie adevãratã în orice context. Multe aplicații verifică dacă valoarea în discuție este 40, în caz contrar apare un mesaj de eroare care ne spune că fișierul nu este de tip BMP. După cum se poate observa aceastã strategie de lucru nu este dintre cele mai corecte.
biWidth ocupã patru octeți și reprezintă lătimea imaginii în pixeli. Această valoare nu reflectã și cazul în care este necesarã "umplerea liniilor"; pentru ca spațiul ocupat sã fie multiplu de 32.
biHeight ocupã patru octeți și se referã la înãlțimea imaginii în pixeli.
biPlanes ocupã doi octeți și indicã numãrul de plane de culoare din fișierul bitmap. Câmpul este setat de obicei pe 1.
biBitCount ocupã doi octeți și reprezintã numãrul de biți folosiți pentru stocarea informației despre un pixel (se mai numește și adâncime). Valorile pentru acest câmp sunt în general 1, 4, 8 și 24.
biCompression este o valoare, stocatã pe patru octeți, ce indicã tipul compresiei folosite în cadrul fișierului respectiv. Valoarea zero indicã de fapt cã nu avem de-a face cu vreo compresie și pare sã fie cam singura valoare care apare aici.
biSizeImage este mãrimea în octeți a bitmap-ului (ocupã patru octeti). Unele aplicatii se bazeazã pe o valoare corectã în acest câmp.
biXPelsPerMeter ocupã patru octeți și se referã la mãrimea pe orizontalã a fișierului în pixeli pe metru.
biYPelsPerMeter ocupã patru octeți și se referã la mãrimea fisierului pe verticalã în pixeli pe metru.
biClrUsed ocupã patru octeti și indicã numãrul de culori folosite.
biClrImportant ocupã patru octeti și indicã numãrul culorilor importante folosite.
În cazul în care imaginile utilizeazã 4 sau 8 octeti pe pixel trebuie sã avem o paletã de culori. Numãrul valid de culori cuprinse în aceastã paleta este 16 și respectiv 256 dar pânã la urmã ele pot fi mai mici dacã imaginea nu face uz de toate culorile posibile. În cazul în care se folosesc 24 de biți pe pixel avem de-a face cu o imagine full-color care nu mai necesitã o paletã de culori.
Formal grafic BMP este răspândit, dar este foarte rar folosit, dat fiind faptul că imaginea este stocată aici fără a fi comprimată, ducând astfel la o mărime considerabilă a fișierului. Reprezentarea imaginilor în memorie este foarte asemănătoare cu formatul BMP. De altfel fișierul în formatul BMP este de fapt o copie pe disc ale imaginii din memorie, astfel că fișierele pot fi afișate foarte rapid pe monitor.
3.5 Formatul grafic GIF
Creație a firmei CompuServe Inc., formatul grafic GIF (Graphic Interchange Format) este folosit în special pentru a stoca mai multe imagini bitmap în același fișier. După multitudinea de imagini GIF aflate pe BBS (sau pe alte site-uri publice) se poate spune că acesta este cel mai utilizat format pentru stocarea imaginilor.
Formal GIF a fost creat ca un protocol de comunicații pentru imagini interactive. El folosește pentru comprimarea datelor algoritmul LZW (Lempel Ziv Welsh), neștiind să stocheze date nearhivate sau arhivate cu un alt algoritm.
O altă caracteristică a formatului GIF este că, în partea de date, el codifică liniiile într-un mod diferit. Dacă în modul obișnuit imaginea bitmap inițiala este codificată în mod continuu, linie după linie, în formatul GIF liniile codificate sunt întrețesute printr-un algoritm în patru pași. În primul pas pornește cu rândul 0 și continuă cu rândul 8. Al doilea pas este identic cu primul cu diferența că el pornește cu rândul 4. Pasul al treilea începe cu rândul 2 și continuă tot cu al patrulea rând. Astfel în primii trei pași s-au codificat rândurile pare. La al patrulea se vor codifica și rândurile impare.
Un utilizator care ar dori să vadă o imagine pe un server la care este conectat prin modem, ar vedea imaginea stocată în mod obișnuit apărând de sus în jos. După ce 50% din imagine ar fi afișată, el ar putea vedea doar partea de sus a imaginii, pe când dacă se folosește tehnica întrețeserii, aceeași jumătate îi poate da utilizatorului o idee despre ceea ce conține imaginea.
3.5.1 Organizarea fișierului GIF
Formatul grafic GIF organizează fișierul în niște pachete de date însoțite de informații adiționale numite blocuri. Anumite blocuri de date pot fi situate în orice parte a fișierului, de aceea fișierele GIF trebuie citite ca și cum ar fi o mulțime de șiruri de date.
Un bloc de date poate conține și sub-blocuri. Un sub-bloc începe întotdeauna cu un contor care are valoarea cuprinsă ntre 1 și 255, și care ne indica numărul de octeți din sub-bloc. Mai multe sub-blocuri pot apărea în grupuri continue terminate de un contor care are valoarea zero. (De menționat că nu există blocuri formate numai din contor, fără date). Rezultatul e de tipul: contor, date, contor, date, contor=0.
Formatul GIF este capabil să stocheze doar imagini în care valoarea unui pixel este stocată pe maxim 8 biți. Întotdeauna într-un fișier GIF vom găsi o paletă de culori, imaginea fiind stocată folosind modelul RGB. Deși un fișier GIF poate stoca mai multe imagini în același timp, această opțiune nu e foarte folosită, astfel că majoritatea programelor care folosesc formatul GIF știu să citească doar prima imagine din fișier.
Exista două tipuri de fișiere GIF, sau mai bine spus două versiuni ale formatului GIF. Acestea sunt GIF87a și GIF89a. Primul este formatul original, iar cel de-al doilea aduce câteva îmbunătățiri (cum ar fi de exemplu posibilitatea de a stoca text și grafică în același fișier).
3.5.2 Versiunea GIF87a
Această versiune a apărut pentru prima dată în mai 1987 și este recunoscută de majoritatea programelor de grafică ce lucrează cu GIF. Fiecare fișier începe cu un Antet urmat de informatii despre Ecran și CuloriPeScurt. Matricea Globală de Culori este opțională. Toate aceste trei secțiuni se află în același offset în oricare din fișierele GIF87a. Fiecare imagine stocată în fișier este alcătuită dintr-un AntetLocal urmat de un MatriceLocalăDeCulori. Fișierul se termină cu un caracter terminator care ne indică sfârșitul fișierului.
Antetul are lungimea de numai 6 bytes și este folosit pentru a identifica un fișier de tipul GIF. Primii trei bytes conțin șirul “GIF”, iar următorii trei versiunea fișierului (“89a” sau “87a”). Unele programe care citesc fișierele “87a” pot să vizualizeze și fișiere “89a” deși imaginea obținută nu va fi corectă.
Secțiunea Ecran și Culori pe Scurt conține informații care descriu ecranul și culorile folosite la crearea fișierului GIF. LatimeEcran si InaltimeEcran contin rezoluția minimă necesară pentru ca imaginea GIF să fie afișată. Dacă placa grafică a monitorului nu suportă o asemenea rezoluție, atunci va trebui ca imaginile să fie scalate pentru ca afișarea lor să fie corectă.
CampImpachetat conține următoarele subcâmpuri de date:
Primii trei biți indică numărul de biți necesari pentru o culoare din Matricea Globală de Culori minus unu. De exemplu, dacă o imagine are 8 biți pe pixel atunci valoarea găsită aici va fi 7 (“111” în reprezentarea binară). Astfel mărimea maximă a matricii globale de culori este 256, formatul GIF neputând înregistra mai multe culori. Acest câmp are valoarea corectă a imaginii, chiar dacă nu există Matricea Globala.
Dacă Indicatorul de Sortare al Culorilor este poziționat, atunci elementele din Matricea Globală de culori sunt sortate în funcție de importanța lor, adică după frecvența apariției lor în imagine. Culoarea cu cea mai mare frecvență de apariție o vom regăsi în primul element din tablou. Această sortare a culorilor este utilă în cazul în care vrem să afișăm imaginea pe un monitor care suportă mai puține culori decât are imaginea. Acest sub-câmp poate fi regăsit doar la versiunea “89a” a fișierului, în versiunea “87a” fiind rezervat.
Rezoluția unei culori conține numărul de biți necesari pentru o culoare în paleta originală minus unu.
Indicatorul de prezență a Matricii Globale de culori e poziționat dacă această matrice globală există. Dacă nu există, atunci în fișier urmeaza antetul local pentru prima imagine.
IndexFond conține un index în Matricea Globală de culori și indică culoarea folosită pentru marginea și fondul ecranului. Dacă nu avem un tabel global de culori acest câmp are valoarea 0.
AspectRatio ne oferă raportul dimensional al pixelilor care alcătuiesc imaginea. Raportul dimensional se obține prin împărțirea lățimii unui pixel la înălțimea lui.
MatriceaGlobalaDeCulori poate fi prezentă sau nu. Dacă nu există, atunci fiecare imagine din fișier va conține o matrice locală, ceea ce mărește mult dimensiunile fișierului.
Fiecare culoare din matrice conține un triplet de trei octeți, fiecare din ei reprezentând o valoare corespunzătoare culorilor de bază roșu, verde, albastru (n această ordine). Numărul de elemente din acest tabel este o putere a lui doi: 2,4,8,16,32,64,128,256.
Aceste 3 secțiuni conțin date globale despre imaginile stocate în fișier. Ele sunt urmate de una sau mai multe imagini. Fiecare imagine are un Antet local urmat eventual de o Matrice locală de culori.
Antetul local apare înaintea fiecarei imagini din fișier și are următoarea structură:
Separatorul are valoarea 2Ch și este identificatorul pentru un început de imagine.
Stânga și sus sunt coordonatele în pixeli ale colțului stânga-sus al imaginii pe ecranul în care se face afișarea. Coordonatele 0,0 corespund colțului stânga-sus al ecranului.
LatimeImagine și ÎnaltimeImagine conțin mărimea imaginii în pixeli.
CmpÎmpachetat este format din următoarele subcâmpuri:
Bitul 0: Indicator de prezență al Matricii Locale de culori
Bitul 1: Indicator al imaginii întrețesute
Bitul 2: Indicator de sortare a culorilor
Biții 3-4: Rezervați
Biții 5-7: Rezervați
Matricea Locală de culori : are aceeași structură ca și Matricea Globală, singura diferența constând în aria în care are efect acest tabel. Astfel, dacă Matricea Globală se referea la toate imaginile din fișier, MatriceaLocală se referă doar la imaginea din care face parte tabelul. Pentru această imagine, Matricea Locală are prioritate în fața celei globale.
Imaginea : fișierele GIF folosesc algoritmul de compresie LZW. În mod normal o imagine comprimată cu LZW este aranjată sub forma unui pachet continuu de date care va fi citit de la un capăt la celălalt. Totuși, în fișier imaginea se regăsește sub forma unor sub-blocuri de date. Fiecare sub-bloc începe cu un contor ce are valori cuprinse între 1 și 255, și indică numărul de octeți din sub-bloc. Acești octeți urmează imediat după contor.
Formatul imaginii necodificate este extrem de simplu. Fiecare octet se va referi la un pixel și va conține un index către una din matricele de culori. Deși structura fișierului GIF ar permite stocarea culorilor direct în imaginea comprimată, specificațiile GIF nu indică că acest lucru ar fi posibil. Oricum, chiar și pentru o imagine doar în alb și negru, cu un pixel pe un bit, trebuie să folosească un index pe 8 biți într-o matrice cu 2 culori.
Remorca : este un octet ce apare ca un ultim octet în fișier. Valoarea lui este 3Bh și indică sfârșitul pachetelor de date din fișierul GIF.
3.5.3 Versiunea GIF89a
Este asemănătoare cu GIF87a, cu deosebirea că conține câteva blocuri de control în plus care nu existau în specificațiile 87a. Aceste blocuri de control sunt: Blocul de Control 89a, Blocul de Text, Blocul Comentariu si Blocul Aplicație. Cu excepția câtorva restricții, aceste blocuri pot apărea oriunde în fișier după Matricea Globală de culori și fără a fi limitate ca număr.
Toate blocurile încep cu un Indicator ce are valoarea 21h. Acest indicator e urmat de o Etichetă care identifică tipul blocului. Toate aceste blocuri sunt opționale și pot să nu apară în fișier. De fapt orice fișier 89a ce nu folosește aceste facilități poate fi etichetat ca fiind de tipul 87a.
Blocul de control 89a
Acest bloc controlează cum va fi afișată imaginea sau textul ce urmează după acest bloc. Informația pe care o conține ne spune dacă acestea vor fi afișate în mod transparent sau opac peste imaginea existentă deja pe ecran, dacă imaginea existentă va fi ștearsă sau nu, și dacă se va aștepta intervenția utilizatorului pentru a se afișa imaginea următoare.
Structura acestui bloc este:
Cmp Bytes Explicații
Indicator 1 Conține valoarea 21h
Eticheta 1 Conține valoarea F9h
MărimeBloc 1 Conține valoarea 04h
CmpÎmpachetat 1 Conține informații grafice
TimpDeAșteptare 2 Timpul de așteptare în sute de secunde.
IndexTransparent 1 Index pentru culoarea transparentă
Terminator 1 Întotdeauna are valoarea 0
CmpulÎmpachetat este alcătuit astfel:
Bitul 0: IndicatorTransparent Dacă e poziționat pe 1 atunci IndexTransparent
va conține o valoare validă pentru culoarea transparentă
Bitul 1: IndicatorulUtilizatorului Dacă e poziționat pe 1 atunci înainte de afișarea
imaginii următoare se va aștepta un semnal din partea utilizatorului.
Biții 2-4: MetodaDeAfisare are următoarele valori:
0 – Nu se folosește nici o metodă.
1 – Peste imaginea anterioară
2 – Se șterge imaginea anterioară cu culoarea de fond
3 – Sub imaginea anterioară
Biții 5-7: Rezervați
Dacă se folosește simultan și IndicatorulUtilizatorului și TimpDeAsteptare, atunci afișarea imaginii următoare se va face la intervenția utilizatorului sau la expirarea timpului de așteptare.
Blocul de Text
Multe imagini bitmap sunt însoțite adesea de comentarii introduse de utilizatori. Deși textul din acest bloc nu aparține imaginii, el poate fi combinat cu imaginea la afișare. Ba chiar, în formatul 89a, există posibilitatea ca fișierul să poată fi format doar din texte.
Pentru a afișa textul din acest bloc, trebuie construit pe ecran un grilaj imaginar format din linii orizontale și verticale, iar în fiecare celulă să se plaseze câte o literă. Culorile de fond și cea de desenare a textului sunt luate din Matricea Globală de culori. Structura acestui bloc este urmatoarea:
Câmp Bytes Explicații
Indicator 1 Conține valoarea 21h
Eticheta 1 Conține valoarea 01h
MarimeBloc 1 Conține valoarea 0Ch
StângaText 2 Colțul stânga-sus al textului
SusText 2
LatimeText 2 Lățimea textului
ÎnaltimeText 2 Înălțimea textului
LatimeCaracter 2 Lățimea unui caracter
ÎnaltimeCaracter 2 Înălțimea unui caracter
CuloareText 1 Index la culoarea textului
CuloareFont 1 Index la culoarea fontului
Text pointer Indicator la sublocuri de text
Terminator 1 Întotdeauna are valoarea 0
Text este un indicator la sub-blocurile de text ce conțin textul de afișat, structurat pe sublocuri identic ca la imagine, dar nearhivat.
Blocul Aplicație
Permite stocarea de informații întelese doar de programele care citesc fișiere GIF. Aceste date pot fi informații adiționale folosite doar pentru a afișa imaginile sau pentru a coordona modul de combinare a lor cu alte imagini din alte fișiere. Datele cuprinse aici includ instrucțiuni de schimbare a modului grafic, procesarea imaginilor sau memorarea de culori ce nu sunt prezente în alte blocuri. Se mai pot include informații referitoare la modem-uri, imprimante, etc.
Structura este următoarea:
Câmp Bytes Explicații
Indicator 1 Conține valoarea 21h
Eticheta 1 Conține valoarea FFh
MarimeBloc 1 Conține valoarea 0Bh
Identificator 8 Șir ce conține identificatorul aplicației
CodIdentificare 3 Șir ce conține o serie, o versiune, un cod ASCII
care identifică aplicația
Aplicatia 1 Conține codul propriu-zis al aplicației
Terminator 1 Conține întotdeauna valoarea 0
Blocul Comentariu
Textul cuprins aici nu apare în imagine, dar poate fi vizualizat cu un editor cum ar fi de exemplu Norton Editor. Acest bloc cuprinde următoarele câmpuri:
Câmp Bytes Explicații
Indicator 1 Conține valoarea 21h
Eticheta 1 Conține valoarea FEh
Comentarii 1 Indicator la sub-blocurile de comentarii
Terminator 1 Conține întotdeauna valoarea 0
Formatul grafic GIF rămâne unul dintre cele mai populare formate folosite pentru stocarea fișierelor imagine. Avantajele sale majore sunt portabilitatea pe un mare număr de sisteme de operare și faptul că folosește întrețeserea liniilor ( lucru util mai ales în transmiterea fișierelor imagine la mare distanță). Dezanvantajul principal al formatului GIF este vechimea sa (ultima versiune datează din 1989). Chiar și așa fișiere GIF găsim pe majoritatea calculatoarelor.
3.6 Formatul grafic PCX
Unul din cele mai utilizate formate grafice este formatul PCX. Este cunoscut de mulți utilizatori ca și formatul PC Paintbrush, deoarece el a fost dezvoltat de firma ZSoft odată cu programul de desenare PC Paintbrush (versiunea MS-DOS). Mai târziu a fost preluat de Microsoft odată cu introducerea lui Paintbrush în kitul Windows. El a mai fost folosit și de către producători de FAX-uri pentru codificarea imaginilor trimise pe linia telefonică.
3.6.1 Organizarea fișierului PCX
Fișierele din formatul PCX folosesc stocarea de tip bitmap a datelor imaginii la
care se aplică algoritmul de compresie RLE (Run Length Encoding).
Un fișier PCX este organizat în trei mari secțiuni: un Antet cu datele referitoare la
imaginea propriu-zisă,o paletă de culori (opțional) și secțiunea de date.
ANTETUL ocupă primii 128 de bytes din fișier. Structura lui conține următoarele
câmpuri:
Identificatorul este o valoare definită de specificațiile PCX, care identifică fișierul PCX și are întotdeauna valoarea 10h.
Versiunea ne dă structura exactă a secțiunii de date, structură ce poate varia în funcție de vechimea programului ce a creat imaginea. În funcție de valoarea acestui câmp se pot identifica urmatoarele versiuni:
Valoare Versiunea PCX Descriere
0 versiunea 2.5 Paletă EGA fixată
2 versiunea 2.8 Paletă EGA modificabilă
3 versiunea 2.8 Fără informații despre paletă
4 PaintBrush for Windows
versiunea 3.0 PaintBrush Plus,
PaintBrush for Windows (Imagini pe 24 biți)
Numarul de biți per plan pentru fiecare pixel are ca valori posibile 1, 2, 4 sau 8, corespunzător la 2, 4, 16 sau 256 de culori.
Valorile Xmin, Ymin, Xmax, Ymax ce reprezintă colțul stânga-sus și respectiv colțul dreapta-jos. Colțul din stânga-sus al ecranului are valorile 0,0. Colțul dreapta-jos poate avea cel mult 65535,65535 (65535×65535 este dimensiunea maximă a unei imagini PCX).
Valorile Orizontal și Vertical ne oferă rezoluția ecranului la care a fost creată imaginea. De exemplu, dacă imaginea a fost creată pe un monitor VGA cu rezoluția 1280×1024, atunci Orizontal=1280 și Vertical=1024.
EGAPalette este un vector de 48 de octeți, în fiecare octet se specifică nuanțele culorilor în cazul adaptoarelor grafice EGA.
Numărul de Plane – este o variabilă care specifică numărul de plane folosite la afișarea imaginii. Valoarea lui este 1, 3 sau 4 și se folosește împreună cu BițiPixelPlan pentru a determina modul grafic cel mai potrivit pentru afișarea imaginii. Avem următoarele combinații posibile:
Modurile video
PaletteInfo poate avea valorile 1 (pentru imagine monocromă) și 2 (pentru o imagine cu nuanțe de gri). Această variabilă ne spune dacă trebuie să afișăm imaginea alb-negru sau color.
MărimeOEcran și MărimeVEcran sunt variabile ce au fost introduse odată cu versiunea 4.0 a PaintBrush-ului și contin rezoluția ecranului sub care a fost creată imaginea.
Bytes Rezervați completează lungimea antetului până la 128 de octeți și au fost introduși pentru dezvoltări ulterioare.
Paleta de culori
Primele versiuni ale fișierului PCX nu aveau incluse o paletă de culori, deoarece la acea vreme monitoarele puteau afișa un număr limitat de culori. Ulterior în fișiere a fost inclusă o asemenea paletă care pentru monitorul EGA este de forma unui masiv cu 48 de elemente: 16 tripleți corespunzători culorilor roșu, verde și albastru. Elementele 0,1,2 din masivul EGAPalette sunt valorile corespunzătoare pentru culoarea 0, elementele 3,4,5 pentru culoarea numărul 1, etc. Elementele acestui masiv au valori cuprinse între 0 și 255. Dar cum placa grafică EGA acceptă numai culori cuprinse între 0 și 3, este nevoie de un calcul al valorilor care se vor scrie în memorie. În acest sens se vor deplasa valorile la stânga cu 6 biți.
Paleta VGA este alcătuită din 256 de culori. Deci, pentru înregistrarea paletei în fișier vom avea nevoie de 256×3=768 de culori. Antetul având doar 128 de octeți, a fost nevoie ca paleta VGA să fie mutată la sfârșitul fișierului. Dacă dorim să aflăm dacă este sau nu o paletă VGA stocată în fisier, ne poziționăm cu 769 de octeți înainte de sfârșitul fișierului. La aceasta poziție ar trebui să se afle un indicator pentru paleta VGA și a cărui valoare este întotdeauna C0h. Dacă nu întâlnim această valoare atunci fișierul nu conține o paletă VGA. Un alt element de recunoaștere a paletei VGA este versiunea, care are valoarea 5 în cazul prezenței paletei. Confuzia poate apărea în două cazuri: dacă se folosesc culori pe 24 de biți, caz în care nu mai avem o paletă VGA (culorile fiind înscrise direct în sectiunea de date), și dacă înaintea sfârșitului de fișier avem valoarea C0h în mod pur întâmplător, fără ca acest lucru să indice prezența paletei VGA.
Secțiunea de date
După antetul fișierului, urmează datele propriu-zise ale imaginii. Imaginea este stocată în format bitmap și este comprimată cu algoritmul RLE (Run Length Encoding). Acest algoritm este relativ simplu: O secvență mai lungă de caractere identice este codificată cu doi bytes. Primul indică numărul de repetiții, iar cel de-al doilea indică caracterul de repetat. De exemplu, dacă avem șirul “AAAmmmmmHHHHHHH” acesta se va reprezenta comprimat astfel: “3A5m7H”. Dacă șirul este format din caractere nerepetitive, atunci acestea vor fi stocate pe un byte. Pentru a face diferența între un byte și o secvență de 2 bytes, se folosesc drept flag primii doi biți mai semnificativi ai primului byte citit.
O linie este stocată în fișierul PCX în două feluri: orientată pixel sau orientată plan. Dacă este orientată pixel atunci, pixelii sunt înregistrați unul după altul. În celălalt caz avem linia desfacută pe componentele lor din fiecare plan, adică fiecare plan în parte este văzut ca formând o singură imagine.
Imaginile care au 4 plane sunt cele cu 16 culori utilizate pentru imagini EGA. Procedura de codificare este următoarea: o linie de imagine este descompusă în patru linii de informație, fiecare linie corespunzând unui plan, iar o linie-plan este văzută ca făcând parte dintr-o imagine monocromă.
3.6.2 Formatul DCX
Formatul PCX este capabil de a stoca doar o singură imagine la un moment dat. Aplicațiile care au nevoie de mai multe imagini, ce aparțin aceluiași grup, pun de multe ori un nume asemănător fișierelor PCX pentru a le identifica. Pentru a înlătura acest inconvenient s-a creat formatul DCX, capabil să stocheze mai multe fișiere PCX. Ele au forma unor containere în care se pot depune pâna la 1023 de imagini PCX. Antetul fișierului PCX arată astfel:
Identificator: Variabilă de tip Longint care contine valoarea 3ADE68B1h
PageTable: Vector maxim 1024 de componente de tip Longint și conține poziția fiecărei imagini PCX în cadrul fișierului.
Formatul PCX este tot atât de răspândit precum este Windows deoarece este formatul folosit de programul de desenare PaintBrush, ce e inclus în kitul Windows. Și chiar dacă el nu este atât de folosit, majoritatea programelor ce lucrează cu imagini îl recunosc.
3.7 Formatul TIFF
Pentru a putea stoca imaginile alb-negru, provenite de la scaner, în anul 1986, Corporația Aldus introduce un nou format grafic, numit Tag Image File Format sau mai pe scurt formatul TIFF. Acesta pornea pe piață cu versiunea 3.0. De-a lungul anilor, la această versiune s-a mai adăugat și posibilitatea de a stoca imagini color RGB, CMYK și YCbCr rezultând astfel versiunea 6.0.
Formatul Tiff se poate regăsi pe diverse platforme ca MS-DOS, MacIntosh, UNIX. El stochează imaginile în format bitmap și folosește până la 24 de biți pentru un pixel. Imaginea poate fi stocată nearhivat sau arhivată cu diverși algoritmi ca RLE, CCITT Group 3 și Group 4, JPEG. Totodată el poate stoca și mai multe imagini într-un fișier.
Organizarea fișierului
Fișierul TIFF este organizat pe trei mari secțiuni: Antetul, Directorul Imaginii și Zona de Date în care sunt stocate imaginile bitmap. Din aceste trei secțiuni avem nevoie doar de primele două, existând posibilitatea ca Zona de Date să lipsească. Un fișier TIFF care conține mai multe imagini are câte o secțiune. Directorul Imagini pentru fiecare imagine stocată.
TIFF are reputația de a fi un format complicat datorită faptului că în interiorul fișierului numai Antetul poate fi găsit la o locație fixă, restul secțiunilor variind de la fișier la fișier. Fiecare Director al Imaginii împreună cu bitmap-ul asociat formează un subfișier. Nu există o limită a numărului de subfișiere pe care le poate contine un fișier TIFF.
Puteți observa mai jos cele trei moduri posibile în care poate fi structurat un fișier TIFF:
În fiecare dintre exemple antetul se găseste la începutul fișierului. În primul caz Directorii Imaginior apar înaintea bitmap-urilor ceea ce face ca acest mod de aranjare să fie mai ușor de citit. În al doilea caz fiecare Director al Imaginii apare înaintea bitmap-ului făcând structura fisierului comodă atât pentru cel care creează fișierul cât și pentru cel care îl citește. Ultimul caz este mai neobișnuit și apare în cazuri mai rare.
Antetul
În ciuda complexității fișierului TIFF, antetul este foarte simplu. El conține 3 câmpuri și are lungimea de 8 octeți:
Ordinea 2 bytes {Ordinea octeților}
Versiunea 2 bytes {Numărul versiunii TIFF}
Primul_DI LongInt {Locația primului Director al Imaginii din fișier}
Ordinea indică dacă datele din fișier sunt scrise cu modelul little-endian ( formatul INTEL – valoarea 4949h ) sau cu big-endian ( formatul Motorola – valoarea 4D4Dh ). Au fost alese aceste valori deoarece ele sunt la fel pe orice tip de mașină, indiferent de ordinea de citire.
Versiunea are întotdeauna valoarea 2Ah și este mai mult un identificator al fișierului decât o versiune propriu-zisă. Ordinea împreună cu Versiunea pot fi considerate și ca indicatori de recunoastere ai unui fișier TIFF.
Directorul Imaginii
Un Director al Imaginii este o colecție de informații folosite pentru a descrie imaginea bitmap care urmează. Poate părea puțin cam confuz deoarece Directorul Imaginii poate apărea oriunde în fișier, însă el este pus într-o listă simplu înlănțuită, împreună cu ceilalți Directori, listă spre care pointează Primul_ID din Antet. Câmpul din ultimul Director al Imaginii indică spre valoarea 00h.
Spre deosebire de structurile întâlnite până acum în această serie de articole, structura Directorului Imaginii nu este fixă, ea depinzând de tipul imaginii căruia îi servește ca și antet. În această structură se pot adăuga sau scoate noi câmpuri, exact ca și paginile dintr-un dosar.
Directorul Imaginii are următorul format:
Câmpuri 2 bytes {Numărul de câmpuri din structură}
CâmpuriTIFF Vector de câmpuri
Următorul_DI LongInt {Locația următorului Director al Imagini din fișier}
Mărimea tabelului CâmpuriTIFF este necunoscută la începutul rulării unui program de afișare. Următorul_DI conține locația în fișier a următorului Director al Imaginii. Dacă nu mai există o altă imagine în fișier atunci acesta va contine valoarea 00h.
Câmpurile
Flexibilitatea crescută a fișierelor TIFF își plătește prețul în cazul câmpurilor, și aceasta deoarece fiecare câmp trebuie să conțină pentru un octet de informație alți 12 octeți pentru alte specificații. Un câmp are următoarea structură de 4 subcâmpuri totalizând 12 octeți:
Identificator 2 bytes {Identificatorul câmpului}
Tipul_Datelor 2 bytes {Tipul obiectelor din câmp}
Contor LongInt {Numărul de obiecte din câmp}
Locatia_Obiectului:LongInt {Locația obiectelor în fișier}
Identificator conține o valoare numerică care identifică tipul de informații pe care le conține câmpul. De obicei un câmp conține informații legate de mărimea imaginii bitmap asociată Directorului Imaginii, mărimea unui pixel, algoritmul de comprimare ș.a.m.d. Câmpurile din Directorul Imaginii trebuie scrise în fișier în ordinea crescătoare a Identificatorului.
Tipul_Datelor contine o valoare care indică tipul obiectelor din câmp. Valorile pe care le poate lua acest subcâmp sunt prezentate în tabelul următor:
Tipul ascii corespunde datelor pe 7 biți, iar tipul rational este format din 2 date de tip long, dintre care primul este numărătorul unei fracții, iar al doilea numitorul.
Locatia_Obiectului conține locul în care se află obiectele în fișier. Dacă mărimea informațiilor din obiecte este mai mică de 4 octeți atunci acest subcâmp va contine în loc de locație, chiar informatiile propriu-zise.
Zona de Date
Fișierele TIFF conțin numai imagini de tip bitmap, deși cu ajutorul câmpurilor nu este exclusă nici stocarea unor imagini vectoriale, sau a unor texte.
Bucățile
Întâlnite și în alte formate grafice sub diferite denumiri, bucățile sunt niște colecții de rânduri consecutive din imaginea bitmap. Pentru a defini o bucată avem nevoie de trei câmpuri, definite în sectiunea anterioară. Acestea sunt RowsPerStrip ( RânduriPerBucată – 278 ), StripOffsets (LocatiaBucătii – 273) și StripByteCounts ( MărimeaBucății – 279).
Primul câmp RowsPerStrip indică numărul de rânduri consecutive din imaginea bitmap, conținute de o bucată. Valoarea implicită a acestui câmp este : 231-1 adică mărimea maximă pe care o poate avea o imagine stocată. Toate bucățile dintr-un subfișier trebuie să folosească aceeași schemă de compresie,iar numărul lor poate fi calculat cu formula:
Nr_Bucăți=[(Înălțimea_imaginii*(RowsPerStrip-1))/RowsPerStrip]
unde prin [x] am luat în considerare partea întreagă din x.
Deoarece putem avea mai multe bucăți, logic ar trebui să avem și mai multe câmpuri StripOffets, dar acest lucru nu se întâmplă deoarece câmpul fiind organizat sub forma unei matrici, fiecare articol este folosit pentru a indica spre o bucată . Dimensiunea tabloului este dată de numărul de plane din care este compusă imaginea ( PlanarConfiguration – 284 ), fiecare coloană reprezentând un plan. Prin acest câmp formatul TIFF împrumută ceva și din formatul PCX, organizat pe plane, cât și din formatul GIF, neorganizat pe plane.
Tot sub forma unei matrici este organizat și ultimul câmp, StripByteCounts. Acesta conține mărimea fiecărei bucăți în octeți după ce aceasta a fost arhivată.
Singura bucată care face excepție de la aceste reguli este ultima. Astfel ea este caracterizată doar de ultimele două câmpuri, StripOffsets si StripByteCounts.
Organizarea pe bucăți a fost introdusă datorită faptului că nu toate calculatoarele care lucrează cu fișierul TIFF au posibilitatea să citească o imagine întreagă în memorie pentru a opera cu ea. Astfel dacă memoria calculatorului nu permite citirea întregii imagini la un moment dat, cu ajutorul celor trei câmpuri descrise mai sus, aceasta poate fi citită și prelucrată pe bucăți. În schimb, dacă bucățile sunt atât de mari încât ele nu încap în bufferul definit de programul de citire, atunci s-ar putea ca citirea imaginii din fișier să eșueze.
Cărămizile
Odată cu apariția versiunii 6.0 a formatului TIFF s-a introdus un nou concept, și anume acela de cărămidă. O cărămidă este de fapt o bucată dreptunghiulară din imaginea bitmap, și a fost introdusă datorită imaginilor TIFF foarte mari, la care chiar și o bucată de un singur rând creează probleme programului de citire, care poate să nu fie pregătit să primească o informație așa de mare. Toată imaginea bitmap este împărțită astfel în cărămizi, care potrivite, una lângă cealaltă ne dau la sfârșit imaginea întreagă.
Mulți algoritmi de compresie, cum ar fi JPEG, pot comprima imaginea bitmap mult mai bine dacă aceasta este organizată în plan, ca în cazul cărămizilor, față de organizarea pe rânduri cum este prezentă în cazul bucăților și cum am mai întâlnit-o și la celelalte formate grafice.
Cele trei câmpuri întâlnite la bucăți sunt înlocuite acum cu altele patru:
TileWidth ( LățimeaCărămizii – 322 ),
TileLength ( LungimeaCărămizii – 323 ),
TileOffsets ( LocațiaCărămizii – 324 ) și
TileByteCounts ( MărimeaCărămizii – 325 ).
Ultimele două au aceeași semnificatie ca și corespondentele lor de la bucăți.
TileWidth și TileLength descriu mărimea cărămizii, și ele trebuie să conțină valori multiplii de 16. Toate cărămizile dintr-un subfișier trebuie să aibă aceleași mărimi, și este indicat ca mărimea imaginii bitmap necomprimată să fie de cel puțin 4K.
Compresia
Probabil că formatul TIFF suportă cei mai mulți algoritmi de compresie. Versiunea 4.0 are incluși numai algoritmii Run Length Encoding, întâlnit la formatul PCX, și CCITT T.4 și T.6. Versiunea 5.0 adaugă la acestea și algoritmul LZW, întâlnit și la formatul GIF, iar versiunea 6.0 adoptă și algoritmul JPEG, algoritm care se descurcă de minune cu împărțirea fisierului în cărămizi.
Astăzi TIFF este unul dintre multele formate grafice standard cu care lucrează majoritatea programelor de desenare și tehnoredactare. Posibilitatea de a stoca imagini bitmap de orice tip, face din TIFF formatul ideal pentru stocarea imaginilor grafice.
Deși este foarte cunoscut datorită flexibilității și puterii de compresie, el este considerat în egală măsură destul de complicat și misterios în același timp. Aceasta deoarece formatul TIFF este extensibil și cu multe caracteristici de care un programator ar putea avea nevoie atunci cînd lucrează cu imagini grafice. Din această cauză el este considerat și ca cel mai confuz format grafic. Acest lucru determină ca multe dintre programele care folosesc formatul TIFF să aibă probleme la citirea fișierelor datorită greșelilor apărute atât în liniile din programul care citește fisierul cât și din programul care a creat fișierul.
3.8 Standardul JPEG
JPEG are rezultate foarte bune de compresie pentru imaginile color [i cele \n nuan]e de gri, \ns# are probleme la lucrul cu imaginile alb-negru. Sunt \nt@lnite probleme [i la lucrul cu imaginile unde culorile au fost mapate(folosirea culorilor implicite ale mediului de lucru; de exemplu folosirea paletei Windows), trebuind ca anterior s# fie expandate \ntr-o reprezentare normal#.
JPEG lucreaz# bine pe imaginile constituite din tonuri continue de culoare. Imaginile care nu se compreseaz# bine sunt cele cu salturi dese ale valorii culorilor.
Exist# mul]i parametrii care influen]eaz# compresia JPEG, \n func]ie de valorile acestora put@ndu-se stabili gradul de compresie al imaginii. Compresia variaz# de la 2:1 p@n# la 100:1 \n cazul aceleia[i imagini. Desigur gradul de compresie are o influen]# foarte mare asupra calit#]ii imaginii. _n cazul unei compresii cu pierdere de informa]ie, de 100:1, calitatea imaginii las# de dorit. _ns# pentru o compresie de 2:1, imaginea compresat# este identic# cu imaginea original#.
Specifica]iile JPEG definesc patru moduri de arhivare a imaginilor, dup# cum urmeaz#:
1) Imaginea este arhivat# dupã modelul st@nga-dreapta, sus-jos, \nt@lnit [i la PCX-uri;
2) Arhivarea folose[te tehnica \ntre]eserii, \nt@lnitã la formatul GIF;
3) O metod# care este mai pu]in folosit# este aceea a arhiv#rii imaginii f#r# pierderi de informa]ie;
4) Se arhiveaz# mai multe copii ale imaginii originale \n acela[i fi[ier, la rezolu]ii diferite, metoda fiind eficientã \n cazul \n care se dore[te afi[area imaginii pe monitoare a c#ror rezolu]ie este sau prea mare, sau prea mic#, pentru o imagine arhivat# la o singurã rezolu]ie.
Formatul JPEG folose[te patru algoritmi de compresie, compresia av@nd rezultate diferite pentru fiecare. Sunt defini]i urm#torii algoritmi:
- compresie secven]ial#;
- codificarea progresiv#;
- compresia f#r# pierdere de informa]ie;
- codificarea ierarhic#.
Majoritatea echipamentelor hardware [i a produselor software implementeaz# numai algoritmul secven]ial de compresie.
3.8.1 Modul secven]ial
Pasul 1: se transform# imaginea din punct de vedere cromatic. Imaginea este transformat# \ntr-un sistem de coordonate tridimensional. Se acord# o importan]# mai mare luminozit#]ii, aceasta fiind considerat# o prim# ax# a compresiei. Celelalte dou# axe sunt folosite pentru codificarea culorii.
Motivul pentru care luminozitatea este considerat# variabil# principal# este pierderea de informa]ie, care la nivel cromatic este mai pu]in sesizat# de ochiul uman dec@t cea la nivelul luminozit#]ii. Aceast# schimbare nu este neap#rat necesar# deoarece algoritmul de compresie ac]ioneaz# asupra oric#rei componente de culoare independent. Oricum rata de compresie va fi mai mic# din moment ce vor trebui codificate toate componentele care contribuie la calitatea luminozit#]ii.
La transform#rile de culoare se ob]in pierderi ca urmare a rotunjirii, dar cantitatea de eroare indus# \n aceast# etap# este cu mult mai mic# dec@t eroarea de la pa[ii urm#tori.
Pasul 2(op]ional): se relizeaz# medii ale grupurilor de pixeli, grupuri denumite [i componente de e[antionare.
Componenta de luminozitate este l#sat# la rezolu]ia ei original#, \n timp ce componenta cromatic# este deseori redus# la 2:1 pe orizontal# [i 2:1 sau 1:1 pe vertical#. Vorbind \n termeni JPEG, vom \nt@lni alternative de e[antionare numite: 2h1v sau 2h2v, dar tot la fel pot fi \nt@lnite [i alternativele 411 [i 422.
La acest pas volumul de date se reduce cu 1/2 sau 1/3 din lungimea ini]ial#. _n termeni numericiieste o pierdere mare, dar pentru cele mai multe acest lucru aproape c# nici nu se observ#, datorit# capacit#]ilor sc#zute ale ochiului uman relativ la informa]ia cromatic#.
De notat c# e[antionarea nu se poate aplica imaginilor \n nuan]e de gri, de aici provenind [i rata de compresie mai mare pentru imaginile color.
Pasul 3: gruparea valorilor pixelilor pentru orice c@mp \n blocuri de 8×8 pixeli cu ajutorul func]iei DCT(Discrete Cosine Transform), o transformat# Fourier, av@nd ca rezultat o hart# a frecven]elor, cu 8×8 componente.
Se reprezint# numeric valoarea medie a fiec#rui grup de 8×8 urmat# succesiv de cele mai mari frecven]e din cadrul blocului p@n# la cele mai mici. Motivul pentru care se face aceast# transformare este c# acum se poate reduce informa]ia de frecven]# mare f#r# a influen]a frecven]ele sc#zute.
Pasul 4: pentru fiecare bloc, se ob]ine un coeficient de cuantificare DC, dup# care rezultatul se rotunje[te la \ntreg. Acesta este pasul fundamental de pierdere a informa]iei. Cu c@t un coeficient DC este mai mare, cu at@t vom avea mai multe pierderi de date. Chiar dac# se ob]ine valoarea minim# pentru DC, adic# 1, tot se va pierde informa]ie deoarece rezultatul transformatei DCT nu este un \ntreg \n mod normal.
Frecven]ele mari sunt cuantificate mai imprecis(ob]in@ndu-se valori mari ale coeficien]ilor DCT), dec@t cele mai mici, din moment ce ele sunt mai pu]in vizibile ochiului.
Datele referitoare la luminozitate sunt cuantificate mai exact dec@t informa]iile cromatice, prin folosirea unor tabele de cuantificare separate. Majoritatea programelor de compresie JPEG folosesc o simpl# scalare liniar# a acestor tabele, conform formatului JPEG standard, folosind doar un factor de calitate introdus de utilizator, pentru a determina multiplicatorul necesar scal#rii.
Pasul 5: dup# cuantificarea coeficien]ilor DC se trece la compresia [irurilor de date folosind algoritmul Huffman, compresia aritmetic# fiind o extensie de optimizare.
_n acest pas nu se pierde informa]ie. Multe implement#ri suport# numai \ns# numai modul Huffman, compresia aritmetic# folosind Q-coding, care este patentat.
Modul de compresie aritmetic# ofer# o compresie cu 5-10% mai bun# dec@t algoritmul Huffman standard folosit.
Pasul 6: \ntr-un fi[ier JPEG to]i parametrii de compresie sunt inclu[i \n headere, astfel \nc@t decompresia s# poat# fi f#cut# rapid, f#r# parcurgerea \ntregului fi[ier.
Printre ace[ti parametri se includ tabelele de compresie ale algoritmului Huffman. Pentru aplica]ii specializate standardul permite eliminarea acestor tabele din fi[ier, acest lucru salv@nd destul de mult# memorie. Acestea pot fi considerate tabele globale la nivelul aplica]iei.
Algoritmul de decompresie este inversul acestui proces. Decompresorul multiplic# coeficien]ii redu[i din tabela de cuantificare pentru a reproduce coeficien]ii DC. Din moment ce toate acestea sunt valori aproximative, valoarea pixelului reconstituit este de asemenea aproximativ#, dar dac# tehnica a fost cea men]ionat#, aceste erori vor fi greu vizibile. Un decompresor performant adaug# ni[te pa[i de netezire pentru a reduce discontinuit#]ile dintre pixeli.
Standardul JPEG nu specific# comportamentul exact al compresorului [i decompresorului, l#s@nd loc pentru implement#ri calitative. _n particular, implement#rile renun]# la vitez# \n favoarea unei calit#]i mai bune a imaginii, pentru alegerea unei aproxim#ri mai exacte a coeficien]ilor DC.
3.8.2 Modul progresiv
Suport# transmisii de imagini \n timp real. Permite coeficien]ilor DCT de a fi trimi[i treptat \n multiple scan#ri ale imaginii.
Cu fiecare scanare, decodorul poate produce o redare de calitate. De aceea, se poate ob]ine imediat o imagine de calitate redus#, urm@nd ca ulterior s# fie \mbun#t#]it# dup# cum permite timpul. Spa]iul total necesar este aproape acela[i ca pentru compresia JPEG secven]ial#, calitatea imaginii f#c@nd diferen]a.
Decodorul face un ciclu de decompresie pentru fiecare scanare. Acest mod de compresie este rar \nt@lnit, viteza procesoarelor fiind suficient de mare pentru o decompresie rapid# a imaginilor compresate \n modul secven]ial.
3.8.3 Modul ierarhic
Se poate compresa o imagine \n rezolu]ii diferite. De exemplu, pentru o imagine color pot exista versiuni de 512×512, 1024×1024 sau 2048×2048 \n acela[i fi[ier.
Imaginile de rezolu]ie \nalt# sunt compresate ca diferen]# fa]# de urm#toarea imagine, necesit@nd mai pu]ini bi]i dec@t dac# ar fi fost compresate independent. Oricum, compresia care se ob]ine este mult mai mic# dec@t cea ob]inut# prin compresia imaginii de dimensiuni maxime \n modul secven]ial.
Cadrele individuale ale imaginilor pot fi compresate \n modul progresiv, dac# se dore[te acest tip de compresie.
Modul ierarhic nu este suportat de toate aplica]iile din prezent.
3.8.4 Compresia f#r# pierderi
Acest mod nu folose[te DCT, folosind o compresie pe baz# de diferen]e. Se stocheaz# doar diferen]ele dintre valoarea pixelului [i o valoare prezis# a acestuia.
Valoarea prezis# a pixelului este dat# de o func]ie care ]ine cont de valorile compresate deja transmise. De exemplu, se pot folosi ca func]ie de predic]ie media pixelilor preceden]i. Standardul JPEG permite 8 func]ii diferite de predic]ie. Compresia diferen]elor se face folosind acelea[i metode ca la modul de compresie secven]ial, adic# algoritmul Huffman sau compresia aritmetic#.
La sf@r[itul anului 1995, au fost aprobate c@teva noi extensii ale standardului JPEG. Cea mai probabil# de a deveni popular# este cuantificarea variabil#, care permite tabelelor de cuantificare, prezentate la modul secven]ial, s# fie scalate la diferite nivele, \n diferite p#r]i ale imaginii.
_n acest fel p#r]ile critice ale imaginii pot fi detaliate mai mult, ob]in@ndu-se o calitate mai mare, dec@t p#r]ile f#r# aspecte semnificative ale aceleia[i imagini.
Se introduce un cod de semnalare la fiecare grani]# de bloc DCT(8×8 pixeli), pentru a seta un nou factor de scalare.
3.8.5 Pa[ii de prelucrare ai compresiei bazate pe DCT
Figurile 1 [i 2 prezint# pa[ii proces#rii care sunt nucleul modurilor de operare bazate pe DCT.
Aceste figuri ilustreaz# cazul special al unei compresii de imagine \n tonuri de gri. Compresia DCT poate fi asem#nat# cu compresia unui [ir de 8×8 blocuri de e[antioane dintr-o imagine grayscale. Compresia imaginilor color poate fi aproximat# cu multiple imagini grayscale, care s# fie compresate \n \ntregime, una c@te una, sau sunt compresate prin intercalarea unor blocuri de e[antioane de dimensiune 8×8 de la fiecare pe r@nd.
Pentru tipurile de compresie care opereaz# \n mod secven]ial, diagramele simplificate indic# cum func]ioneaz# compresia unei singure componente. Fiecare bloc 8×8 de intrare, \[i urmeaz# drumul prin to]i pa[ii de procesare [i produce o ie[ire \n form# comprimat# \n interiorul unui [ir de date.
Pentru modul progresiv, exist# apriori un buffer astfel \nc@t imaginea s# poat# fi stocat# [i \mp#r]it# \n multiple scan#ri cu \mbun#t#]iri succesive ale calit#]ii. Pentru modul ierarhic de-a lungul pa[ilor se construiesc blocuri cu un c@t mai mare cadru de ac]iune.
8×8 FDCT [i IDCT
La intrarea \n procesul de compresie, e[antioanele din imaginea surs# sunt grupate \n blocuri de 8×8, transformate din \ntregi f#r# semn apar]in@nd intervalului [0,2P-1] \n \ntregi cu semn \n intervalul [-2P-1,2P-1] [i introdu[i \n Forward DCT(FDCT).
La ie[ire, Inverse DCT(IDCT) se ob]in blocuri de dimensiuni 8×8 utilizate pentru a reconstitui imaginea ini]ial#.
Urm#toarele ecua]ii reprezint# defini]iile matematice ale FDCT [i IDCT:
DCT este asem#n#tor cu Transformata Fourier Discret# DFT. Se poate deduce compresia bazat# pe DCT prin analogie cu FDCT ca analizor armonic sau IDCT ca sintetizator armonic. Orice bloc de 8×8 din e[antioanele imaginii surs# este de fapt un semnal discret de 64 de puncte care este o func]ie de dou# argumente x [i y. FDCT consider# acest semnal ca o intrare [i o descompune \n 64 de semnale ortogonale de baz#. Fiecare con]ine una din cele 64 de frecven]e spa]iale unice, care cuprind spectrul semnalului de intrare. Ie[irea FDCT-ului este setul de 64 de amplitudini bazate pe semnalul sau coeficien]ii DCT ale c#ror valori sunt unic determinate de c#tre semnalul de intrare de 64 de pixeli. Valorile coeficien]ilor DCT pot fi privite drept cantit#]i relative de frecven]e spa]iale bidimensionale con]inute \n semnalul de intrare de 64 de pixeli.
Coeficientul cu frecven]a 0 \n ambele dimensiuni este numit coeficient DC, iar ceilal]i 63 de coeficien]i sunt numi]i coeficien]i AC. Deoarece valorile e[antioanelor variaz# foarte \ncet de la un punct la altul de-a lungul unei imagini, pa[ii algoritmului FDCT reprezint# temelia pentru ob]inerea rezultatelor comprimate prin concentrarea celor mai multe semnale \n frecven]e spa]iale reduse.
Pentru un bloc 8×8 dintr-o imagine surs# tipic#, cele mai multe dintre frecven]ele spa]iale au amplitudinea zero sau aproape zero [i nu este necesar# codificarea lor.
Fig.1 Pa[ii de procesare \n cadrul compresiei bazate pe DCT
Fig.2 Pa[ii de procesare \n cadrul decompresiei bazate pe DCT
La decompresie, IDCT parcurge pa[ii de procesare \n ordine invers#. Se iau cei 64 de coeficien]i DCT, care \n acest moment au fost cuantifica]i, [i reconstituie o ie[ire de 64 de puncte sub forma unui semnal prin \nsumarea semnalelor de baz#. Matematic, DCT este o punere \n coresponden]# 1 la 1 a unor vectori de 64 de pixeli \ntre imagine [i domeniul de frecven]e. Dac# FDCT [i IDCT ar putea fi calculate cu o precizie perfect# [i dac# coeficien]ii DCT nu ar fi cuantifica]i ca \n descrierea urm#toare, semnalul original de 64 de puncte ar putea fi reconstituit exact.
_n principiu, DCT nu introduce nici o pierdere \n e[antioanele din imaginea surs#, el doar dac# reu[e[te o transformare \ntr-un domeniu \n care nu pot fi efectiv codate.
C@teva propriet#]i ale implement#rilor practice pentru FDCT [i IDCT ridic# problema: ce anume ar trebui s# se cear# de la standardul JPEG. O proprietate fundamental# este c# ecua]iile FDCT [i IDCT con]in func]ii transcendentale. Nici o implementare fizic# nu le poate calcula cu precizie perfect#. Datorit# importan]ei aplica]iilor bazate pe DCT [i a rela]iilor lui DFT, au fost prezenta]i mul]i algoritmi prin care FDCT [i IDCT pot fi aproximativ calculate. Cercet#rile din domeniul algoritmilor DCT rapizi continu# [i nu exist# un singur algoritm care s# fie optim pentru toate implement#rile. Ce este optim \n software pentru un procesor pentru scopuri generale este pu]in probabil s# fie optimal \n firmware pentru un DSP programabil [i sigur este suboptimal pentru un VLSI dedicat.
Chiar \n lumina preciziei finite a intr#rilor [i a ie[irilor DCT, implement#ri ale aceluia[i algoritm, FDCT sau IDCT, care sunt proiectate independent, cu toate c# acestea difer# minimal, doar \n modul de reprezentare a termenilor cosinus, a rezultatelor intermediare sau \n felul \n care se rotunjesc valorile frac]ionare, vor produce ie[iri u[or diferite pentru acelea[i intr#ri.
Pentru a conserva libertatea de inovare sau modificare a implement#rilor, JPEG nu specific# un algoritm unic de calcul a FDCT sau IDCT \n standardul propus. Acest lucru \mpiedic# oarecum, deoarece dou# codoare sau decodoare diferite vor produce \n general ie[iri diferite pentru intr#ri identice.
Standardul JPEG rezolv# aceast# problem# prin specificarea unui test de precizie, ca parte component# a codoarelor sau decodoarelor bazate pe DCT; acest lucru pentru a se asigura \mpotriva func]iilor de baz# cosinus calculate imprecis, ceea ce ar duce la degradarea calit#]ii imaginii.
Pentru orice model bazat pe DCT, propunerea JPEG specific# modelele de compresie/decompresie separate pentru imagini cu e[antioane de 8 bi]i respectiv de 12 bi]i pe component#. Modelele pe 12 bi]i sunt folosite pentru imagini care necesit# resurse de calcul mai mari pentru a se putea ob]ine precizia dorit# a FDCT sau IDCT. Imaginile cu alte precizii de e[antion pot fi potrivite prin modele pe 8 sau pe 12 bi]i dar acest lucru trebuie f#cut \n afara standardului JPEG. Este responsibilitatea aplica]iilor s# decid# cum s# includ# sau s# \ncadreze un e[antion de 6 bi]i \ntr-o interfa]# a cadrului de 8 bi]i, cum s# despacheteze ie[irile decodorului [i cum s# codifice informa]iile necesare.
Cuantificarea
_n cadrul acestei faze, fiecare frecven]# va fi cuantificat#, utiliz@nd un coeficient care se g#se[te \ntr-unul dintre tabelele JPEG de cuantificare. Aceste tabele JPEG au fost \n a[a fel elaborate \nc@t, dac# frecven]ele au valori apropiate ele s# r#m@n# neschimbate, iar \n cazul existen]ei vreunei frecven]e cu o valoare mai “deosebit#” aceasta s# fie transformat# spre o valoare convenabil# algoritmului de compresie binar# care va urma. Exist# un tabel de coeficien]i pentru cuantificarea luminozit#]ii [i un altul pentru intensitate.
Prin aceast# opera]ie, anumite puncte ale imaginii, care difer# substan]ial fa]# de cele \nconjur#toare, \[i pierd valoarea real#, ajung@ndu-se ca, \n urma decomprim#rii, s# avem diferen]e fa]# de imaginea real#. Spuneam la \nceput c# prin comprimarea JPEG imaginea se modific#, av@nd diferen]e neobservabile cu ochiul liber. La acest pas apar cele mai semnificative diferen]e fa]# de imaginea original#, \ns# felul \n care se execut# cuantificarea face ca aceste diferente s# nu fie semnificative din punctul de vedere al utilizatorului.
Dac# este important ca imaginea s# fie intact# [i dup# decompresie, este indicat# folosirea unui alt algoritm, chiar dac# performan]ele nu sunt la fel de bune.
Codificarea binar#
Ultimul pas care a mai r#mas este acela al codific#rii propriu-zise a datelor. Codificarea se poate face \n dou# moduri: cu ajutorul unui algoritm matematic sau cu algoritmul Huffman. Algoritmul matematic poate fi foarte eficient \n anumite cazuri, dar este extrem de complex, astfel \nc@t majoritatea programelor care folosesc arhivarea JPEG nici m#car nu l-au mai implementat. _n schimb algoritmul Huffman este folosit mai des, codificarea f#c@ndu-se pe bi]i [i nu pe octe]i. Algoritmul de compresie Huffman se bazeaz# pe \nlocuirea simbolurilor frecvent \nt@lnite cu buc#]i de cod mai scurte. Pentru a se face apel la specificul anumitor imagini, \n algoritmul de arhivare JPEG s-au introdus mai multe tabele Huffman de compresie a simbolurilor frecvent folosite.
Dar 64 de frecven]e erau ordonate, dup# prelucrarea \n zig-zag, \ntr-o ordine descresc#toare relativ#, iar dup# cuantificare multe dintre frecven]e au ajuns s# aib# valoarea zero. Acum putem, \n sf@r[it, s# apreciem de ce a fost nevoie de toate aceste opera]ii pentru a codifica o simpl# imagine. Av@nd toate aceste valori de zero [i lucr@nd la nivel de bit algoritmul Huffman se poate dovedi foarte eficient. Ba mai mult, pentru c# prima frecven]# reprezint# valoarea medie a punctelor din blocul 8×8 [i pentru c# aceast# valoare este destul de mare, se recurge la codificarea separat# a primelor valori din fiecare bloc 8×8. {i pentru c# dou# valori medii, a dou# blocuri 8×8 adiacente, nu difer# prea mult \ntre ele, nu se vor lua \n calcul valorile medii propriu-zise, ci diferen]a dintre ele.
3.8.6 Organizarea fi[ierului JPEG
Mai \nt@i s# men]ion#m c# fi[ierul JPEG este organizat pe blocuri de informa]ii. Primul bloc, numit [i \nceputul imaginii, este alc#tuit din numai doi octe]i (FFh D8h ) [i reprezint# identificatorul de fi[ier. Urm#torii octe]i sunt aloca]i antetului de fi[ier [i sunt prezenta]i \n structura de mai jos:
_ncep_mag 2 bytes
APP0 2 bytes
LungimeAPP0 2 bytes
IdentificatorJFIF {ir de 4 bytes
Versiunea {ir de 2 bytes
UM 1 byte
RezolOriz 2 bytes
RezolVert 2 bytes
L#]imeUnghie 1 byte
In#l]imeUnghie 1 byte
APP0 con]ine \ntodeauna valorile FFh [i E0h [i este un marcator de aplicatie JPEG. LungimeAPP0 con]ine lungimea antetului \mpreun# cu Unghia, o imagine \n miniatur# a originalului [i se calculeaz# cu formula:
LungimeAPP0 = 16 + 3*L#]imeUnghie*In#l]imeUnghie
Cunoscute [i ca JFIF (JPEG File Interchange Format) fi[ierele JPG mai au, pe l@ng# \nceputul imaginii, [i un alt identificator, IdentificatorJFIF, care con]ine tocmai literele ‘JFIF’. Versiunea fi[ierului o reg#sim \n c@mpul Versiunea, cei doi octe]i pe care-i posed# semnific@nd 01h [i 02h pentru versiunea 1.02. Alte versiuni mai vechi sunt [i versiunile 1.0 [i 1.1. UM, RezolOriz [i RezolVert sunt folosite pentru a specifica unit#]ile de m#sur# care se folosesc pentru afi[area imaginii. UM poate avea valorile 01h pentru a specifica puncte per inch sau 02 pentru puncte per centimetru, iar RezolOriz si RezolVert con]in rezolu]ia imaginii. Dac# UM are valoarea 00h, atunci RezolOriz:RezolVert va da raportul orizontal/vertical al unui punct din imagine.
Imaginea arhivat# poate con]ine [i o copie a ei \n miniatur# numit# Unghie [i care are dimensiunile \n L#]imeUnghie [i In#l]imeUnghie. Dac# nu exist# Unghia, atunci aceste c@mpuri vor avea valoarea 0. Unghia este folosit# pentru a ar#ta utilizatorului o preimagine, \nainte de a se face cererea explicit# de extragere a imaginii din fi[ier. Ea nu este arhivat# deloc, este stocat# \ntodeauna \n format RGB, iar m#rimea ei nu poate dep#[i 64K.
Dup# antet pot exista unul sau mai multe segmente de extensie JFIF op]ionale. Aceste segmente sunt folosite pentru a stoca informa]ii adi]ionale la fi[ierul JPG [i pot fi \nt@lnite numai \ncep@nd de la versiunea 1.02. Structura acestor segmente este urm#toarea:
APP0 2 byte { con]ine valorile FFh E0h }
Lungime 2 bytes
Identificator {ir de 4 bytes { con]ine [irul ‘JFXX’ }
CodExtensie byte
CodExtensie este folosit pentru a indica tipul de informa]ie pe care o con]ine acest segment. La versiunea 1.02 a fi[ierului reg#sim doar urm#toarele coduri: 10h (Unghia este codificat# folosindu-se algoritmul JPEG); 11h (Unghia este stocat# folosindu-se un octet pentru un punct [i este \nso]it# de o palet# de culori); 13h (Unghia folose[te puncte pe 3 octe]i RGB). Informa]iile propriu-zise pe care le poart# segmentul, se reg#sesc imediat dup# structur#, iar lungimea lor poate fi determinat# prin diferen]a \ntre Lungime [i lungimea structurii. Multe dintre programele care lucreaz# cu fi[iere JPG consider# Antetul ca fiind op]ional, [i astfel se poate \nt@mpla ca anumite fi[iere s# nu aib# acest segment. _n interiorul lor se reg#se[te doar codul rezultat dup# codificarea JPEG. Totu[i, cele mai noi specifica]ii ale formatului JFIF nu mai permit excluderea acestuia. Pentru a identifica codul JPEG al imaginii, trebuie c#utat segmentul care \ncepe cu valorile FFh [i D8h ( \nceputul imaginii ).
<ööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööööö unul dintre cei mai faimos>Renumit pentru puterea lui de compresie, formatul JPEG s-a r#sp@ndit extrem de mult. Cel mai deosebit lucru este faptul c# formatul JPEG poate stoca imagini la diferite rezolu]ii cu gradul de compresie ales de utilizator. Calitatea imaginii rezultate dup# decompresie este variabil#, \n func]ie de gradul de comprimare specificat. Deosebita apreciere o datoreaz# modului \n care algoritmul [tie s# fac# uz, nu numai de valorile pe care le con]ine o imagine grafic#, acestea fiind numai ni[te date aleatoare, ci [i de particularit#]ile pe care ni le ofer# o imagine grafic#, \n special dac# aceasta con]ine o fotografie.
=== 4_95_1 ===
Cap 4. Compresia/decompresia: no]iuni [i algoritmi
Compresia datelor este procedeul prin care se realizeaz# reducerea spa]iului ocupat pe suport de un fi[ier. Decompresia este procedeul care asigur# revenirea la forma ini]ial# a unui fi[ier
4.1 Concepte
4.1.1 Seturi de date
Procesoarele de texte conduc la crearea de fi[iere \n care se utilizeaz# de regul# litere mici [i litere mari, semne de punctua]ie [i spa]ii.
Din cele 256 de simboluri sunt utilizate cel mult 90 de caractere. Mai mult frecven]ele de apari]ie ale caracterelor In text sunt diferite. _n aplica]iile economice se lucreaz# cu fi[iere ce con]in date alfanumerice pentru identificare materiale, furnizori, persoane [i date numerice asociate unor cantit#]i, pre]uri, valori.
C@nd se lucreaz# cu colectivit#]I foarte mari( eviden]a popula]iei, eviden]a automobilelor, eviden]a ac]ionarilor [i a ac]iunilor) se aleg sisteme de codificare convenabile a[a fel \nc@t seturile de date ob]inute s# poat# fi manipulate eficient.
Deseori pentru stocarea imaginilor \n fi[iere, nu sunt folosite tehnici de compresie adecvate. Spre exemplu, \n cazul imaginilor \n 16 culori, memorarea acestora se face 1 pixel pe bait. Aceast# tehnic# conduce la nefolosirea a 240 de coduri din alfabet.
Fi[ierele ce con]in programe \n C,C++, Assembler, Pascal au [i ele o redundan]# mare \n ceea ce prive[te folosirea eficient# a spa]iului de memorare pentru programele surs#.
_n toate cazurile se utilizeaz# conceptul de fi[ier, de[i seturile de date con]in [i rela]ii.
Fi[ierul brut reprezint# ceea ce este introdus, direct sau indirect, de la tastatur# [i scris pe suport folosind codul ASCII f#r# a se face modific#ri.
Fi[ierul modificat se ob]ine pornind de la un fi[ier brut prin efectuarea unor modific#ri care elimin# unele inconveniente.
4.1.2 Clasificarea seturilor de date
Exist# mai multe criterii de clasificare a seturilor de date.
Criteriul semnifica]iei grupeaz# seturile de date \n:
fi[iere text \n care datele se reprezint# fie direct cu ajutorul codului ASCII, fie dup# efectuarea unei conversii(binar, virgul# mobil#, zecimal Impachetat sau despachetat);
fi[iere imagine color sau alb-negru, materializare a matricei de pixeli;
fi[iere de sunet, care con]in codificarea frecven]elor, duratele [i intensit#]ile lor.
Criteriul structurii:
fi[iere continue care con]in date ce reflect# elementele unui masiv, textul unei c#r]i; pentru aceste fi[iere unitatea de baz# este cuv@ntul(2 bai]i) dac# reprezentarea elementelor masivului corespunde cu tipul \ntreg (int), dublu cuv@nt(4 bai]i) dac# elementele masivului au tipul virgul# mobil#(float), respectiv 1 bait corespunz#tor tipului caracter;
fi[ier format din articole, datele grupate descriu indivizii unei colectivit#]i dup# un [ablon. Exist# posibilitatea ca [abloanele s# fie identice(fi[iere cu articole de lungime fix#), sau [abloanele s# difere(fi[iere cu articole de lungime variabil#).
Un alt criteriu, de care se ]ine seama \n cazul compresiei de date este criteriul naturii informa]iei:
fi[iere \n care totul reprezint# informa]ie ce nu trebuie pierdut#, asupra acestor fi[iere put@ndu-se operadoar cu algoritmi f#r# pierdere de informa]ie;
fi[iere care pot face obiectul unor rafin#ri, seturi de date \n care spa]iile consecutive, folosite ca separatori pot fi eliminate.
4.1.3 Conceptul de compresie
Necesitatea compresiei de date este impus# de stocarea unui volum c@t mai mare de informa]ii \n memoria intern# a calculatorului. Problema devine cu at@t mai important# cu c@t procesul de informatizare a societ#]ii devine mai profund.
Compresia este un procedeu prin care se ob]ine o reducere a spa]iului de memorare ocupat de un set de date. Prin compresie datele ini]iale sunt transformate ob]in@ndu-se reprezent#ri echivalente numite [i date compresate.
Decompresia este procedeul prin care datele compresate sunt aduse la o form# c@t mai apropiat# sau chiar identic# cu forma pe care au avut-o \naintea compresiei.
4.2 Defini]ii
Compresia de date este de multe ori privit# ca un exerci]iu de reducere a redundan]ei \n fi[ier. De fapt, reprezint# cu mult mai mult.
Compresia de date rezolv# una din problemele majore ale teoriei informa]iei: cum se poate codifica \n mod eficient un mesaj, din punct de vedere al num#rului de bi]i pe simbol putem spune c# a avut loc o compresie optimal#. Nu sunt bi]i folosi]i ineficient. Acesta este scopul compresiei de date.
_n literatura de specialitate, sunt folosi]i deseori termenii de compresie [i compactare. _ntre ace[ti doi termeni exist# o diferen]#.
Prin defini]ie, compactarea datelor este o tehnic# de reducere a lungimii reprezent#rilor fizice a datelor, cu p#strarea numai a unui subset din setul de date ini]ial. Acest subset poart# denumirea de “informa]ie relevant#”.
Prin compresie de date se \n]elege un procedeu de compactare care este complet reversibil. Adic# din fi[ierul compresat, se poate ob]ine prin aplicarea decompresiei fi[ierul ini]ial.
Deci dac# la compresie fi[ierul se poate recupera integral dup# compresie, la compactare acest lucru nu mai este valabil. De exemplu, prin compactarea unui text se poate \n]elege eliminarea spa]iilor nesemnificative.
Pentru m#surarea lungimii fi[ierelor, \n compresia de date se folosesc dou# tipuri de lungimi:
– lungimea fi[ierului ca num#r de simboluri, L(F);
– lungimea fi[ierului ca num#r de bi]i,, unde reprezint# simbolul din fi[ierul F de pe pozi]ia i.
4.3 Indicatori de performan]# ai algoritmilor
Performan]a unui algoritm de compresie este privit# sub trei aspecte semnificative:
modul \n care se face compresia fiec#rui text \n parte;
comportamentul statistic global al produsului care implementeaz# algoritmul;
resursele necesare pentru realizarea compresiei.
4.3.1 Gradul de compresie
O expresie cantitativ# a puterii algoritmului de a reduce lungimea fa]# de, dat# prin:
4.3.2 Rata de compresie
Reprezint# ponderea pe care o are textul compresat \n textul ini]ial:
4.3.3 Lungimea minimal#
Reprezint# lungimea sub care textul compresat devine mai lung dec@t textul ini]ial. Se identific# deci situa]ia limit# c@nd algoritmul de compresie \[i pierde eficien]a.
Pentru coduri de compresie cu lungime fix#, se consider#:
– n num#rul de simboluri distincte din text;
– k lungimea codului;
– m lungimea textului ca num#r de simboluri; presupunem c# aceast# lungime se memoreaz# pe patru bai]i.
– h(a) – lungimea codului ini]ial, av@nd valoarea 8.
Din inegalitatea:
rezult#:
[i se ob]ine lungimea minimal#:
Dac# se consider# coduri de lungime variabil# asociate simbolurilor, se calculeaz# o lungime medie a codurilor [i deasemenea se determin# intervalul . Vor rezulta valori limit# precum [i un nivel mediu de \ncadrare a lungimii peste care compresia devine eficient#.
Lungimea minimal# se poate ob]ine \n acest caz prin tehnici de simulare.
4.3.4 {iruri comune
Fi[ierul F con]ine [iruri pentru care sunt stabili]i delimitatori sau separatori. {irurile comune sunt cele care apar de mai multe ori \n fi[iere. De exemplu, fi[ierul F:
F=<abc#abc#abc#aaa#ccc>
Av@nd delimitatorul # con]ine cinci [iruri, dintre care trei sunt identice. Ponderea iP a [irurilor comune este dat# de raportul:
unde:
B – num#rul total al [irurilor din fi[ierul F;
A num#rul [irurilor comune, unde prin [ir comun se \n]elege acel [ir care are cel pu]in dou# apari]ii.
Se noteaz# cu Ii lungimea unui [ir din fi[ier [i cu fI frecven]a de apari]ie a sa. Se determin# indicatorul:
Unde:
n – num#rul total de [iruri diferite;
fi – frecven]a de apari]ie a tipului i de [ir.
Pentru fi[ierul:
F1=<abcd#abcd#abcd#abcd#abcd>
n=1 ,f1=5
Pentru fi[ierul:
F2=<abcd#abcd#xyz#xyz#abcd#xyz#xyz>
n=2
Frecven]a de apari]ie a [irului abcd este f1=3, iar frecven]a de apari]ie a [irului xyz este f2=4
Pentru fi[ierul:
F3=<abcd#xyz#mno#12345#xaby>
I(F3)=0
n=5, iar [irurile au frecven]e de apari]ie egale cu unitatea.
Num#rul de schimb#ri n(F) din fi[ierul F ofer# o imagine a caracterului variabil de dispunere a simbolurilor.
Pentru fi[ierul F1=<aaaaaaaaaaaaaaa> num#rul de schimb#ri n(F1)=0
Pentru fi[ierul F2=<aaaaabbbbbbb> num#rul de schimb#ri n(F2)=1, \ntruc@t singura schimbare este trecerea de la simbolul a la simbolul b.
Pentru fi[ierul F3=<ababababababababab> num#rul de schimb#ri n(F3)=17.
Pentru fi[ierul F4=<abcdefghijklmmn> num#rul de schimb#ri n(F4)=12 iar pentru fi[ierul F5=<aaabbbcccdddaaaaabbbcdaaaa> num#rul de schimb#ri n(F5)=8. Exist# o leg#tur# \ntre num#rul de schimb#ri, lungimea alfabetului utilizat, lungimea fi[ierului [i rata de compresie.
Se consider# un fi[ier format din [irurile s1,s2,…,sn av@nd lungimile m1,m2,…,mn. Analiza acestor [iruri poate pune \n eviden]# sub[iruri comune. Fie [irurile sI [i sj formate din simbolurile:
#ai1 ai2……aimi #
respectiv
#aj1 aj2 ….ajmj #
Se constituie [irul yij av@nd un num#r de coomponente egal cu mij=min{mi,mj} unde ykij=1 dac# aik=ajk [i 0 \n caz contrar. Se calculeaz#
Se consider# fi[ierul:
F=<abcdef#abcdefg#xycdef#abc#cdef>
av@nd cinci [iruri.
s1=#abcdef# m1=6
s2=#abcdefg# m2=7
s3=#xycdef# m3=6
s4=#abc# m4=3
s5=#cdef# m5=4
iar valorile matricei y sunt:
y11=111111 y21=111111 y13=001111
y14=111 y15=0000 y22=1111111
y23=001111 y24=111 y25=0000
y33=111111 y34=001 y35=0000
y44=111 y45=000 y55=1111
Din aceste valori se poate determina matricea: B=(bij).
Matricea B este simetric#.
_n cazul \n care fi[ierele F con]in n cadre de film de anima]ie se analizeaz# secven]ele. Fi[ierul cadru se constituie ca un [ir. Matricea B scoate \n eviden]# gradul de asem#nare a cadrelor concentr@nd \n jurul diagonalei principale valori apropiate de 1.
Se calculeaz# suma elementelor matricei B pe linii [i pe coloane. Se stabile[te un prag de asem#nare cu luarea \n considerare a elementelor pe linie/coloan# care acoper# 70% din sumele calculate.
Aceast# tehnic# st# la baza determin#rii seturilor de modele de antrenare pentru algoritmii de compresie cu re]ele neuronale. Este posibil ca vectorii yij s# se construiasc# [i prin glisare, ceea ce presupune introducerea unui simbol neutru \n raport cu alfabetul ini]ial. De exemplu, [irul sI=#abcdefghijklm# [i [irul sj=#defghijklm# conduc la mij=0 iar yij=000000000, deci bij=0.
Prin introducerea secven]ei de simboluri neutre &&& [irul sj va deveni sj=#&&&defghijklm# [I yij=000111111111 iar bij=9/13. Aceste transla]ii se reg#sesc frecvent \n fi[ierele multimedia(film).
4.4 Erori ale compresiei de date
Transmisia datelor folosind canalele discrete f#r# zgomot este din nefericire, un model nu prea realistic \n sistemele de comunica]ie. Actualele sisteme de transmisii de date, sunt \nclinate spre dou# tipuri de erori:
erori de faz#, \n care codul unui simbol este pierdut sau ad#ugat;
erori de amplitudine, \n care codul este un simbol corupt.
Gradul \n care erorile de canal degradeaz# transmisia, este un parametru important \n alegerea metodeu de compresie a datelor.
Susceptibilitatea la erori a algoritmilor de compresie, depinde \n mare m#sur# de tipul algoritmului: static sau dinamic.
4.4.1 Algoritmi statici
Este \n general cunoscut c# algoritmii de tip Huffman, au proprietatea de autocorec]ie, prin care erorile de transmisie tind s# nu se propage mai departe. _ntr-un algoritm static, prin sincronizare se \n]elege identificarea \n acela[i mod, la emisie [i la recep]ie, a cuv@ntului transmis.
_n continuare este prezentat exemplul pentru ilustrarea abilit#]ii algoritmului Huffman standard de recuperare a erorii de faz#. Se folose[te mesajul “bcdaeb”, codificat dup# algoritmul Huffman astfel:
a 1
b 011
c 010
d 001
e 000
{irul va fi codificat 0110100011000011. Urm#torul tabel demonstreaz# impactul pierderii primului bit, celui de-al doilea sau celui de-al patrulea. Punctele delimiteaz# codurile simbolurilor interpretate.
_n cazul pierderii primului bit, are loc resincronizarea dup# cel de-al treilea bit. _n [ir s-a \nlocuit simbolul b cu grupul aa.
C@nd al doilea bit este pierdut, primii 8 bi]I ai [irului sunt gre[it interpreta]i. Resincronizarea se produce la bitul nou#.
_n cazul pierderii bitului 4, au loc acelea[i efecte ca \n cazul pierderii bitului al doilea. _ns#, primul simbol al [irului, b este citit corect.
Efectul erorilor de amplitudine asupra [irului bcdaeb este prezentat \n urm#torul tabel. La erorile de amplitudine, bi]ii 1,2 [i 4 sunt schimba]i, nu pierdu]i ca \n cazul erorilor de faz#. Astfel bitul 1 este transformat din 0 \n 1, bitul 2 din 1 \n 0, iar bitul 4 din 0 \n 1.
_n cazul schimb#rilor bi]ilor 1 [i 2, sunt perturba]i numai primii 3 bi]i, respectiv primul simbol al [irului. Schimbarea bitului 4 cauzeaz# pierderea sincroniz#rii p@n# la bitul 9.
Autosincronizarea poate fi discutat# mai bine prin urm#toarele defini]ii.
Dac# s este un sufix al unei codific#ri [i exist# secven]ele de cod [i , putem spune: s=, unde desemneaz# secven]a de sincronizare a lui s. De exemplu, pentru [irul prezentat mai sus, 1 este secven]# de sincronizare pentru 01, \n timp ce 000001 [i 011 sunt secven]e de sincronizare pentru sufixul 10.
Dac# fiecare sufix are o secven]# de sincronizare, atunci codul este complet autosincronizat. Dac# niciunul sau doar c@teva din sufixele indicate au secven]e de sincronizare atunci codul este nesincronizat sau par]ial sincronizat.
De asemenea, orice cod prefix care este complet autosincronizat se va sincroniza cu probabilitate 1, dac# [irul surs# const# \ntr-o succesiune de mesaje independente.
4.4.2 Algoritmi dinamici
Algoritmii dinamici sunt de departe cei mai afecta]i de erorile de transmisie. De exemplu, \n cazul algoritmului Huffman adaptiv, de[i receptorul se poate resincroniza cu emi]#torul, \n scopul localiz#rii corecte a unui \nceput de codificare, informa]ia pierdut# reprezint# c@]iva bi]i sau c@teva simboluri.
Din aceast# cauz#, la recep]ie vor fi redefinite gre[it codific#rile simbolurilor din punctul \nt@lnirii erorii, fiind compromis \ntreg [irul.
Lempel [i Ziv recunosc c# principalul dezavantaj al algoritmului lor este susceptibilitatea la eroare.
4.5 Compresia/decompresia imaginilor
Reducerea volumului de date pentru stocare [i transfer se datoreaz# tehnicilor de comprimare [i decomprimare. Compresia se aplic# tuturor tipurilor de date: textuale, grafice, vectoriale, imagini bitmap, imagini fixe sau animate [i sunet. _n conformitate cu specificul fiec#rui tip de dat#, se aleg algoritmi potrivi]i, specifici sau norma]i. Tabelul urm#tor prezint# o sintez# a algoritmilor de compresie a imaginilor.
_n func]ie de caracteristicile amintite pentru cele dou# clase de formate, imagine bitmap [i imagine vectorial#, procesul de compresie este diferit. Astfel, la imaginea bitmap codajul se aplic# asupra datelor proprii ale acesteia, \n timp ce la imaginea vectorial# el este rareori aplicat. _nsu[i procesul de vectorizare poate fi considerat ca o opera]ie de comprimare, datorit# stoc#rii imaginii prin punctele specifice ale componentelor sale. Deci, chiar necomprimate, fi[ierele ce con]in imagine vectorial# sunt mult mai mici \n compara]ie cu cele matriciale, ele fiind generate sub form# matematic#.
Pentru a putea evalua cantitatea de informa]ii din fi[ierele imagine, \n tabelul urm#tor vom prezenta necesarul de resurse de memorie \n MB, pentru o imagine \n format A4 necomprimat#. De asemenea tabelul arat# c# dimensiunea unui fi[ier este dependent# de sistemul de codaj al culorilor utilizate.
Analiz@nd tipurile de algoritmi folosi]i la comprimarea imaginilor, observ#m c# ace[tia se por grupa dup# urm#toarele criterii:
a) dup# volumul de informa]ii recuperate la decomprimare:
algoritmi f#r# pierdere la decomprimare;
algoritmi cu pierderi(informa]ia recuperat# nu este \n totalitate acea de la comprimare).
b) dup# timpul necesar decompresiei:
comprimare simetric#, timpii de comprimare [I decomprimare sunt egali;
comprimare asimetric#, \n sensul c# timpul necesar comprim#rii este mult superior timpului de decompresie.
Printre numero[ii algoritmi de compresie a imaginii fixe s-a impus ca standard JPEG(Joint Photographic Experts Group), creat la ini]iativa ISO a CCITT. Acest standard se \ncadreaz# \n clasa metodelor de comprimare cu pierdere [i utilizeaz# algoritmi hibrizi, baza]i pe transformarea cosinus discret# [i pe codajul Huffman. Principiul sub care func]ioneaz# JPEG este stabilirea unei rela]ii \ntre pixelii unei imagini [i codificarea lor, iar prin aplicarea sa se poate ob]ine o imagine comprimat# \ntr-un raport 71:1, f#r# o degradare vizibil# a calit#]ii acesteia.
Structura standardului a fost finalizat# \n 1989 [i poart# \n clar denumirea de “compresie numeric# a imaginilor fixe de natur# fotografic#”. Realizarea normei JPEG este condi]ionat# de existen]a a trei elemente necesare:
un codor, care prime[te datele numerice ale imaginii surs# [i genereaz#, conform unui ansamblu de proceduri, datele imaginii comprimate;
un decodor, care transform# datele imagine comprimat# \n datele imaginii reconstruite, folosind un ansamblu de proceduri;
un format de transfer, care prezint# datele imagine comprimat#, precum [i specifica]iile ob]inute din procesul de codaj.
=== 4_95_2 ===
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\par }}{\*\pnseclvl1\pnucrm\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl2\pnucltr\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl3\pndec\pnstart1\pnindent720\pnhang{\pntxta .}}{\*\pnseclvl4\pnlcltr\pnstart1\pnindent720\pnhang{\pntxta )}}{\*\pnseclvl5\pndec\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl6\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl7\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl8\pnlcltr\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}{\*\pnseclvl9\pnlcrm\pnstart1\pnindent720\pnhang{\pntxtb (}{\pntxta )}}\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {\tab Reducerea cantit#]ii de date se bazeaz# pe eliminarea acelor aspecte din imagine care nu afecteaz# perceperea vizual# a acesteia.
\par \tab Informa]ii suplimentare asupra algoritmului de comprimare a imaginii cu ajutorul acestei metode au fost prezentate \\n capitolul anterior.
\par \tab }{\b Metoda de compresie RGB 5-5-5}{ se bazeaz# pe reducerea num#rului de bi]i necesari pentru codificarea fiec#rei culori de baz#(RGB) a imaginii. Trecerea la codificarea pe 5 bi]i \\n loc de 8 bi]i a culorilor de baz# ale semnalului produce un raport de compresie de 8:5. Pierderea de 3 bi]i face ca informa]ia \ldblquote culoare\rdblquote s# fie restituit# cu mai pu]ine nuan]e. Este o metod# indicat# atunci c@nd se dore[te lucrul cu mai mult de 256 de culori simultan.
\par \tab }{\b Metoda CLUT(Color Look-Up Table)}{ lucreaz# cu paletele de culoare prin care se define[te culoarea unui pixel afi[at pe ecran [i se re]ine un num#r n de culori corespunz#toare unei puteri \\ntregi a lui 2. Astfel, se construie[te un tabel de coresponden]# \\ntre fiecare culoare [i valoarea sa \\nregistrat# pe 24 de bi]i. Fiecare pixel din imagine este reprezentat pe n bi]i, stabili]i ini]ial, pornind de la adresa din tabel [I cu indexul \\n tabela sa de culoare. de exemplu pentru puterea 8 a lui n se re]in deci 256 de culori.
\par \tab }{\b Metoda de compresie DYUV(Delta YUV)}{ const# \\n codificarea diferen]elor de luminan]# [i crominan]# ale unui punct, \\n raport cu urm#torul. Ea preia principiul codajului video compus, Y reprezent@nd luminan]a, U [i V codific@nd informa]ia de culoare. Calitatea ob]inut# prin aceast# metod# este apropiat# codific#rii pe 24 bi]i/pixel, dar folosind numai 8 bi]i/pixel, adic# reducerea volumului este de 3:1. Acest codaj este recunoscut pentru ob]inerea imaginilor naturale, fotografice sau reproducerea picturilor, de bun# calitate cu multe nuan]e de culori [i cu conturul elementelor nu foarte bine delimitat.
\par \tab Tehnica de compresie }{\b RLE(Run Length Encoding)}{ este avantajoas# \\n cazul datelor care con]in secven]e lungi [i pu]ine valori diferite. Pentru o imagine \\n culori codificarea se face prin identificarea unei culori, apoi prin indicarea acesteia [i a num#rului de pixeli din aceast# culoare. Raportul de compresie ob]inut prin aceast# metod# variaz# de la imagine la imagine, nefiind \\ns# foarte mare.
\par \tab }{\b Codajul Huffman}{ const# \\n a c#uta informa]ia redundant# [i \\n a o codifica \\n func]ie de frecven]a sa de apari]ie. Astfel, bai]i sau grupuri de bai]i care apar mai des se codific# cu un num#r mai mic de bi]i, coresponden]a dintre aceste informa]ii codificate [i codul propriu-zis ]in@ndu-se \\ntr-un tabel de coresponden]#, tabel care este necesar receptorului pentru a decodifica informa]ia. Aceast# tehnic# este folosit# nu numai pentru codificarea imaginilor, \\n special a imaginilor monocrome, ci [i pentru codificarea datelor textuale. Codajul propriu-zis este precedat de o analiz# a datelor [i de calculul frecven]elor de apari]ie.
\par \tab O alt# transformare a imaginilor matriciale, \\n afar# de vectorizare, este recunoa[terea optic# a caracterelor(}{\b Optical Character Recognition(OCR)}{). Aceast# transformare converte[te o imagine bitmap \\ntr-un text ASCII, recunoscut electronic. Tehnologia de recunoa[tere a formelor a fost introdus# ca sistem \\nc# din 1960, pentru font-uri cu dimensiune fix# [i se bazeaz# pe potrivirea caracterelor scanate ca fotografie, cu un set de caractere prestabilit, p#strat \\n memorie. Pentru a recunoa]te textul scris cu un anumit font, caracterele lui trebuiau s# fie mai \\nt@i memorate. Ulterior aceste tehnici s-au \\mbog#]it cu unele noi, specifice inteligen]ei artificiale. Conform acestora, identificarea caracterelor se face \\n baza prezen]ei sau absen]ei unui set de caracteristici, care nu mai ]in seama de dimensiunea ini]ial# sau aparen]a lor.
\par }\pard \fi720\nowidctlpar\widctlpar\adjustright {Denumite }{\b Intelligent Character Recognition(ICR)}{, datorit# folosirii algoritmilor inteligen]ei artificiale, aceste tehnici permit identificarea caracterelor dintr-un text, care este scris cu mai multe tipuri de font-uri [I chiar [I dintr-un text scris de m@n#. De asemenea, se pot examina formulare fie c# sunt completate manual sau automat, fie c# sunt cu alegeri de multiple op]iuni, sau se poate verifica autenticitatea unei semn#turi. Folosind aceste tehnici rata de recunoa[tere este mult mai mare dec@t \\n cazul tehnicilor cu fonturi fixe, \\ntruc@t ele sunt mai pu]in sensibile la degrad#rile de calitate ale originalului.
\par }\pard \nowidctlpar\widctlpar\adjustright {\tab _n plus, avantajul acestei conversii \\n text ASCII const# \\ntr-o economie de la 1:5 p@n# la 1:10, pentru o imagine necomprimat#. Aceast# transformare se realizeaz# prin tehnici diverse, toate cunoscute ca tehnici OCR:
\par {\pntext\pard\plain\f3\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi1440\nowidctlpar\widctlpar\jclisttab\tx1800{\*\pn \pnlvlblt\ilvl0\ls11\pnrnot0\pnf3\pnstart1\pnindent1800 {\pntxtb \'b7}}\ls11\adjustright {\b Compararea matricilor.}{ Este considerat# o metod# rigid#, limit@ndu-se numai la anumite font-uri specificate. Metoda const# \\n descompunerea fiec#rui caracter \\ntr-o matrice de puncte, care se compar# cu un model pre\\nregistrat. Dup# cum se poate constata aceast# tehnic# nu permite identificarea unor caractere unice sau inedite, de asemenea nici a caracterelor cu corpuri prea mici sau prea mari.}{\b
\par {\pntext\pard\plain\f3\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi1440\nowidctlpar\widctlpar\jclisttab\tx1800{\*\pn \pnlvlblt\ilvl0\ls11\pnrnot0\pnf3\pnstart1\pnindent1800 {\pntxtb \'b7}}\ls11\adjustright {\b Analiza formelor cu sau f#r# semnifica]ie.}{ _n compara]ie cu tehnica de comparare a matricilor, aceasta permite recunoa[terea unei mari variet#]i de caractere. _n loc de descompune \\n matrici de puncte fiecare caracter, aceast# tehnic# descompune fiecare caracter \\n forme geometrice care se compar# deasemenea cu modele pre\\nregistrate.}{\b
\par {\pntext\pard\plain\f3\lang1048\cgrid \loch\af3\dbch\af0\hich\f3 \'b7\tab}}\pard \fi1440\nowidctlpar\widctlpar\jclisttab\tx1800{\*\pn \pnlvlblt\ilvl0\ls11\pnrnot0\pnf3\pnstart1\pnindent1800 {\pntxtb \'b7}}\ls11\adjustright {\b Re]ele neuronale.}{ Cu toate c# re]elele neuronale furnizeaz# sistemelor OCR capacit#]i de \\n]elegere ridicate, aceast# metod# necesit# timpi mari de tratament. Utilizatorul \\ndic# programului OCR coresponden]a de stabilit \\ntre formele imprimate [i caracterele ASCII. Societatea Mimetics este cea care a experimentat aceast# nou# metod#, introduc@nd tehnica re]elelor neuronale \\n OCR. Ea se dore[te a fi [i mai sensibil# la pierderile de calitate ale originalului.}{\b
\par }\pard \nowidctlpar\widctlpar\adjustright {\b \tab }{_n mod obi[nuit, produsele software OCR de]in [i posibilit#]i integrate pentru corectarea ortografic# a cuvintelor analizate. Caracteristicile acestor produse, ce folosesc tehnicile de recunoa[tere a formelor sunt legate de fiabilitatea la citire a datelor [i de viteza sau rata de recunoa[tere. Rata de recunoa[tere a caracterelor depinde mult de calitatea originalului ob]inut la citire. De asemenea, este oferit# [i posibilitatea ca utilizatorul s# indice corespondentul ASCII al unui semn grafic ne\\n]eles automat, dup# care sistemul \\l va \\nv#]a [i recunoa[te \\n toate situa]iile viitoare. Recunoa[terea optic# a caracterelor este o activitate care consum# o cantitate relativ mare de resurse ma[in#, lucru de care trebuie s# se ]in# seama.
\par }\pard\plain \s3\keepn\nowidctlpar\outlinelevel2\adjustright \b\f16\fs20 {\fs24
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\par }{\b 4.6 Compresia cu fractali
\par }{\fs16
\par \tab }{Tehnicile de compresie a imaginilor cu ajutorul fractalilor au fost introduse \\n urm# cu aproximativ 10-15 ani. Ideile care au stat la baza acestor tehnici au provenit din teoria matematic# numit# Sisteme de Func]ii Iterative (Iterated Function Systems(IFS)). Aceast# teorie a fost dezvoltat# \\n 1981 de c#tre John Hutchinson.
\par \tab Michael Barnsley un cercet#tor al Institutului Tehnologic din Georgia a scris o carte, care a devenit foarte popular#( \ldblquote Fractals Everywhere\rdblquote ) \\n care prezenta mecanismul matematic bazat pe IFS al unei noi metode de modelare \\n grafica pe calculator, metod# ce este cunoscut# sub numele de }{\b teorema colajului}{(dac# se dore[te compresia unei imagini arbitrare pe baza IFS, trebuie mai \\nt@i s# g#sim o familie de contrac]ii astfel \\nc@t atractorul imaginii s# fie cea mai fidel# aproximare posibil# a imaginii).
\par \tab Aceast# teorem# a dezv#luit o posibilitate uimitoare. Se punea \\ntrebarea \ldblquote dac# matematica fractal# poate genera imagini care s# reprezinte elemente naturale ca de exemplu nori, mun]i, etc., nu poate fi folosit# aceasta [i pentru a comprima imagini. Problema invers# este definit# astfel \ldblquote pornind de la o imagine s# se ajung# la un IFS care s# semene c@t mai bine cu imaginea original#\rdblquote .
\par \tab Aceast# problem# a ridicat o mul]ime de dificult#]i. Barnsley, a crezut c# a g#sit r#spunsul la aceast# problem#. A p#r#sit institutul [i a fondat Iterated Systems Incorporated \\mpreun# cu Alan Sloan.
\par \tab _n ianuarie 1988, scrie un articol \\n revista BYTE \\n care public# ni[te idei despre modul \\n care se poate compresa o imagine cu scopul de a ob]ine o rat# de compresie mare, dar nu era oferit# solu]ia problemei inverse. La acel moment nimeni nu [tia exact cum pot fi comprimate/decomprimate imaginile cu ajutorul IFS la o rat# de compresie rezonabil#.
\par \tab _n martie 1988 Arnaud Jaquin public# cu ajutorul lui Barnsley o schem# modificat# numit# Partitioned Iterated Function Systems(PIFS), schem# care a dat un nou av@nt cercet#rilor \\n acest domeniu.
\par \tab Ideea lui Jaquin era foarte simpl#. O imagine s# nu fie considerat# ca un colaj de copii al acesteia, ci ca ni[te copii a unor p#r]i mai mici ale acesteia. De exemplu, o parte a unui nor nu poate ar#ta ca un \\ntreg peisaj cu mun]i, cer [i nori, dar este foarte probabil ca aceasta s# arate la fel cu o alt# parte a norilor sau cu o alt# por]iune din imagine. Astfel tehnica compresiei de imagini cu ajutorul fractalilor a luat na[tere.
\par }{\fs16
\par
\par }{\b 4.6.1 Ce}{ }{\b este compresia de imagini cu fractali ?}{
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caracteristic# este una din cheile tehnicilor de compresie cu fractali.Dar cum vom descrie comportamentul acestei ma[ini ? Transform#ri de genul celor de mai jos ne vor ajuta:
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {\tab Astfel de transform#ri sunt numite transform#ri afine. Transform#rile afine sunt capabile s# modifice, \\ntind#,roteasc#,scaleze [i s# translateze o imagine.
\par }\pard \fi720\nowidctlpar\widctlpar\adjustright {Barnsley a sugerat c# probabil stocarea imaginii ca o colec]ie de transform#ri ar putea duce la compresia imaginii. Argumentul sau a fost dat de posibilitatea de a descrie o imagine sub forma c@torva parametri de transform#ri afine. Dac# avem parametrii transform#rilor afine (ori ai ma[inii de copiat) vom putea produce aceast# imagine.
\par Ma[ina produce imagini auto-similare numite fractali. Nu avem nevoie dec@t de c@]iva parametri pentru pentru a produce un triunghi Sierpinski \\n locul imaginii ini]iale. Dar, cum s# g#sim ace[ti parametrii pentru orice imagine ? (problema inversa men]ionat# anterior). Visul lui Barnsley era de a descoperi un algoritm care s# calculeze ace[ti parametri, vis care nu a c#p#tat contur niciodat#.
\par Oricum o metod# exist#. Dar pentru aceasta trebuie s# descriem mai int@i IFS deoarece compresia de imagini cu fractali se bazeaz# pe acesta.
\par }\pard \nowidctlpar\widctlpar\adjustright {
\par }\pard\plain \s1\keepn\nowidctlpar\widctlpar\outlinelevel0\adjustright \b\f16\cgrid {4.6.2 Sisteme de func]ii iterative
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {\fs16
\par }\pard\plain \s17\nowidctlpar\widctlpar\adjustright \f16\lang2057\cgrid {\tab Comportamentul ma[inii de copiat poate fi descris cu ajutorul unui model matematic numit IFS. Un IFS const# \\ntr-o colec]ie de contrac]ii ale transform#rilor afine. Aceast# colec]ie de transform#ri define[te o familie de contrac]ii:
\par }{\lang1024 {\shp{\*\shpinst\shpleft3618\shptop159\shpright5223\shpbottom819\shpfhdr0\shpbxcolumn\shpbypara\shpwr1\shpwrk0\shpfblwtxt0\shpz9\shplid1029{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pib}{\sv {\pict\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw2831\pich1164\picwgoal1605\pichgoal660\pngblip\bliptag123086634{\*\blipuid 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\par }\pard\plain \s4\keepn\nowidctlpar\outlinelevel3\adjustright \f16 {\tab Pentru un set de intrare S, putem calcula w}{\sub i}{ pentru fiecare i [i s# ob]inem o reuniune a
\par }\pard\plain \s17\nowidctlpar\widctlpar\adjustright \f16\lang2057\cgrid {acestor seturi, notat# W(S). Hutchinson a demonstrat o caracteristic# important# \\n IFS : dac# w}{\sub i}{
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {sunt contrac]ii atunci [i W este o contrac]ie. Teorema lui Hutchinson specific#: \ldblquote fiind dat# o familie W vom avea un unic punct fix \\n spa]iul tuturor imaginilor\rdblquote . Ceea ce inseamn# c# oricare ar fi imaginea (sau setul) cu care \\ncepem, putem aplica repetat W , [i imaginea noastr# ini]ial# va converge c#tre o imagine fix#. _n acest mod W define[te complet o singur# imagine.
\par \tab Cu alte cuvinte, fiind dat# o imagine (sau un set), f}{\sub 0}{
\par
\par putem aplica repetat W(ori ma[ina de copiat descris# cu W) ob]in@nd f}{\sub 1=}{W(f}{\sub 0}{) ca o prim# copie, f}{\sub 2}{=W(f}{\sub 1}{)=W(W(f}{\sub 0}{))=W}{\super (2)}{(f}{\sub 0}{) ca o a doua copie [I a[a mai departe.
\par \tab Atractorul(imaginea unic#) este rezultat al transform#rii:
\par }{\lang1024 {\shp{\*\shpinst\shpleft3906\shptop334\shpright6291\shpbottom769\shpfhdr0\shpbxcolumn\shpbypara\shpwr1\shpwrk0\shpfblwtxt0\shpz10\shplid1030{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}{\sp{\sn fFlipV}{\sv 0}}{\sp{\sn pib}{\sv {\pict\picscalex100\picscaley100\piccropl0\piccropr0\piccropt0\piccropb0\picw4207\pich767\picwgoal2385\pichgoal435\pngblip\bliptag1897868858{\*\blipuid 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\par }\pard\plain \s2\keepn\nowidctlpar\widctlpar\outlinelevel1\adjustright \b\f16\lang2057\cgrid {
\par 4.6.3 Auto-similaritatea \\n imagini
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {
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Acum vom dori s# g#sim o harta(map) W care ia o imagine [i o transform#(comprim#). Dac# dorim s# [tim c@nd W este contrac]ie atunci va trebui s# definim distan]a dintre dou# imagini. Distan]a este definit# ca:
\par
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {unde f [i g sunt valori ale nivelului \\n tonuri de gri ale unui pixel(pentru imaginile cu palet# \\n nuan]e de gri), P este spa]iul imaginii, x [i y sunt coordonatele unui pixel oarecare.
\par \tab Aceast# distan]# define[te pozi]ia (x,y) \\n care imaginile f [i g difer# cel mai mult.
\par \tab Imaginile naturale nu sunt auto-similare. Imaginea utilizat# mai jos nu con]ine genul de auto-similaritate pe care \\l g#sim \\n triunghiul lui Sierpinski. Dar urmatoarea imagine va demonstra c# putem g#si por]iuni auto-similare \\ntr-o imagine.
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\par Imaginea original# \tab \tab Por]iuni auto-similare \\n imagine.
\par }\pard\plain \s17\nowidctlpar\widctlpar\adjustright \f16\lang2057\cgrid {\fs16
\par }{Observ#m c# o parte a p#l#riei este similar# cu por]iune din reflexia acestei p#l#rii \\n oglinda. Principala distinc]ie \\ntre cele dou# tipuri de auto-similarit#]i(din triunghiul lui Sierpinsky [i din imaginea prezentat# anterior) este aceea c# triunghiul este alc#tuit dintr-un num#r de copii ale \\ntregului pe c@nd imaginea prezentat# va fi format# dintr-un num#r de copii ale unor p#r]i identificate \\n cadrul imaginii analizate. Aceste p#r]i nu sunt exact la fel ceea ce \\nseamn# c# imaginea pe care o vom comprima cu ajutorul setului de transform#ri determinat nu duce la ob]inerea unei copii identice a imaginii originale atunci c@nd vom folosi procedeul invers(decompresia).
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {\tab Experimentele au ar#tat c# majoritatea imaginilor ca de exemplu copaci, fe]e, case, nori etc. con]in por]iuni auto-similare.
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\par }\pard\plain \s2\keepn\nowidctlpar\widctlpar\outlinelevel1\adjustright \b\f16\lang2057\cgrid {4.6.4 Sisteme de func]ii iterative parti]ionate (Partitioned Iterated Function Systems – PIFS)
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {\b\fs16
\par }\pard\plain \s17\nowidctlpar\widctlpar\adjustright \f16\lang2057\cgrid {\tab Pentru comprimarea unei imagini este necesar# divizarea acesteia \\n por]iuni care s# nu se suprapun# (numite blocuri domeniu [i notate cu D), por]iuni care vor fi transformate separat. Prin \\mp#r]irea imaginii \\n buc#]i, va trebui s# comprim#m multe forme ceea ce se poate dovedi dificil daca folosim IFS. Din acest motiv vom folosi PIFS. PIFS sunt deasemenea bazate pe transform#rile afine dar acum acestea vor descrie transformarea dintre o por]iune a imaginii [i por]iunea urm#toare(numite blocuri rang [i notate cu R). Transform#rile afine men]ionate anterior sunt capabile s# transforme geometric p#r]ile imaginii dar nu pot transforma nuan]a de gri a pixelului. Din acest motiv va trebui s# adaug#m o nou# dimensiune transform#rilor afine.
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\par }{unde s}{\sub i}{ reprezint# contrastul [i o}{\sub i}{ str#lucirea transform#rii iar z=f(x,y).
\par \tab Transformarea W trebuie s# fie o contrac]ie \\n toate cele trei dimensiuni: x,y, [i z. Aceast# transformare va fi o contrac]ie c@nd distan]a z va fi mic[orat# cu un factor mai mic dec@t 1 sau chiar dac# fiecare s}{\sub i}{<1 (unele experimente arat# c# [i o valoare de 1.2 este bun#).
\par
\par }{\b 4.6.5 Comprimarea imaginii}{
\par }{\fs16
\par \tab }{S# presupunem c# avem o imagine f pe care dorim s# o comprim#m. Cu alte cuvinte dorim s# g#sim o familie de contrac]ii w}{\sub 1}{,w}{\sub 2}{, , w}{\sub N}{ cu
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Vom c#uta o parti]ie a lui f \\n cele N suprafe]e nesuprapuse ale imaginii c#reia s#-i aplic#m transformarea w}{\sub i}{ [i s# ob]inem f. Acest lucru este prea optimist, deoarece imaginea nu este compus# din buc#]i care pot fi transformate [i potrivite \\n alta parte a imaginii. Putem spera doar s# ob]inem o imagine
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\par c@t mai mic#. Cu alte cuvinte, vom c#uta o transformare W al carei punct fix nu este f, dar este apropiat de acesta.
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\par Ecua]ia de aproximare a distan]ei este:
\par *********
\par \tab Ar trebui s# g#sim buc#]i D}{\sub i}{ [i contrac]ii w}{\sub i,}{ astfel \\nc@t s# putem aplica w}{\sub i}{ blocului domeniu D}{\sub i, }{[i ar trebui s# ob]inem ceva foarte apropiat cu o alt# parte R}{\sub i}{(cu un bloc rang). G#sirea blocului rang R}{\sub i}{ corespunz#tor blocului domeniu D}{\sub i}{ se face prin minimizarea distan]ei dintre cele dou# tipuri de blocuri(acesta este scopul problemei).
\par \tab Fie urm#torul exemplu:
\par \tab \tab Presupunem c# trebuie s# comprim#m o imagine cu dimensiunile 256×256 pixeli cu tonuri de gri(grayscale image). Fie R}{\sub 1}{,R}{\sub 2}{,.,R}{\sub 1024}{ blocurile rang ale imaginii(dimensiune 8×8 pixeli), [i fie D colec]ia de blocuri domeniu ale imaginii(dimensiune 16×16 pixeli). Deci un bloc domeniu este de 4 ori mai mare dec@t un bloc rang. Colec]ia D con]ine (256-16+1)*(256-16+1)=58081 p#trate. Pentru fiecare R}{\sub i}{ vom c#uta \\n toat# mul]imea D pentru a g#si acel bloc domeniu D}{\sub i}{ care minimizeaz# distan]a. Sunt 8 moduri(un p#trat poate fi aranjat \\n patru moduri sau inversat [i aranjat \\n alte patru moduri) \\n care putem coontracta un p#trat \\n altul. Ceea ce \\nseamn# c# va trebui s# compar#m 8*58081=464648 blocuri cu fiecare din cele 1024 blocuri rang. Deasemenea un bloc din D este de patru ori mai mare dec@t un bloc din R}{\sub i}{, astfel \\nc@t va trebui s# reducem blocul domeniu pentru a-l putea compara cu un bloc rang.
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\par }}\dpx2466\dpy2469\dpxsize5472\dpysize1440\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat0\dplinehollow}}}}{Fiecare transformare va necesita 8 bi]i pentru a determina pozi]ia lui D}{\sub i}{ (pozi]ia pe axele x [i y), 7 bi]i pentru o}{\sub i}{, 5 bi]i pentru s}{\sub i}{ [i trei bi]i pentru a determina transformarea necesar# a blocului pentru a mapa blocul domeniu sub forma unui bloc rang. Avem deci nevoie de 31 de bi]I pentru a transforma un bloc domeniu \\ntr-un bloc rang(8x8x8=512 bi]i) ceea ce duce la o rata a compresiei de 16,5:1. Rata de compresie este unul din parametrii ce determin# eficien]a unui algoritm. Un alt parametru este eroarea introdus# de algoritmul de compresie care pentru o imagine cu o palet# de gri definit# pe 8 bi]I este dat# de urm#toarea formul#(PNSR \endash peak to peak signal to noise ratio):
\par
\par Exist# o mul]ime de variante ale acestui algoritm ce de exemplu quadtree partitioning, HV-partitioning, triangular partitioning, self vector quantization, convolution transform coding etc.
\par }{\b
\par 4.6.6 Comprimarea imaginilor color
\par }{
\par Exist# foarte pu]ine lucr#ri publicate care se ocup# de comprimarea imaginilor color cu ajutorul tehnicilor fractale. Se recomand# ca o imagine color s# nu fie comprimat# pe baza componentelor de culoare RGB ci separat prin intermediul canalelor YIQ . Canalele YIQ pot fi comprimate individual iar canalele I [I Q ar trebui comprimate la o rat# mai mic# dec@t canalul Y.
\par Alte sugestii ar fi s# se comprime separat componenta verde din grupul RGB [I s# se \\ncerce prezicerea apari]iei celorlalte componente(R [I B).
\par
\par }{\b 4.6.7 O compara]ie \\ntre compresia imaginilor cu ajutorul fractalilor [I compresia cu ajutorul standardului JPEG}{
\par
\par La rate mari ale compresiei compresia cu ajutorul fractalilor are aproape acelea[I avantaje compresia pe care o ob]inem cu ajutorul algoritmului JPEG. _n fig.A putem vedea c@teva compara]ii \\ntre compresia imaginilor cu ajutorul algorimului JPEG [I cea cu ajutorul tehnicii fractalilor. Rezultatele care apar mai jos sunt cele la care trebuie s# ne a[tept#m atunci c@nd folosim compresia cu fractali. Totu[I pentru unele imagini se pot ob]ine rate de compresie de peste 80:1. Pentru rate de compresii mari calitatea imaginii poate l#sa de dorit.
\par Compresia cu fractali este un proces asimetric, compresia unei imagini dur@nd foarte mult comparativ cu decompresia acesteia. Prin compara]ie algoritmul ce utilizeaz# JPEG este un algoritm simetric compresia/decompresia av#nd durate aproximativ egale.
\par Durata de compresie \\n cazul utiliz#rii fractalilor poate fi redus# prin utilizarea unui hardware dedicat. Un avantaj al tehnicii cu fractali este c# decompresia va lua mai pu]in timp dec@t decompresia \\n cazul utiliz#rii algoritmului JPEG.
\par O caracteristic# foarte important# a compresiei cu fractali este aceea c# permite decomprimarea imaginii la o dimensiune mai mare dec@t cea ini]ial# f#r# a afecta foarte mult calitatea imaginii(fenomenul de pixelizare a imaginii dispare aproape complet). Detaliile care apar la m#rirea imaginii vor fi generate prin intermediul algoritmului de decomprimare. _n fig.B putem vedea un astfel de exemplu.
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\par }}\dpx2034\dpy-177\dpxsize3177\dpysize1480\dpfillfgcr255\dpfillfgcg255\dpfillfgcb255\dpfillbgcr255\dpfillbgcg255\dpfillbgcb255\dpfillpat0\dplinehollow}}}}{\tab \tab
\par
\par }\pard\plain \s18\nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {
\par }\pard\plain \nowidctlpar\widctlpar\adjustright \f16\lang1048\cgrid {
\par }\pard\plain \s5\keepn\nowidctlpar\widctlpar\outlinelevel4\adjustright \b\f16\lang1048\cgrid {Fig.B}{\b0
\par }}
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[ ARM94]
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[FAL92]
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[BAR88]
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[DUB91]
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[GOU92]
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Armin Bunde “Fractals in science” – 298 pag.
[BAX94]
Gregory A Baxes “Digital image processing: principles and aplications” – New York Ed.John Wiley&Sons 1994, 452 pag.
[YOU94]
H.Youlal “Modelesation parametrique en traitment d’images” – Paris Ed.Mason 1994, 207 pag.
Adrese Internet
HTTP://INLS.UCSD.EDU/RESEARCH/FISHER/FRACTALS/NEWSP
HTTP://WWW.ITERATED.COM
HTTP://SPROTT.PHYSICS.WISC.EDU/FRACTALS.HTM
HTTP://WWW.ECE.CONCONDIA.CA/~SORIAL/IMAGE.HTML
HTTP://SPANKY.TRIUMF.CA (Baza de date ce contine programe si teorie fractala precum si legaturi catre alte servere ce au un specific asemanator)
HTTP://SPANKY.TRIUMF.CA/WWW/OTHER_PROGS.HTML
HTTP://SPANKY.TRIUMF.CA.PUB/FRACTALS/CODE/
HTTP://IPSUN1.CS.NTHU.EDU.TW/~MR834342/FIC.HTML
HTTP://INLS.UCSD.EDU/Y/FRACTALS
HTTP://WWW.FAQS.ORG/FAQS/COMPRESSION_FAQ/PART2/PREAMBLE.HTML
HTTP://WWW.CTR.COLUMBIA.EDU/~ALEX/PUBLICATIONS.HTML
HTTP://WWW.CS.UBCLUJ.RO/TT~7316/FRACTALIA (Publicatie in limba romana cu fractali)
HTTP://WWW.IGLOBAL.NET/PUB/CAMELOT/CHAOS/CHAOS.HTM (articol despre teoria haosului si fractali)
HTTP://WWW.RASIP.FER.HR/RESEARCH/COMPRESS/ALGHORITMS/ADV/FRACOMP/INDEX.HTML
=== CAP5S1 ===
Capitolul 5 Aplica]ii
_n acest capitol voi descrie aplica]iile ce au fost realizate pe baza ideilor [i algoritmilor prezenta]i \n lucrare.
Prima aplica]ie prezentat# este o implementare \n C a algoritmului de compresie/decompresie cu fractali propus de Barnsley.
Fi[ierele sursa trebuie s# fie \n format TGA alb-negru(gray scale), reprezentate pe 8 bi]i(imaginea surs# nu este comprimata initial). _n cazul \n care se dore[te crearea unor fi[iere imagine \n acest format, se va salva imaginea \n format TGA grayscale. Pentru conversia fi[ierelor imagine testate am utilizat programul “Sea” produs de Harold de Laat & Bart Wakee. Header-ul imaginii are 18 bai]i [i nu 198 cum este generat de alte utilitare grafice(vezi Photoshop).
Dac# s-au generat fi[ierele cu un header de 198 bai]i, se vor [terge ultimii 180 de bai]i din fi[ier utiliz@nd un editor de texte.
Aplica]ia este structurat# \n trei sec]iuni de baz#:
o sec]iune ce realizeaz# interfa]a programului cu utilizatorul [i vizualizeaz# fi[ierele selectate de acesta;
o sec]iune ce realizeaz# compresia imaginii;
o sec]iune ce realizeaz# decompresia imaginii.
Pentru test#ri s-au folosit variante simplificate ale acestor sec]iuni, deoarece compresia de imagini cu ajutorul fractalilor este un proces cu o durat# destul de mare (durat# care este influen]at# desigur [i de viteze de lucru a procesorului pe care rulez acest program ).
Textele surs# ale aplica]iei sunt prezentate \n anexa A a lucr#rii. Acestea au fost compilate utiliz@nd Borlandc 3.1. Dac# se dore[te recompilarea surselor se vor utiliza cele trei proiecte deja generate, ]in@ndu-se cont de faptul c# \n anumite situa]ii vor trebui setate din nou directoarele \n care se afla resursele mediului Borland.
Prezent#m mai jos interfa]a programului cu utilizatorul.
Pa[ii algoritmilor de compresie/decompresie sunt delimita]i prin comentariu \n codul surs# pentru o mai bun# \n]elegere a implement#rii algoritmului. Deasemenea, programul con]ine o serie de comentarii, pentru delimitarea opera]iilor ce se execut# asupra imaginilor.
_n timpul execu]iei algoritmului, programul afi[eaz# dimensiunile blocului curent pe care \l testeaz# precum [i valoarea erorii globale pentru acest bloc.
Algoritmul de decompresie spre deosebire de cel de compresie este foarte rapid. Acesta este unul din avantajele compresiei cu fractali, avantaj ce contrabalanseaza pierderea de informa]ie a algoritmului [i timpul foarte mare necesar compresiei.
Performan]ele compresiei cu fractali au fost testate pe urm#toarele fi[iere imagine:
– aaboot.tga;
– castel.tga;
– lita.tga;
– tech.tga;
– tech2.tga;
– test.
_n urm#torul tabel furniz#m c@teva date despre aceste fi[iere:
Procesul de decompresie se realizeaz# iterativ. Lafiecare itera]ie se realizeaz# o \mbun#t#]ire a calit#]ii imaginii.
Unele imagini au o refacere mai rapid# \n procesul de decompresie. Pierderea de informa]ie este de circa 2-3% \n medie dar se constat# c# pentru fi[ierele de dimensiuni mici imaginea reconstituit# \[i pierde semnificativ din calitate.
Rata medie de compresie a imaginilor se apropie \n aceste cazuri de rata de compresie ob]inut# cu ajutorul unor algoritmi f#r# pierdere de informa]ie.
Performan]ele acestui algoritm devin vizibile abia c@nd \l utilizeze \n cazul fi[ierelor de dimensiuni foarte mari. Astfel pentru un fi[ier TGA cu dimensiunile 1024×768 \n nuan]e de gri (grayscale) cu o lungime de 786450 bai]i, se ob]ine un fi[ier compresat cu lungimea de 4.578% adic# aproape 22:1. Compresia \n acest caz este foarte bun#, dar timpul necesar a crescut de aproximativ 15 ori fa]# de timpul necesar compresiei uneia din imaginile studiate.
Prezent#m mai jos imaginea ini]ial# [i imaginea decompresat# pentru imaginile studiate (imaginile au fost decompresate \ntr-un ciclu de 28 de iteratii).
Concluzii:
Compresia cu fractali prezint# dejavantaje din urm#toarele puncte de vedere:
durat# de compresie foarte mare;
algoritmul de compresie este unul cu pierdere de informa]ie.
Pe de alt# parte avantajele acestui tip de compresie sunt:
rat# de compresie bun# pentru fi[ierele imagine de dimensiuni mari;
timp de decompresie mic comparativ cu al]i algoritmi de decompresie.
Totu[i prin implement#ri hard ale acestui algoritm, sau utiliz@nd calculul paralel al erorilor(acest algoritm se preteaz# la un astfel de calcul) se poate ajunge la durate cu mult mai mici ale compresiei pentru aceast# metod#.
Exist# de asemenea posibilitatea utiliz#rii unui algoritm hibrid(ce combin# diver[ii algoritmi de compresie a imaginii cunoscu]i dar implementarea acestora este dificil#).
O caracteristic# important# ce decurge din propriet#]ile fractalilor este aceea c# se permite decomprimarea imaginii la o dimensiune mai mare dec@t cea ini]ial# f#r# a afecta foarte mult calitatea imaginii(fenomenul de pixelizare a imaginii dispare aproape complet, detaliile necesare fiind generate prin intermediul algoritmului).
=== CAP5S2 ===
A doua aplica]ie este o implementare \n assembler a unui generator teren montan cu ajutorul fractalilor func]iile implementate put@nd s# fie apelate din C pentru o urm#rire mai u[oar# o modului de generare [i desenare. Acest algoritm a ap#rut pentru prima dat# \n jurul anului 1990.
Versiunea curent# permite deplasarea virtual# \n terenul generat. De asemenea putem roti punctul de vedere al utilizatorului utiliz@nd un increment de 15 grade.
Tastele utilizate sunt:
– “H” merg spre st@nga;
– “J” merg \napoi;
– “K” merg \nainte;
– “L” merg spre dreapta;
– “A” rota]ie spre st@nga;
– “S” rota]ie spre dreapta.
– “X”,”Q” ie[ire
Punctul de vedere al celui ce prive[te este considerat ca av@nd un arc de 180 de grade.
Programul utilizeaz# instruc]iuni 386 astfel \nc@t nu va putea fi rulat pe un calculator 286.
Imaginea rezultat# este:
Programul a fost compilat utiliz@nd instrumentele specifice Borland versiunea 3.1.
Programul “Plasma.pas” ilustreaz# modul \n care se genereaz# terenul pentru aplica]ie. Unele din generate sunt utilizate drept \n#l]imi iar altele sunt considerate v#i [i datorit# continuit#]ii nuan]elor se ob]ine respectivul peisaj .
Prin algoritmi de netezire se rafineaz# muchiile mai ascu]ite ce ar putea rezulta \n urma unei diferen]e prea mari \ntre nuan]ele rezultate. De asemenea mai aplic algoritmi ce trateaz# imaginea din punctul de vedere al celui ce “survoleaz#” terenul, precum [i algoritmi de rotire a imaginii rezultate, [i de trasare a \n#l]imilor, utiliz@nd apoi din nou algoritmi de neteziire a imaginii rezultate.
Concluzii:
Avantaje:
nu mai este necesar# stocarea imaginilor intermediare, anima]ia fiind realizat# prin intermediul algoritmului;
algoritmul poate fi utilizat \n crearea de jocuri (mai ales simulatoare de zbor) fiind totu[i necesar s# gener#m pe parcurs [i alte elemente de imagine ca de exemplu copaci,r@uri,nori etc., lucru posibil dac# utiliz#m algoritmii specifici fractalilor.
Dezavantaje:
datorit# num#rului mare de calcule necesar pentru generarea [i desenarea terenului, pentru a ob]ine un efect de anima]ie continuu este necesar un calculator mai puternic.
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