8-Valent Fuzzy Logic for Iris Recognition and [602195]

8-Valent Fuzzy Logic for Iris Recognition and
Biometry

N. Popescu -Bodorin*, Member , V.E. Balas**, Senior Member , and I.M. Motoc*, Student: [anonimizat] , IEEE
*Artificial Intelligence & Computational Logic Lab., Math. & Comp. Sci. Dept., „Spiru Haret ‟ University, Bucharest, Rom ânia
**Faculty of Engineering, „Aurel Vlaicu‟ University, Arad, România
[anonimizat], [anonimizat], [anonimizat]

Abstract – This paper shows that maintaining logical
consistency of an iris recognition system is a matter of finding a
suitable partitioning of the input space in enrollable and
unenrollable pairs by negotiating the user comfort and the safety
of the biometric system. I n other words, consistent enrollment is
mandatory in order to preserve system consistency. A fuzzy
3­valent disambiguated model of iris recognition is proposed and
analyzed in terms of completeness, consistency, user comfort and
biometric safety . It is also shown here that the fuzzy 3 ­valent
model of iris recognition is hosted by an 8 ­valent Boole an
algebr a of modulo 8 integers that represents the computational
formalization in which a biometric system (a software agent ) can
achieve the artificial understanding of iris recognition in a
logically consistent manner .
I. INTRODUCTION
Because the visual acuity of the human agent is doubled by its
intelligence – both of them together ensuring an excellent
quality in indentifying the (dis)similarity of iris images, the
geometry that illustrate s the binary decisions given by the
human agent during a Turing test [11] of iris recognition is
very simple (Fig. 1.a): it consist s of one collection of crisp
points (0 and 1 ) and one histogram that counts how many
times a decision of unitary score ( 1 – for the case of similar
irides) or a n ull decision ( 0 – for the pairs of non -similar
irides) was given by the human agent. Still, the geometry that
illustrate s the fuzzy binary decisions given by a software
agent ([6]-[8]) during a Turing test of iris recognition is not
that simple: in th is case, the fuzzy biometric decisions given
by the software agent define (draw) a f-geometry [13] in
which the intra – and inter -class score distributions could be a
little bit confused (Fig. 1.c, Fig. 2.a, Fig. 2.b), or confused
much stronger (Fig. 1.b, Fig. 1.c in [6], Fig. 10 in [4]), or not
confused at all. (Fig . 1.b from here, and Fig. 4.a, Fig. 4.b,
Fig. 4.c in [6]).
A. Crisp / Fuzzy Iris Recognition
In fact, Fig. 1.a illustrates that iris recognition is crisp for a
human agent, and consequently, the recognition function R
(as it is perceived by the human agent) is a crisp indicator of
the imposter (0) and genuine (1) classes of iris pairs (P):

R(•,•): P →{0,1},

In concordance with the terminology introduced in [13], the
function R (Fig. 1.a) will be referred to as the prototype
recognition function and it is a crisp concept. The goal of
designing automated iris recognition systems is to find fuzzy
approximat ions f-R for the prototype recognition function R, as close as possible to R. Such an approximation f -R will be
further referred to as a fuzzy recognition function . The fuzzy
approximations f -R obtained by applying automated iris
recognition methods are of the same types a s those presented
in Fig. 1.b (an excellent approximation, [7]), Fig. 1.c, Fig. 2.a,
Fig. 2.b (very good approximations, [8]), Fig. 10 in [4] (good
approximation), Fig. 1.b – Fig. 1.c and Fig. 4.a – Fig. 4.c in [6]
(good approximations), wh ere the marks (good, very good,
excellent) were given using as a reference the result obtained
in an approach considered nowadays as being the “state of the
art” in iris recognition (and marked here as “good
approximation ” [4]).
B. Why Crisp , Why Fuzzy?
In the case in which the recognition is made using artificial
agents and good quality eye images, the fact that the
approximations f -R depart from the prototype R ( situation
illustrated in Fig. 1.b – Fig. 1.b.c and Fig. 4.a – Fig. 4.c from
[6] and in Fig. 10 from [4]) can not be caused by the lack of
visual acuity of the system, but only by the less intelligent
manner in which the system decides (understands) iris
similarity or dissimilarity. Practically, the artificial agent
fuzzifies the prototype R and the separation between genuine
and imposter score distributions . More inadequate and
unintelligent the image processing is, much confusion it
introduces in the biometric decision model. There are two
significant differences between the ways in which the human
agent and software agent decide th e similarity or dissimilarity
of two iris images:
– Ordinary people are not aware of the numerical reality of an
image but only of certain meanings „decoded‟ accordingly to
their experience from the chromatic variation captured in the
image. For the human agent the iris image is not a numerical
data but a set of complex knowledge about the iris texture and
the image quality (given by the technical acquisition
conditions and the posture in which the eye is captured). The
similarity/dissimilarity decision given by the human agent for
a pair of iris images is based on ad-hoc technique s of
compari ng two such sets of knowledge , techniques which are
adaptive in relation with the pair of images analyzed .
– An artificial agent makes the biometric decision using only
numerical support. Fr om its point of view , the iris image is
numeric al data in the first place. Depending on the
intelligence with which it is endowed , the artificial agent can
extract (artificial) knowledge about the numerical data, which

is usually referred to as „features‟, and further encoded in a
numeric format. For example, the binary iris code ( [1] – [4],
[6]) is a binary encoding of the features extracted from a uint8
(8-bit unsigned integer) iris image. The artificial agent
performs the comparison of two iris images indirec tly, by
comparing encoded features of the two iris images.
In short, the human agent operates in a rich knowledge space,
whereas an artificial agent usually encodes the actual
knowledge space in a relatively poor, partial and often
imprecise numeric data, in a manner very similar to los sy
compression. This is why the fuzzification is almost inherent
in the ordinary practice of automated iris recognition.
C. The problem
As it was described above, a ny simple Turing test of iris
recognition undertaken by usi ng good quality images [9]
confirms that different or identical iris images are easily and
correctly recognized by human agents and fuzzy recognized
by software agents. The cause of this happening is that the
same problem is represented (projected) in different spaces of
knowledge, or in other words, as it is intuitively illustrated in
Fig. 2, human s and software agents see the iris recognition
from different perspectives. For the human agent „genuine‟
and „imposter‟ are crisp and disjoint concepts, whereas for the
artificial agent they are fuzzy concepts which sometimes
share a confusion zone. The problem is how to reconcile these
two different views that humans and artificial agents have on
iris recognition. The solution is to find a suitable defuzzifica –
tion of the imposter and ge nuine score distributions which
guarantees that the fuzzy (and consequently the crisp )
concepts „genuine‟ and „imposter‟ are disjoint, the
appartenence of a pair of irides to these fuzzy or crisp sets
being , in this case, mutually exclusive events .
D. Related Works
The papers investigating logical aspects of iris recognition or
logical aspects of biometry in general are indeed very few.
The situation when a pair of irides ambiguously belongs to
both imposter and genuine fuzzy sets is investigated in [8]. It
is shown there that in such case, artificial understanding of
iris recognition experimental data is logically inconsistent. It
is the case of wolf -lamb pai r discussed in [12].
It is not the first time when we say it, what iris recognition
really is and how different providers of biometric solutions
compete to each o ther are two very different things. Still, due
to this competition a lot of commented experimental data was
published ( [5], for example) , all of them together
involuntarily proving that Equal Error Rate ( EER ) is a crisp
concept only in theory. The negative aspect of this
competition is the fact that a lot of resources were invested to
minimize EER value without a preliminary proper
investigation of the suitable means of doing that, but as
illustrated in Fig. 1.a and Fig. 1.d, the real improvement of iris
recognition technology depends on finding an d accepting a
major change of perspective which implicitly leads to EER
minimization. The first steps in this directi on were undertaken
with very good results in [8] and [7] (see Fig. 2.a, Fig. 2.b) by
defining and simulating Intelligent Iris Verifier (IIV)
(a)

(b)

(c)

(d)

Fig. 1. (a) – The crisp geometry (0 -1) of a biometric problem which is
decidable in a consistent binary logic. (b) – The f -geometry of a biometric
problem which is decidable in a fuzzy but still consistent binary logic. (c)
The f -geometry of a biometric problem decidable in a fuzzy binary logic with
very weak -confused fuzzy logical values. (d) – Viewing biometric decisions
in iris recognition from different perspective: hypothetical genuine (I”) and
imposter (D”) fuzzy clusters in real world as perceived by the hum an agent
(I‟, D‟) and by the software agent (I, D) .

and IIV Distributed System (IIVDS) . However, the artificial
(automated) understanding of the experimental data obtained
in these simulations proved to be quite a difficult but
rewarding task . The last section of th is paper shows that
clarifying the logical model of iris recognition allowed us to
define iris recognition theory and the problem of designing
improved iris recognition systems as classical problems of
system identification [10]. In this perspective, designing iris
recognition systems means identifying possible variables
([10], type I.a structure identification problem ), relevant
variables ( [10], type I.b structure identification problem),
input -output relation as a collection of fuzzy if -then Sugeno
rules ([10], type II.a structure identification problem) , the
partitioning of the premise space ( [10], type II.b structure
identification problem) , and doing all of these accordingly to
the results of a Turing test (Fig. 1.a) and in a logically
consistent manner .
The experimental data used i n this paper is obtained in [8]
and illustrated in Fig. 2.a and Fig. 2.b. The reason for using
these data is that the impost er and genuine score distributions
obtained in this case are much closer to the original
recognition prototype function R (Fig. 1.a) than those
obtained in other approaches.

II. A FUZZY 3-VALENT DISAMBIGUATED MODEL OF IRIS RECOGN ITION

When „genuine‟ and „imposter‟ (pairs / comparisons) are
fuzzy concepts / sets that share a (narrower or a wider)
confusion zone, there are elements of vocabulary (irides / iris
codes / digital identities) which ambiguously belong in both
of them. Such a 2 -valent fuzzy model of iris recognition is
ambiguous and logically inconsistent [8]. Disambiguation is
achieved by introducing a third fuzzy set, namely the fuzzy
EER interval (f -EER) as a separator between the „genuine‟
and „imposter‟ fuzzy sets which in this way become disjoint,
the appartenence of a recognition score to them being , in this
case, mutually exclusive events.
A. A Practical Example
Disambiguation is a matter of system calibration and design
(a type II.b st ructure identification problem, [10]) which must
be carried out with respect to the desired FAR / FRR (False
Accept / Reject Rate) specification . For instance, let us
consider the requirement that a system must have very low
FAR and FRR rates, 1E -10 to be more precise. Let us
consider that we know a way to pessimistically estimate the
FAR and FRR for scores where experimental data is not
dense enough, or is completel y missing, as POFA and POFR
(Pessimistic Odds of False Accept / Reject) . Hence, the
recognition of identical / different irides will take place for all
similarity scores t for which POFA(t)<1E -10, respectively for
which POFR(t)<1E -10. Hence, the fuzzy EER interval
(f­EER) is determined as (n,p), n =POFA­1(1E-10),
p=POFR­1(1E-10)) which for the case considered is well
approximated by the interval (0.3725, 0.55). Because FAR
and POFA are both decreasing functions, and because FRR
and POFR are both increasing f unctions (with respect to the

(b)

(a)

Fig. 3. (a) – The f -geometry of a biometric problem decidable in fuzzy binary
logic with very weak confused logical values. (b) – Zoom within FAR -FRR
curves corresponding to the iris recognition tests undertaken in [8]

TABLE 1: THE FUZZY 3-VALENT DISAMBIGUATED MODEL OF IRIS
RECOGNITION APPLIED ON A PARTICULAR CASE [8]
User attempt :
Negative (1) or Positive (2)
Claim System response quantified as recognition score: [0.55,1] –
positive claim accepted , [0,0.3725] – positive claim rejected ,
[0.3725, 0.55] – uncertainty interval (f-EER).
Score Decision / Interval Decision / Meaning
1 I’m not X.
Decline an enrolled
digital identity
(negative claim) . t,
recognition /
similarity
score I ≡ [0.55, 1] False / Claim Rejected
O ≡ (0.3725, 0.55) Uncertain / Claim Repeat
D ≡ [0, 0.3725] True / Claim Accepted
2 I am X.
Claim an enrolled
digital identity
(positive claim) . I ≡ [0.55, 1] True / Claim Accepted
O ≡ (0.3725, 0.55) Uncertain / Claim Repeat
D ≡ [0, 0.3725] False / Claim Rejected
similarity score), the comparisons scored in [0, 0.3725] and
those scored in [0.55, 1] will be recognized by the system as
being imposter / genuine comparisons, respectively. Hence,
fuzzy decisions of the system can be encoded in three fuzzy
values, I (Identical), D (Different) and O (Otherwise),
corresponding to the partitioning ( [10], type II.b structure
identification problem) of the premise space (irides / iris
codes / digital identities) as preimages of three intervals
through a fuzzy recognition function f -R (Table 1, Table 2).
Hence in a fuzzy 3 -valent disambiguated model of iris
recognition there are three kinds of iris code pairs: genuine,
imposter a nd undecidable . The iris codes of an undecidable

pair are, in fact, unenrollable , or else, the restriction of the
system to the set of these pairs is logically inconsistent. This
shows that in order to preserve logical consistency, each time
when an iris code attempt to enroll in the system , one -to-all
(one candidate iris code to all previously enrolled iris codes)
comparisons are mandatory.

III. 8-VALENT ISOMORPHIC BOOLE AN ALGEBR AS WHICH CAN HOST A FUZZY 3-
VALENT DISAMBIGUATED MODEL OF IRIS RECOGN ITION

The Boole an algebr a I3 = ((Ø, [0,n] , (n,p) , [p,1] ), U, ∩, C)
generated by the empty set, imposter, genuine and incertitude
intervals with the reunion, intersection and complement,
induces a formal logic that hosts fuzzy values I, O, and D.
Still, this characterization is rathe r symbolic than
computational. From a set of Boole an algebr as isomorphic to
I3 we will choose one that can be expressed computationally
(arithmetically). The candidates to choos e from are illustrated
in Table 5, where the empty fuzzy value (E) encodes (the
empty set as the set of) the impossible state s/decision s of a
logically consistent biometric system : the state s/decision s in
which the system accepts both the positive and the negative
claim regarding a n iris code candidate and an identity is not
observable in a logically consistent biometric system .
An algebr a isomorphic to I3 is S3=((E, I, O, D), U, ∩, C) –
the Boole an algebr a of strings (unsorted an d with no
repetition) generated through concatenation, intersection and
complementary by the empty string E=‟ ‟ and the distinct
characters „I‟, „O‟, „D‟, corresp onding to the modal values I,
O and D.
The following function :

is the isomorphism between the algebra of binary codes
B3=({0,1}3, And, Or, Neg) and S3=((E, I, O, D), U, ∩, C),
where:
– , , is the canonical bas is of R3,
– function „ va‟ returns in binary digit the truth values
of its argument,
– And, Or and Neg are the bit -wise logical operators.
In its turn, the Boole an algebr a B3=({0,1}3, And, Or, Neg) is
isomorphic to the Boole an algebr a V3=({0,1}3, ⨁,*, !) of the
vectors on the unit cube in R3, (Fig. 4.c), generated by the
vectors of the canonical basis of R3, where :
– „+‟, „ -‟ are the sum and the difference of two vectors,
– „*‟ extracts the common (dependent) part of two
vectors with respect to the canonical basis,
– ⨁ is defined by the relation a ⨁ b = a + b – a*b,
– „!a‟ is the difference from vector „a‟ to the main
diagonal of the unit cube.
The following function :

transforms the algebra B3
= ({0,1}3, And, Or, Neg) in the
algebra of modulo 8 integers denoted (Z8, P, S, N) where:
– N(a) is the complement of „a‟ relative to 7, –
TABLE 2: BIOMETRIC DECISION IN A FUZZY 3-VALENT DISAMBIGUATED
MODEL OF IRIS RECOGN ITION
I
[0.55, 1]
Genuine
pairs False Accept Rate:
FAR(0.55) ≈ POFA(0.55) = 1E -10.
True Accept Safety:
1-FAR(0.55) ≈ 1 -POFA(0.55) = 1 –(1E-10).
O
(0.3725, 0.55)
Undecidable
pairs
INCERTITUDE Genuine Discomfort Rate :
FRR(0.55) ≈ 2.7E-4;
Imposter Discomfort Rate :
FAR(0.375) ≈ 1.42E -4.
–––––––––––––––––––
Total Discomfort Rate :
4.12E -4 DISCO MFORT SECURIT Y
D
[0, 0.3725]
Imposter
pairs False Reject Rate:
FRR(0.3725) ≈ POFR(0.3725) = 1E -10.
False Reject Safety:
1-FRR(0.3725) ≈ 1 -POFR(0.3725) = 1 –(1E-10).

TABLE 3: BINARY ENCODING FOR I NPUT AND FOR
THE BIOMETRIC DECISI ON

P Positive Claim “I am X” , N Negative Claim “I am not X”

A‟ Accepted Input , R‟ Rejected Input

TABLE 4: INPUT -OUTPUT RELATION IN A FUZZY 3-VALENT DISAMBIGUATED
MODEL OF IRIS RECOGN ITION
(1)
Input (2)
Input
encoding (3)
Similarity
score (4)
Fuzzy /Modal
encoding (5)
Output (6)
Output
encoding
Positive /
Negative
claim P / N p ≤ t ≤ 1 I Accept ed P,
Rejected N PA’&NR’
n < t < p O Rejected P,
Rejected N PR’&NR’
0 ≤ t ≤ n D Rejected P,
Accepted N PR’&NA’

– P(•,•) and S( •,•) are defined in Table 6 and in column
[c] of Table 5.

The table of additive operation S (supremum) of the Boole an
algebr a (Z8, P, S, N) can be read also from Fig. 4.a, if it is
taken into account that for each pair of modulo 8
integers :
– Or and then ,
– Or and are comparable in the partial order of the
Boole an algebr a (Z 8, P, S, N) and then
,
– Or and are not comparable in the partial order of
the Boole an algebr a (Z 8, P, S, N) and then their
„sum‟ is the first (the lowest) common successor, i.e.
≤ ≤

The table of multiplicative operation P (infimum) of the
Boole an algebr a (Z 8, P, S, N) can be read also, from Fig. 4.a, if
it is taken into account that for each pair of modulo 8
integers:
– Or and then ,
– Or and are comparable in the purpose of partial
order of the Boole an algebr a (Z 8, P, S, N) and then:

– Or and are not co mparable in the partial order of
the Boole an algebr a (Z 8, P, S, N) and then their

product is the last (the highest) common predecessor,
i.e.:

The t otally ordered subsets of partially ordered algebra
(Z8, P, S, N) are represented in Fig. 4.d against two
coordinates of entropy: the vertical coordinate encodes
entropy as natural (arithmetic) order of modulo 8 integers,
whereas the horizontal coordinate encodes what we called the
absolute entropy of modulo 8 integers w ith respect to the
product operation within the Boole an algebr a (Z 8, P, S, N).
The absolute entropy of an element „a‟ with respect to an
operation „P‟ within a Boole an algebr a B is defined here as
the number o distinct elements of the set {P(a, b) | b B}:
E(a) = card( unique( {P(a, b) | b B})).
Analyzing the order in the Boole an algebr a (Z8, P, S, N)
helped us figuring that the table of product operation is block –
recursive (in three steps with blocks of dimension 1, 2, and 4),
fact which further allowed the determination of an explicit
formula for product calculus:

P a , b = c c = 2n aM2n+1 2n bM2n+1 2n ,2
n=0
where M stands for modulo and the operator „ ‟ is considered
to be a logico -arithmetical operator which return s logical
value s as natural numbers 0 and 1 . The explicit formula of the
sum operation calculus has been further defined through
complementarity:

and ver ified against data within Table 6 and Fig. 4.d.
At this stage the following question appears: in what
formal language are well -formed the strings that define the
product and the sum within the Boole an algebr a (Z8, P, S, N)?
They are well -defined in a formal language obtained by
overloading Peano Arithme tic with the first degree logic of
the propositions about the natural order between modulo 8 –
integers („a b‟, „a ≤ b‟). In this language, the expression
(aM2n+1 2n) which interferes in the calculation of P and S
returns the natural value 0 or 1 accordingly with the true
value associated to the inequality . Hence, it has been
illustrated that, in order to describe the 8 -valent fuzzy logic of
iris recognition, Peano Arithmetic must be extended with
logical support. Of course, we could see this in a reversed
perspective: since it is normal that the arithmetic to describe
the Boole an algebr as generated by finite subsets of natural
numbers, it is also normal to consider that the study of a fuzzy
logic model of iris reco gnition has lead us to an improved
model of arithmetic. However, this paper is not concerned
with establishing these pure theoretical aspects. If the
arithmetic should or shouldn‟t be overloaded with logical
support, it is a question for theoreticians. Her e in this paper
the overloaded model of arithmetic is called Peano -2
Arithmetic and it is used t o compute the operations within
8­valent Boole an algebr a of iris recognition (Z8, P, S, N).
TABLE 5: ENCODING THE OUTPUT I N A FUZZY 3-VALENT DISAMBIGUATED
MODEL (ISOMORPHIC REPRESENT ATIONS OF THE BOOLE AN ALGEBR A I3)
[a]
Symbolic
encoding [b]
Binary
labels [c]
Octal
labels [d]
Octal
labels [e]
Binary
labels [f]
Meaning
IOD 111 7 7 111 PR‟| NR‟
OD 011 3 6 110 PR‟| NA‟
IO 110 6 5 101 PA‟| NR‟
ID 101 5 4 100 PA‟| NA‟
O 010 2 3 011 PR‟& NR‟
D 001 1 2 010 PR‟& NA‟
I 100 4 1 001 PA‟& NR‟
E 000 0 0 000 PA‟& NA‟

(a)
(b)

(c)
(d)

Fig. 4. Isomorphic representations of the Boole an algebr a S3: (a) – Boolean
algebra B3; (b) – Boolean algebra I3; (c) – Boolean algebra V3; (d) – Boolean
algebra (Z8, P, S, N).

TABLE 6: PRODUCT AND SUM WITHI N BOOLEAN ALGEBRA (Z8, P, S, N)

S 0 1 2 3 4 5 6 7 E
0 0 1 2 3 4 5 6 7 8
1 1 1 3 3 5 5 7 7 4
2 2 3 2 3 6 7 6 7 4
3 3 3 3 3 7 7 7 7 2
4 4 5 6 7 4 5 6 7 4
5 5 5 7 7 5 5 7 7 2
6 6 7 6 7 6 7 6 7 2
7 7 7 7 7 7 7 7 7 1

P 0 1 2 3 4 5 6 7 E
0 0 0 0 0 0 0 0 0 1
1 0 1 0 1 0 1 0 1 2
2 0 0 2 2 0 0 2 2 2
3 0 1 2 3 0 1 2 3 4
4 0 0 0 0 4 4 4 4 2
5 0 1 0 1 4 5 4 5 4
6 0 0 2 2 4 4 6 6 4
7 0 1 2 3 4 5 6 7 8

(a) (b)

IV. BACK TO THE IRIS RECOGNITION PRACTICE
First of all, the artificial understanding of the experimental
data obtained in iris recognition tests illustrated in Fig. 3 is
expressed in the following theorem (N. Popescu -Bodorin,
V.E. Balas, [8]):

Theorem 1: The correspondence
: E D O I OD ID IO IOD
0 1 2 4 3 5 6 7
achieve s the defuzzi fication of the fuzzy sets I, D and O as
the elements 4, 1 and 2 from the 8 -valent Boole an algebr a
(Z8, P, S, N), where:

and „M‟ stands for modulo.

Secondly, the artificial understanding of the experimental data
obtained in iris recognition tests illustrated in Fig. 3 is
logically consistent and reflected in the following theorem:

Theorem 2: System structure and consistency of the fuzzy
3-valent disambiguated model of iris recognition (N.
Popescu -Bodorin):
Let I CP a set of iris code pairs fuzzy assigned to the fuzzy
sets D, O and I by a recognition function f-R as in Fig. 3.b and
Table 4. Let EICP be the set of enrollable iri s code pairs,
EICP = f-R­1(I) U f-R­1(D), i.e. the support of the fuzzy
concepts I and D as they appear through the fuzzy recognition
function f-R, and let f-K = (EICP, f-R, {I, D}) the fuzzy
formal theory of iris recognition defined over the vocabulary
of enrollable iris code pairs (a restriction of the fuzzy 3 -valent
disambiguated model to the vocabulary of enrollable pairs) .
Then the fuzzy theory f -K is f-consistent as a theory of
recogniti on (i.e. its defuzzified form is a consistent theory of
recognition ).
Proof :
Let us consider the following partitioning of the input space:

Input
(octal) Partitioning
(octal) State
(octal) Output
(octal) Output
(binary)
ICP ≡ 7
IOD ≡ 7 EICP ≡ 5 D ≡ 1 1 0
I ≡ 4 4 1
f-K theory
K theory
UICP ≡ 2 O ≡ 2
Fuzzy 3 -valent disambiguated model of iris recognition

P(4,1) = 0, and S(4,1) = 5, or equivalently , the fuzzy sets 4 and
1 are mutually exclusive and complementary to each other in
the output space , which in its turn generates a subalgebra
({0,4,1 ,5}, P, S, N) of (Z 8, P, S, N). Conse quently, the fuzzy
theory f-K = (EICP, f -R, {I, D}) illustrated in Fig. 3.b can by
defuzzified as a crisp theory K = (EICP, R, {1, 0}) like that
illustrated in Fig. 1.a. In other words, in the vocabulary of f -K
theory there is no support for the concept of wolf -lamb [12]
pair (there is no support for impersonation). □ V. CONCL USION
Maintaining consistency of a biometric system is a matter
of partitioning the input space into two classes: enrollable and
unenrollable pairs. Consistent enrollment is mandatory in
order to preserve system consistency.
The fact that the fuzzy 3 -valent disambiguated model of
iris recognition is incomplete (there are, indeed, undecidable
pairs in the input space) is reflected in the user discomfort
(undecidable pairs must be discarded and the user must repeat
the authentication at tempt) but also in the system safety .
As a theory of recognition, f -K is f-consistent and
complete: for any pair of its vocabulary there is only one
biometric decision to be given, specifically the correct one.
Consistency in iris recognition is not achievable just by
setting a thresho ld and doing one -to-one comparisons. It can
be guaranteed only by establishing a safety band (f -EER
interval) and practicing one -to-all comparisons for each
authentication / identif ication / enrollment attempt .
ACKNOWLEDGMENT
The authors would like to thank Professor Donald Monro (Dept. of
Electronic and Electrical Engineering, University of Bath, UK) for
granting the access to the Bath University Iris Image Database.
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