Abstract This paper shows that the k-means [601312]

Abstract — This paper shows that the k-means
quantization of a signal can be interpreted both as a crisp
indicator function and as a fuzzy membership assign ment
describing fuzzy clusters and fuzzy boundaries. Com bined
crisp and fuzzy indicator functions are defined her e as
natural generalizations of the ordinary crisp and f uzzy
indicator functions, respectively. An application t o iris
segmentation is presented together with a demo prog ram.
Keywords — circular fuzzy iris ring, circular fuzzy limbic
boundary, combined crisp indicator function, combin ed
fuzzy indicator function, fast k-means quantization , fuzzy
clusters, fuzzy boundaries, iris recognition, iris
segmentation, k-means.
I. INTRODUCTION
ELATIVELY few iris segmentation techniques have been
reported in the last two decades. In the classical iris
segmentation procedures, like those in Wildes’s [1] and
Daugman’s approaches [2]-[10], iris segmentation me ans
fitting (nearly) circular contours by solving 3-dim ensional
optimization problems to find a radius and two center
coordinates via gradient ascent or by using edge detectors
and Hough transform [11] or by iterating active con tours
[9,10]. For more details, we would like to refer to Bowyer
et al. for a survey of iris recognition [12].
All of the previous iris segmentation approaches ar e 3-
dimensional optimization problems. They also assume that
the segmentation is done before iris unwrapping. In this
context two questions must be answered:

Is it possible to formulate the finding of the limb ic
boundary as a 1-dimensional optimization problem or as
a search in a 1-dimensional parameter space? If yes ,
would the resulting iris segments be accurate enoug h to
guarantee strong recognition results?

We affirmatively answer these questions by giving, in the
same time, four reasons to work with circular
approximation of the iris:
i) It is clear that only by knowing the center coordin ates

Nicolaie Popescu-Bodorin is with the Mathematics an d Computer Science
Department – ‘Spiru Haret’ University of Bucharest (România), where he
teaches Computational Logic and Artificial Intellig ence Labs. As a PhD
Candidate at Mathematics and Computer Science Depar tment – University of
Pitesti (România), he works in the field of Digital Signal/Image Processing and
specifically on Iris Recognition related subjects. Correspondence address:
Nicolaie Popescu-Bodorin, OP 19, CP 77, Bucharest 3 , RO. and by unwrapping the iris region in the first plac e, limbic
boundary finding could become a 1-dimensional searc h for
a radius i.e. a search for a line within the unwrap ped iris
region. Consequently, assuming a rough approximatio n of
the actual iris as a circular ring concentric with the pupil is
a choice [13]-[15] which guarantees an affirmative answer
to the first question from above.
ii) An anatomic argument for using circular approximati on
of the iris is that since the pupillary boundary is nearly
circular, there must be a circular concentric iris ring
controlling the pupil movements. Such a circular ir is ring is
expected to play the most important role in iris re cognition,
despite the fact that it appears to be a rough appr oximation
of the actual iris.
iii) A system requirement sustaining the use of concentr ic
circular iris ring is that the segmentation routine must be
fast and energy-efficient. Nearly lossless unwrappi ng of the
iris can be computed using a polar or a bipolar coo rdinate
transform, depending on the type assumed for the ir is:
concentric or eccentric circular ring. The latter i s
computationally more expensive than the former beca use
the eccentricity varies from a sample to another an d
consequently, one bipolar mapping must be (re)compu ted
for each sample (eye image). When the iris ring is assumed
to be concentric, the polar mapping is computed onl y once
for all samples, during program initialization.
iv) At last, but not the least, a practical argument fo r using
circular approximation of the iris is given by the quality of
the recognition results presented in [13], [14] and by the
iris segmentation results illustrated in [15].
A harder question regarding the Circular Fuzzy Iris
Segmentation procedure [13]-[15] is the following: why the
various operations (within the segmentation procedu re)
are needed or expected to work well?
The short answer to the above question came as a re sult
of our experimental works and is stated here as a p rinciple:
detecting a certain feature of a signal (of an imag e) is
always a matter of finding a suitable quantization space
and a suitable quantization function to enhance the target
feature against ‘unwanted noise’ (against the
surrounding neighbours in the feature space). In th e best
case scenario feature discovery would be nothing mo re
than a well chosen binary encoding (compression) of the
feature space.
Another four principles of k-means optimal signal A Fuzzy View on k-Means Based Signal
Quantization with Application in Iris
Segmentation
Nicolaie Popescu-Bodorin, IEEE Member,
http://fmi.spiruharet.ro/bodorin/
R

quantization can be found in [16] 1 together with the Fast k-
Means Image Quantization algorithm (FKMQ).
A longer and more detailed answer to the last quest ion
from above will be given further in this paper.
II. COMBINED CRISP AND FUZZY INDICATORS OF A DISJOINT
REUNION
Generally speaking, a segmentation technique workin g on
discrete signals is a semantic operator encoding th e input
signal using a finite set of labels (symbols) which are
somehow meaningful in human understanding of the in put
signal. The first difficulty in interpreting a segm entation as
being fuzzy is the lack of the instruments that cou ld enable
us to view the result of a segmentation as a crisp or as a
fuzzy membership function defined from the input si gnal to a
collection of segments encoded as a list of arbitrary
symbols , possibly non-numeric , and more often found
outside [0, 1] interval. This section is meant to work
around this issue by extending the definition of th e ordinary
crisp and fuzzy indicator functions to cover the ab ove
described situation.
In fuzzy set theory [18], a membership function tha t only
takes binary values is called a crisp indicator fun ction. We
extend the meaning of this definition by making the
following considerations: a crisp indicator is, in fact, the
ordinary indicator function of an ordinary sub-set within a
set:
)Aa (icallog) a (I ,Xa};1 , 0 {X:I A A ∈ = ∈∀ → (1)
For any sub-set A of X, AAX ∪= (where Adenotes the
complement of A in X), hence we may consider that t he
crisp indicator of A is nothing more than an encodi ng (in
two symbols) of a disjoint cover of X containing tw o sets:
A and A (regardless the nature or the values of those two
symbols and the nature of the sets A and X). Conseq uently,
it is naturally to define combined crisp indicator of a
disjoint reunion :
Un
1jjAX
==, (2)
as being the sum:
∑
==n
1jA xjI * j CCI , (3)
or more generally, as follows:
kn
1jA Xk s) a (I * jS) a (CCI,Aa, n , 1kj=



= ∈∀ =∀ ∑
=, (4)
where n , 1kk}s {S== is a sequence of distinct symbols. It

1 In [16], some considerations regarding the speed o f FKMQ are already
outdated by the newer and faster implementation [17 ]. Also, the iris
segmentation procedure proposed in [16] was tempora rily abandoned for the
following reason: despite its accuracy in finding t he iris segment available in the
eye image (see Fig.6-7 in [16]), ‘guessing’ the bes t eccentric circular ring that
matches the available iris segment proved to be a t ough challenge (an ill-posed
inverse problem with 6 variables) and therefore, wa s impossible to formulate
recognition results based on that segmentation tech nique. Still, future solutions
to this problem are not excluded. On the other hand , it must be mentioned that
the difference between the segmentation procedure p roposed in [16] and
Circular Fuzzy Iris Segmentation [13]-[15] is that the latter searches directly
for the line approximating limbic boundary in the u nwrapped iris region (it
searches for a line number in a different and small er feature space). means that a combined crisp indicator of a disjoint reunion
is unique up to a bijective correspondence between the
sequences of symbols that are used to encode the
memberships to each set within the reunion. Hence, if X is
restricted to R, the combined crisp indicator of a disjoint
cover of X is exactly the equivalence class of all step
functions that can be defined using the sets of tha t cover. If
X is a discrete signal, then we talk in terms of di screte step
functions. Consequently, any discrete step function (and in
particular, any k-means quantization of a discrete signal) is
equivalent (in the above defined sense) to a combin ed crisp
indicator (3). Therefore, it doesn’t really matter what
symbols (or values) are used to encode the crisp in dicator
function. Chromatic k-means centroids and cluster i ndices
{1,…,n} are both equally suitable to encode a crisp
indicator function describing the k-means clusters.
The ordinary crisp indicator of a set is unique (up to a
bijection, as described above), but the ordinary fu zzy
membership assignment functions are not. The combined
fuzzy indicators of a disjoint reunion inherit this property
and they are defined here as follows: given a combined
crisp indicator of the form (3), any monotone function
XCFIsatisfying the relation:
[ ]X XCCICFI =, (5)
where ] [⋅denotes the integer part function, is a combined
fuzzy indicator of the given disjoint reunion (2). In other
words, the function:
( )X X X CCICFIabs*2FIB − = (6)
is an ordinary fuzzy indicator of the boundaries between
the sets of the reunion (2). Naturally, the combined crisp
and fuzzy indicators (3, 5) of a disjoint reunion (2) and the
ordinary fuzzy indicator of the boundaries (6) form an
interdependent triplet.
III. CIRCULAR FUZZY IRIS RING AND CIRCULAR FUZZY
LIMBIC BOUNDARY
Finding the pupil [11],[13],[14] enables us to unwr ap a
circular pupil-concentric region of the eye image ( Fig.1.a)
in polar coordinates (Fig.1.b), to localize the lim bic
boundary in the rectangular unwrapped eye image (Fi g.1.c),
and to obtain an iris segment as in Fig.1.e.

Circular Fuzzy Iris Segmentation (CFIS, N. Popescu- Bodorin):
INPUT: the eye image IM;
1. Apply RLE-FKMQ Based Pupil Finder procedure;
2. Unwrap the eye image in polar coordinates (UI – Fig.1.b);
3. Stretch the unwrapped eye image UI to a rectangl e (RUI – Fig.1.c);
4. Compute three column vectors: A, B, C, where A a nd B contain the
means of the lines within UI and RUI, respectively. C is the mean of
the lines within the [A B] matrix;
5. Compute P, Q, R as being 3-means quantizations o f A, B, C;
6. For each line of the unwrapped eye image count t he votes given by P,
Q and R. All lines voted at least twice as members of an iris band are
assumed to belong to the actual iris segment. Find limbic boundary
and extract iris segment (Fig.1.d, Fig.1.e);
OUTPUT: pupil center/radius, line number of the cir cular fuzzy limbic
boundary, circular fuzzy iris segment;
END.

Fig.1. Iris segmentation stages (CFIS)

Fig.2. Fuzzy iris segment and fuzzy iris boundaries

Fig.3. Circular Fuzzy Iris Segmentation Demo Progra m,
http://fmi.spiruharet.ro/bodorin/arch/cffis.zip
Fig.1 shows iris segmentation stages. The transform from
Fig1.a to Fig.1.b is a lossless pixel-to-pixel tran scoding.
The unwrapped iris region is further stretched and
interpolated in order to obtain rectangular unwrapp ed iris
(RUI – Fig.1.c). All together, Fig1.a-c illustrates a three-
step reversible polar mapping (lossless pixel-to-pi xel polar
transcoding, stretching and interpolation).
One advantage of using such a polar mapping is that the
original pixels within the initial circular iris ri ng can be
traced at any time in the unwrapped versions of the iris.
On the other hand, the extent of the black regions in
Fig.1.b is a measure of the difference between an i deal
polar mapping (in continuous geometry) and a practi cal
lossless pixel-to-pixel polar mapping defined for d igitized
images.
The third advantage is the fact that, here, the inf luence of
the pupil dilation on the recognition performance
(documented in [19]) is explained and illustrated
graphically: comparison of two irides means to over lap
two trapezes through an elastic deformation. At lea st
because of the collarette, the deformation in the r adial
direction is far from uniform. This is the reason f or which
our Gabor Analytic Iris Texture Binary Encoder [13] ,[14]
parses iris features only in the angular direction.

Fig.2 shows what happens to the vectors A, B and C at
steps 4-5 of the CFIS procedure: behind the combine d crisp
indicator function (crisp membership assignment) of a 3-
means quantization (Fig.2), there are fuzzy members hip
assignment functions defined from the set of lines within the
rectangular unwrapped iris area (RUI-Fig.1) to the pupil, to
the iris, to the area outside the iris and even to the iris
boundaries. The area delimited between the fuzzy ir is
boundaries is a fuzzy iris band. Its preimage throu gh the
polar mapping is a circular fuzzy iris ring.
Three fuzzy iris bands are determined using vectors A, B,
C. The final result is computed evaluating the odds that the
lines within the unwrapped iris area belong to the actual
iris segment. This is done in step 6 of the CFIS pr ocedure
by counting the votes received for each line within the
unwrapped iris area as a member of a fuzzy iris ban d.

The most important aspect of the CFIS procedure is that
it performs three searches within a 1-dimensional s ignal
whose length equals the radius of the initial iris circular
region (Fig.1.a). For example, the dimension of the
parameter space which is needed to be searched in o rder to
find the limbic boundary in Fig.2 is: 3*112=336 pix els. On
the other hand, using a Hough accumulator with 343= 7 3
cells to extract a circle (limbic boundary) from th e edges of
an eye image of dimension 240×320 pixels will be to tally
insufficient, but still computationally more expens ive.
IV. THE DEMO PROGRAM
Circular Fuzzy Iris Segmentation demo version [15] is
currently implemented in Matlab and can be tested a gainst
the entire Bath University Iris Database (free vers ion [20])
which contains 1000 eye images. Basically, the demo
program is an implementation of the CFIS procedure with
some speciffic practical adaptations: fault-toleran ce,

timing, graphical display, etc.

CFIS demo program enables pupil localization in 12
frames per second and limbic boundary localization in 5
frames per second, for eye images of dimension 240x 320
pixels. It also leads to the following iris segment ation error
rates:
– Total number of failure cases: 6;
– Pupil finder failures: 1;
– Limbic boundary detection failures: 5.

The demo program proves that iris segmentation can be
treated as being a 1-dimensional optimization probl em if
there is enough accurate morphological information stored
as chromatic variation.
Another important aspect is that the segmentation r esults
obtained by applying CFIS procedure are confirmed b y the
recognition results in [13], [14].

V. CONCLUSION
This paper introduced combined crisp and fuzzy
indicators of a disjoint reunion which are meant to allow a
unified dual description of the k-means quantizatio n as a
crisp and as a fuzzy entity, respectively.
Both of them are instruments that enable us to view the
result of a segmentation as a crisp indicator defin ed from
the input signal to a collection of segments encode d as a list
of arbitrary symbols (possibly non-numeric , and more
often found outside [0, 1] interval), but also as a fuzzy
membership function.
A practical example is the case of Circular Fuzzy I ris
Segmentation procedure in which combined crisp and fuzzy
indicators are encoded in [1, 3] interval (Fig.2).

ACKNOWLEDGMENT
I wish to thank my PhD Coordinator, Professor Luminita
State (University of Pitesti, RO) – for the comments,
criticism, and constant moral and scientific suppor t during
the last two years, and Professor Donald Monro
(University of Bath, UK) – for granting the access to the
Bath University Iris Database.
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