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```					17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

A Fuzzy View on
k-Means Based Signal Quantization
with Application in Iris Segmentation
Nicolaie Popescu-Bodorin, IEEE Member,

Artificial Intelligence & Computational Logic Laboratory,
Mathematics and Computer Science Department,
‘Spiru Haret’ University of Bucharest, ROMANIA,
bodorin☺ieee.org,
http://fmi.spiruharet.ro/bodorin/
November 2009
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

1. Outline
Here we show that k-means quantization of a signal can be
interpreted both as a crisp indicator function and as a fuzzy
membership assignment describing fuzzy clusters and fuzzy
boundaries.

Combined crisp and fuzzy indicator functions are defined here
as natural generalizations of the ordinary crisp and fuzzy
indicator functions, respectively.

An application to iris segmentation is presented together with a
demo program.
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

2. Why a new type of fuzzification?
We consider that a segmentation technique working on discrete
signals is a semantic operator encoding the input signal
using a finite set of labels (symbols) which are somehow
meaningful in human understanding of the input signal.

The first difficulty in interpreting a segmentation as being fuzzy
is the lack of instruments that could enable us to view the
result of a segmentation as a crisp or a fuzzy membership
function defined from the input signal to a collection of
segments encoded as a list of arbitrary symbols, possibly
non-numeric, and more often found outside [0, 1] interval.
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

3. Ordinary Crisp Indicator of a Set

If A is a subset of X, the ordinary crisp indicator of A is:

IA : X  {0,1}; a  X, IA (a)  log ical(a  A)

We may consider that the crisp indicator of A is nothing more
than an encoding (in two symbols) of a disjoint cover of X
containing two sets: A and its complement (regardless of the
nature or the values of those two symbols and of the nature
of sets A and X).
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

4. Combined Crisp Indicator of a Disjoint
Reunion
n
If X   A j                                                                      (1)
j1

it is natural to define the combined crisp indicator of X as a
linear combination of ordinary indicator functions:
n
CCI x   j * I A j                                  (2)
j1
or more generaly, as follows:              n              
k  1, n , a  A k , CCI X (a )  S   j * I A (a )   s k                (3)
 j1      j     
                
where S  {s k }        is a sequence of distinct symbols.
k 1,n
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

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.

Consequently, any discrete step function (in particular, any k-means
quantization of a discrete signal) is equivalent to a combined crisp
indicator.

Therefore, it doesn’t really matter what symbols (or values) are used
to encode the crisp indicator function. Chromatic k-means
centroids and cluster indices {1,…,n} are both equally suitable to
encode a crisp indicator function describing the k-means clusters.
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

6. Combined Fuzzy Indicator of a Disjoint
Reunion
The combined fuzzy indicator of a disjoint reunion is defined
here as follows: given a combined crisp indicator of the form
(2), any monotone function satisfying the relation:

CFI X  CCI X                       (4)
where [·] denotes the integer part function, is a combined fuzzy
indicator (combined fuzzy membership assignment). In other
words, the function:

FIBX  2 * abs CFI X  CCI X                  (5)
is an ordinary fuzzy indicator of the boundaries between the
sets of the reunion (1).
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

7. A fuzzy view on k-means quantized images – fuzzy
clusters, fuzzy boundaries

n
a) k-means quantized image: X   A j
b) Combined crisp indicator        j1
n
for k-means clusters:      CCI x   j * I A j
c) Combined fuzzy indicator            j1
for k-means clusters:       CFI X  CCI X           
d) Ordinary fuzzy indicator
for cluster boundaries: FIBX  2 * abs CFI X  CCI X 
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

8. Circular Fuzzy Iris Ring, Circular Fuzzy
Iris Boundaries
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

9. Circular Fuzzy Iris Segmentation
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

10. Where to find the demo program

Circular Fuzzy Iris Segmentation Demo Program,
http://fmi.spiruharet.ro/bodorin/arch/cffis.zip
A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

11. Where to find implementation details
[1] Nicolaie Popescu-Bodorin, Circular Fuzzy Iris Segmentation , 4th
Annual South-East European Doctoral Student Conference (DSC
2009), 6-7 July 2009, Thessaloniki, GREECE.
[2] Nicolaie Popescu-Bodorin, Searching for 'Fragile Bits' in Iris Codes
Generated with Gabor Analytic Iris Texture Binary Encoder , 17th
Conference on Applied and Industrial Mathematics ( CAIM 2009 ), 17-
20 September 2009, Constantza, ROMANIA.
[3] Nicolaie Popescu-Bodorin, Exploring New Directions in Iris Recognition
11th International Symposium on Symbolic and Numeric Algorithms
for Scientific Computing ( SYNASC 2009 ), 26-29 September 2009,
Timisoara, ROMANIA.
[4] Nicolaie Popescu-Bodorin, A Fuzzy View on k-Means Based Signal
Quantization with Application in Iris Segmentation , 17th Telecommuni-
cations Forum (TELFOR 2009), 24-26 November 2009, Belgrade,
SERBIA.
17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

Acknowledgment
I wish to thank my PhD Coordinator, Professor Luminita State (University
of Pitesti, RO) - for her comments, criticism, and constant moral and
academic support during the last two years, and Professor Donald Monro
(University of Bath, UK) - for granting me access to the Bath University
Iris Database.