short overview here SPIRU HARET University of Bucharest

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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


    5. What About k-Means Quantization?
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.




             Thank you for your attention!

				
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