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

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. A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

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. 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 3. Ordinary Crisp Indicator of a Set If A is a subset of X, the ordinary crisp indicator of A is: 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).

4. Combined Crisp Indicator of a Disjoint Reunion If (1) it is natural to define the combined crisp indicator of X as a linear combination of ordinary indicator functions: (2) or more generaly, as follows: (3) where is a sequence of distinct symbols. 17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

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. A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

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: (4) where [·] denotes the integer part function, is a combined fuzzy indicator (combined fuzzy membership assignment). In other words, the function: (5) is an ordinary fuzzy indicator of the boundaries between the sets of the reunion (1). 17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

7. A fuzzy view on k-means quantized images – fuzzy clusters, fuzzy boundaries a) k-means quantized image: b) Combined crisp indicator for k-means clusters: c) Combined fuzzy indicator for k-means clusters: d) Ordinary fuzzy indicator for cluster boundaries: A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

8. Circular Fuzzy Iris Ring, Circular Fuzzy Iris Boundaries 17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

9. Circular Fuzzy Iris Segmentation A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

10. Where to find the demo program Circular Fuzzy Iris Segmentation Demo Program, 17th Telecommunications Forum, TELFOR 2009, Belgrade, SERBIA, November 24-26, 2009

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 ), 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 ), 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), November 2009, Belgrade, SERBIA. A Fuzzy View on k-Means Based Signal Quantization with Application in Iris Segmentation

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