FUZZY MODELS
AND ALGORITHMS FOR
PATTERN RECOGNITION
AND IMAGE PROCESSING
THE HANDBOOKS
OF FUZZY SETS SERIES
Series Editors
Didier Dubois and Henri Prade
IRIT, Universite Paul Sabatier, Toulouse, ...
FUZZY MODELS
AND ALGORITHMS FOR
PATTERN RECOGNITION
AND IMAGE PROCESSING
James C. Bezdek
University of West Florida
James ...
Library of Congress Cataloging-in-Publication Data
Fuzzy models and algorithms for pattern recognition and image processin...
Contents
Series Foreword v
Preface vii
1 Pattern Recognition 1
1.1 Fuzzy models for pattern recognition 1
1.2 Why fuzzy pa...
vi FUZZY PATTERN RECOGNITION
3 Cluster Analysis for Relational Data 137
3.1 Relational Data 137
A. Crisp Relations 138
B. ...
Contents vii
4.8 Adaptive resonance models 413
A. The ARTl algorithm 414
B. Fuzzy relatives of ART 421
C. Radial basis fun...
Series Foreword
Fuzzy sets were introduced in 1965 by Lotfi Zadeh with a view to
reconcile mathematical modeling and human...
FUZZY PATTERN RECOGNITION
The core set of volumes are respectively devoted to fundamentals of
fuzzy sets, mathematics of f...
Preface
The authors Rather than compile many chapters written by various
authors who use different notations and semantic ...
xii FUZZY PATTERN RECOGNITION
Almost all algorithms have parameters that affect their
performance. Science is about quanti...
Preface xiii
pattern recognition community has used Iris in perhaps a thousand
papers for just this reason or have we?
Dur...
xiv FUZZY PATTERN RECOGNITION
for you, the pattern recognition aficionado is this: pattern
recognition is data, and not al...
Preface xv
We also want to express our thanks to Andrea Baraldi, Alma Blonda,
Larry Hall, Lucy Kuncheva and Thomas Runkler...
1 Pattern Recognition
1.1 Fuzzy models for pattern recognition
There is no lack of definitions for the term pattern recogn...
FUZZY PATTERN RECOGNITION
Regardless of how it is defined, there are two major approaches to
pattern recognition, nunaeric...
INTRODUCTION TO PATTERN RECOGNITION
^ Notation Vectors are boldface (x, v, V, etc.); x e 5RP is the px 1
matrix x = (x, x ...
FUZZY PATTERN RECOGNITION
One of the most basic structures in pattern recognition is the label
vector. No matter what kind...
INTRODUCTION TO PATTERN RECOGNITION
The set N , a piece of a hyperplane, is the convex hull of N . The
vector y = (0.1, 0....
FUZZY PATTERN RECOGNITION
||x - V||Q-I = Vfx - v)^D ^(x - v) Diagonal, A=D-1
X - V M -i=S x-v)'M"nx-v) Mahalanobis, A=M 1
...
INTRODUCTION TO PATTERN RECOGNITION
A classifier is any function D:?^^ h-> N . The value y = D(z) is the
label vector for ...
8 FUZZY PATTERN RECOGNITION
scattered because feature analysis is very data and problem-
dependent, and hence, is almost a...
INTRODUCTION TO PATTERN RECOGNITION 9
validate volumetric and shell type clusters; and this section
concludes with a discu...
10 FUZZY PATTERN RECOGNITION
and several fuzzy generalizations of adaptive resonance theory
(ART), including a short subse...
2 Cluster Analysis for Object Data
2.1 Cluster analysis
Figure 2.1 portrays cluster analysis. This field comprises three
p...
12 FUZZY PATTERN RECOGNITION
Different algorithms produce different partitions of the data, and it
is never clear which on...
CLUSTER ANALYSIS 13
Choosing c=l or c=n rejects the h3T)othesis that X contains clusters
in these cases. The lack of a col...
14 FUZZY PATTERN RECOGNITION
avoid such mistakes. The last column of U allocates most (0.6) of
the membership of Xg to the...
CLUSTER ANALYSIS 15
parameters too. The most common parameters besides U that are
associated with clustering are sets of v...
16 FUZZY PATTERN RECOGNITION
conditions for minimization of the appropriate c-means objective
function.
A. The c-means mod...
CLUSTER ANALYSIS 17
extrema of any objective function. Keep this in mind as you try to
understand your disappointment abou...
18 FUZZY PATTERN RECOGNITION
LaGrangian of J with respect to U. The details for HCM and FCM
can be found in Bezdek (1981),...
CLUSTER ANALYSIS 19
other. Consequently, Identical rows (and their prototypes) will
remain identical unless computational ...
20 FUZZY PATTERN RECOGNITION
w,= IDfk /|U(i,„| , (2.9b)
where U is an a-cut of U,,, the i-th row of the initiaUzing c-
(i)...
CLUSTER ANALYSIS 21
terminal FCM-AO values. The PCM-AO weights were w^=0.20,
FCM-AO and PCM-AO both used m = 2 for theW2=0...
22 FUZZY PATTERN RECOGNITION
many memberships being near 1, but rather, by many memberships
being near zero.
Table 2.3 Ter...
CLUSTER ANALYSIS 23
losing the ability to distinguish between different clusters - for the
advantage of getting possibilis...
24 FUZZY PATTERN RECOGNITION
X^
labeled
^US
„ u _ u —V
2L. , i . „ , . . . , * .
unlabeled
y-x^ux". (2.11)
Pedrycz (1985) ...
CLUSTER ANALYSIS 25
In ordinary HCM-AO or FCM-AO, once U is determined, the next
step is to compute cluster centers {v } u...
26 FUZZY PATTERN RECOGNITION
algorithm towards a solution that avoids the problem of population
balancing that is illustra...
CLUSTER ANALYSIS 27
View 2.3(b) shows a partition obtained by hardening the terminal
ssFCM (Bensald et al., 1996a) partiti...
28 FUZZY PATTERN RECOGNITION
© Supervising points
Figure 2.3(c) A hardened ssfem-AO partition of X43
Supervising points
Fi...
CLUSTER ANALYSIS 29
if you have enough time to find the parameters that yield the same
solution. For p > 3, the luxury of ...
30 FUZZY PATTERN RECOGNITION
Z. is the (positive definite) population covariance matrix of class i.
^'ijk - cov(Xij, Xjjj)...
CLUSTER ANALYSIS 31
m = i P..X./i n ; l < i < c ;
1 k=i ^ik k/ k=i ^ik
p^g(x,^|i:(p,,m,,S^))
'ik
l < i < c l < k < n .
IPj...
32 FUZZY PATTERN RECOGNITION
probabilistic clustering {that is, estimates of the matrix P) is
popular. The efficacy of thi...
CLUSTER ANALYSIS 33
Table 2.5 Terminal EM and FCM partitions and prototypes for X30
Pt. Data Pt. GMD-AO FCM-AO (m=2)
^ 1 ^...
34 FUZZY PATTERN RECOGNITION
Si
"0.11
-0.05
-.05"
0.07 • ' % =
"0.16 0.01"
0.01 0.02 ;and S^ = '0.84 0.24"
0.24 0.57
and e...
CLUSTER ANALYSIS 35
disparities in the ranges of the measured features because it rotates
the basis of 9^^ so that the dat...
36 FUZZY PATTERN RECOGNITION
successive differences when initializing and terminating with V,
and storage for successive i...
CLUSTER ANALYSIS 37
Now consider the computation of, say, the i-th cluster center for
FCM with equation (2.7b) when the im...
Book fuzzy models and algorithms for pattern recognition and image processing
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Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
Book fuzzy models and algorithms for pattern recognition and image processing
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Book fuzzy models and algorithms for pattern recognition and image processing
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