Digital Image Processing&

Pattern Analysis(CSCE 563)

Introduction to Pattern Analysis

Digital Image Processing&

Pattern Analysis(CSCE 563)

Introduction to Pattern Analysis

Prof. Amr Goneid

Department of Computer Science & EngineeringThe American University in Cairo

Prof. Amr Goneid, AUC 2

Introduction to Pattern Analysis

Patterns & Pattern Analysis

Pattern Classes

Minimum Distance (Nearest Neighbor) Classifiers

Structural Matching Classifiers

Information Measure Classifiers

Learning-Based Classifiers

Prof. Amr Goneid, AUC 3

1. Patterns & Pattern Analysis

Patterns are Object Descriptors

The objective of pattern analysis is to classify objects into distinct and separable classes

A pattern consists of a finite set of “features” or “attributes”

Such set is usually called a “Feature Vector”

The “Dimesionality” of a pattern is the number (n) of features in the feature vector

Prof. Amr Goneid, AUC 4

Patterns & Pattern Analysis

Features can be:

Quantitative or Qualitative

Numerical or Binary

Ordinal or Non-Ordinal

The process of mapping object descriptors

to a Feature Vector is called “Feature Extraction”

The process of reducing the number of attributes while maximizing the seprability between classes is called “Dimensionality Reduction”

Prof. Amr Goneid, AUC 5

Feature Vector

object Mapper

Dimensionality

Reduction

Feature

Extraction

k-dimensional

Feature Vector

n-dimensional

Feature Vectorx1

x2

.

.

xn

X = n < k

Prof. Amr Goneid, AUC 6

Feature Vector (Example)

Boundary

Fourier

Descriptors

Sort &

Truncate

Boundary

Pixels

k-dimensional

Feature Vector

n-dimensional

Feature Vectorx1

x2

.

.

xn

X = n < k

Prof. Amr Goneid, AUC 7

Dimensionality Reduction

Removal of features with insignificant weights (ordering or sorting then truncation)Removal of non-orthogonal features (discovering association)Selecting features that maximize inter-class to intra-class variances.Discriminant AnalysisPCA

Prof. Amr Goneid, AUC 8

2. Pattern Classes

Classifiersw1

w2

wM

Classifierx

Prof. Amr Goneid, AUC 9

Feature Space & Seprability

Feature Space

Clusters (Classes)

Linearly Separable Classes

Non-Separable Classes

w1w2

w1w2

w1w2

Prof. Amr Goneid, AUC 10

3. Minimum Distance (Nearest Neighbor) Classifiers

Distance Measures:

Search for minimum

distance to class “center”

“Nearest Neighbor”

w1w2

**x

Prof. Amr Goneid, AUC 11

Class Means

n Features: x1 , x2 , ….., xn

Feature Vector of unknown pattern: X

M Pattern Classes: w1 , w2 , …. wM

Sample Patterns for class (j) L = 1,2,…, Nj have sample feature vectors SjL

Class Means for class (j):

mj = (1/ Nj) L SjL = =

m1

m2

.

mnj

<x1>

<x2>

.

<xn> j

Prof. Amr Goneid, AUC 12

Euclidean Distance

For the unknown pattern X , the Euclidean Distance from the “center” of class wj is:

Dj2(X) = || X – mj||2 = (X – mj)T (X – mj)

= XT X - 2 { XT mj – (1/2) mjT mj }

The “Affinity Function”:

aj(X) = XT mj – (1/2) mjT mj

changes with class (j) and should be maximum

Prof. Amr Goneid, AUC 13

Decision Boundary

The decision boundary between class wi and class wj is:

aij(X) = ai(X) – aj(X) = 0

If aij(X) > 0 pattern X belongs to class wi

If aij(X) < 0 pattern X belongs to class wj

Prof. Amr Goneid, AUC 14

Example:

w1: Ballet dancers, w2 : weight lifters

Features: x1= weight in kg, x2 = height in cm

Samples have means:

m1 = ( 60 170)T , m2 = ( 120 160)T

Unknown pattern X = (x1 x2)T

The affinity functions are:

a1(X) = 60 x1 + 170 x2 – 16250

a2(X) = 120 x1 + 160 x2 - 20000

Prof. Amr Goneid, AUC 15

Example (continued)

The decision boundary is:

a12(X) = - 60 x1 + 10 x2 + 3750 = 0

For the unknown pattern X,

if a12(X) > 0 , it belongs to ballet dancers

if a12(X) < 0 , it belongs to weight lifters

e.g. if x1 = 100 kg , x2 = 165 cm

then it is a weight lifter

weight

height

w1w2

a12(X) = 0

Prof. Amr Goneid, AUC 16

4. Structural Matching Classifiers

String Matching

Two boundaries are coded as strings A , B.

A match occurs at position k if A(k) = B(k).

M = # of matches

Number of mismatches is:

Q = max (length(A) , length(B)) – M

Q = 0 if A and B are identical

Degree of matching is R = M / Q

Prof. Amr Goneid, AUC 17

Example: Matching Polygons

Boundaries can be approximated by polygons

Traversing anti-clockwise, we compute the interior angles between segments.

Angles are quantized into regions. Quantized angles are coded by symbols.

Successive angles of a polygon are represented by a string of symbols that can be matched to another string (polygon)

Prof. Amr Goneid, AUC 18

5. Information Measure Classifiers(Information Content)

For Binary Attributes

Information Content:

A set of n objects with only one attribute:

a = # of objects having attribute

(n-a) = # of objects missing attribute

Information Content is:

I = n log n – { a log a + (n-a) log (n-a)} for n > 1

= 0 for n = 1

Prof. Amr Goneid, AUC 19

Diversity & Similarity

A set of n objects i = 1 , 2 , .. N

and m attributes j = 1 , 2 , … , m

Bij = 1 if object (i) has attribute (j) and zero otherwise

aj = # of objects having attribute (j)

(n – aj) = # of objects missing attribute (j) = i Bij

Information Content:

I (m,n,a) = m n log n - j {aj log aj + (n – aj) log (n – aj)}

for n > 1

I (m,n,a) = 0 for n = 1

Prof. Amr Goneid, AUC 20

Diversity & Similarity

Let m = n

For complete similarity, aj = n for all j

I = n2 log n - j { n log n + 0 } = 0

Least info content

For complete diversity, aj = 1 for all j

I = n2 log n - n(n-1) log (n-1) > 0

Max info content

e.g. for n = m = 2, I = 4

Prof. Amr Goneid, AUC 21

Fusion & Division

Two object sets A , B

Objects in A are similar. Same for B

IA = IB = 0

Fusing A,B into C = AB we gain info by

I = IC – [IA + IB]

Dividing C into A and B with AB = 0, we lose info by I = IC – [IA + IB]

Prof. Amr Goneid, AUC 22

Selecting Division Attributes

Dividing set C on attribute (j) produces two sets A with nj

+ = aj objects and B with nj- = (n-aj) objects.

Let Ij+ = I (m , nj

+ , a) , Ij- = I (m , nj

- , a) , then the gain from fusing A and B into C is

Ij = IC – [IA + IB] = I (m,n,a) – {Ij+ + Ij

- }

The division attribute (j) corresponds to max(Ij )

The process is repeated for the net attribute after removing the first one.

Prof. Amr Goneid, AUC 23

Classification Tree

Recursive computation of info gain produces a classification tree

4 5 6 7

2 3

11

1 0

0

1 0

Prof. Amr Goneid, AUC 24

6. Learning-Based Classifiers(Neural Networks)

Neurons

Learning

Feed-Forward NN

Backpropagation NN

Other NN