Lecture 13,14,15 object recognition

Digital Image Processing 13III.1 Matching by minimun distance classifier Suppose that we define the prototype of each pattern class to be the mean vector of the pattern of that clas

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Digital Image Processing

The scope covered by out treatment of digital image

processing to include recognition of individual image

regions, which we called objects or patterns.

The approaches to pattern recognition are divided

into two principal areas:

Decision-theoretic: This catogory deals with patterns

described using quantitative descriptors, such as length,

area, texture …

Structural: Deals with patterns best described by qualitative

descriptors, such as the relational descriptors

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Digital Image Processing 3

II Patterns and pattern classes

A pattern is an arrangement of descriptors The

name feature is used often in pattern recognition to

denote a descriptor.

A pattern class is a family of patterns that share

some common properties.

w1, w2, , wK denotes pattern classes, Where K is the

number of classes

Pattern recognition by machine involves techniques for

assining patterns to their respective classes automatically

(and with as little human intervention as possible)

Three common pattern arrangements used in

practice are:

Vectors: for quantitative descriptions

Strings and trees: for qualitative descriptions

Pattern vectors are represented by bold

lowercase letters, such as z, y and z, and

take a form or

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Another Example: We can form pattern vectors by

letting x1=r(θ1),…xn=r(θn) The vectors became

points in n-dimensions space.

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In some applications pattern characteristics are best

described by structural relationships.

For example: fingerprint recognition is based on the

interrelationships of print features Together with

their relatives sizes and locations, these features are

primitive components that describe fingerprint ridge

properties, such as abrupt ending, branching, and

disconnected segments.

Recognition problems of this type, in which not only

quantitative mearsures about each feature but also

the spatial relationships between the features

determine class menbership, generally are best

solved by structural approachs.

Example

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Example

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III Recognition Based on

Decision-Theoretic Methods

Decision-theoretic appoaches to recognition

are based on the use of decision functions.

Let x=(x1, x2, , xn)T represent an n-dimensional

pattern vector

ω1, ω2, , ωW denote W pattern classes

The basic problem in decision-theoretic pattern

recognition is to find decision functions d1(x),

d2(x), , dw(x) with property that, if pattern x

belongs to class ωithen:

III Recognition Based on

Decision-Theoretic Methods

The decision boundary separating class ωi

from ωj is given by values of x for which

di(x)=dj(x), or equivalently, by value of x for

which:

Common practice is to identify the decision

boundary between two classes by the single

function dij(x)=di(x)-dj(x)=0

Thus dij(x)>0 for pattern of class ωi and

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III.1 Matching by minimun distance

classifier

Suppose that we define the prototype of each

pattern class to be the mean vector of the pattern of

that class

where Njis the number of pattern vector from class ωj

Using the Euclidean distance to determine closeness reduces

the problem to computing the distance measures:

classifier

Assign x to class ωj if Di(x) is smallest distance

It is not difficult to show that selecting the

smallest distance is equivalent to evaluating the

functions

And assign x to class ωj if Di(x) is largest numerical

value This formalation agrees with the concept of a

decision function as define in Eq (12.2-1)

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classifier

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III.2 Matching by Correlation

Problem is finding matches of

subimage w(s,t) of size JxK

within a image f(x,y) of size

MxN, assume that J<=M,

K<=N

In its simplest form, the

correlation between f(x,y) and

w(x,y) is

For x=0,1, , M-1, y=0,1, , N-1 and the summation is taken

over the image region where w and f overlap

III.2 Matching by Correlation

Move w around the image area, giving

the function c(x,y) The maximum

value(s) of c indicates the position(s)

where w best matches f

The correlation function given in 12.2-7

has disadvantages of being sensitive to

changes in the amplitude of f and w For

example, doubling all values of f doubles

the value of c(x,y).

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An other approach is to perform matching via the correlation

coefficient, which is defined as:

where x=0,1, ,M-1, y=0,1, ,N-1,

w Nis average value of the pixels in w,

f Nis average of f in the region coincident with the current location of

w, and

The summation are taken over the coordinates common to both f

and w.

The correlation coefficient γ(x,y) is scaled in the range -1 to 1,

independent of scale changes in amplitude of f and w

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III.4 Optimum Statistical Classifiers

Foundation

Denote:

p(ωj/x) is probability that a particular pattern x comes from

class ωj

Lkj is coefficient of loss when pattern x actually comes from

class ωjbut classifier decides that x came from class ωk

Then, the average loss incurred if assign x to class ωk,

rj(x):

This equantion often is called conditional average risk or

loss in decision theory terminology

Foundation

We know that p(A/B)=[p(A)*p(B/A)]/p(B) Using this

expression, we write 12.2-9 in the form:

where p(x /ωk) is the probability density function of the

patterns from class ωk and p(ωk) is probability of

occurrence of class ωk

Bacause 1/p(x) is positive and common to all rj(x), so it

can be dropped from 12.2-10 then rj(x) can be:

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Foundation

The classifier has W possible classes to choose from for

any given unknow pattern If it computers r1(x),

r2(x),…,rW(x) for each pattern x and assigns the pattern to

class with the smallest loss, the total average loss with

respect to all decisions will be minimum

The classifier that minimizes the total average loss is

called the Bayes classifier

Thus Bayes classifier assigns an unknown pattern x to

class ωi if ri(x)<rj(x) for j=1,2, ,W; j<>i In other words, x is

assigned to class ωi if

Foundation

The “loss” for correct decision is assigned value 0

and the loss for incorrect decision is assigned

value 1 Under these conditions, the loss function

becames:

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Foundation

The decision functions given in 12.2-17 are

optimal in the sense that they minimize the

average loss in misclassification

However, we have to know:

The probability density functions of the patterns in each

class, and

The probability of occurrence of each class

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Foundation

The second requirement is not problem For instance, if all classes

are equally likely to occur then p(ωi) = 1/M Even if this condition is

not true, these probabilities generally can be inferred from

knowledge of the problem.

Estimation of the probability density functions p(x/ωi) is another

matter If the pattern vectors, x, are n-dimensional, then p(x/ωi) is a

function of n variables, which, if its form is not know, requires

methods from multivariate probability theory for its estimation.

These methods are difficult to apply in practice.

For these reasons, use for Bayes classifier generally is based on

the assumation of an analytic expression for the various density

functions and then an estimation of the necessary parameters from

samples patterns from each class By far the most prevalent form

assumed for p(x/ωi) is Gaussian probability density function

Bayes classifier for Gaussian pattern

classes

Let consider a 1-D problem (n=1) involving two

pattern classes (W=2) governed by Gaussian

densities, with means m1 and m2 and standard

deviations σ1 and σ2, respectively From Eq

12.2-17 the Bayes decision functions have the form:

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Bayes classifier for Gaussian pattern classes

Fig 12.10 show a plot of the probability density functions for the

two classes The boundary between the two classes is a single

point, denoted x0suchs that d1(x0)=d2(x0)

If the two classes are equally likely to occur, then p(ω1)= p(ω2)

=1/2, and the decision boundary is the value of x0 for which

p(x0/ω1)= p(x0/ω2)

In the n-demensional case, the Gaussian density

of the vectors in the jth pattern class has the form

where, mjand Cjare approximated as

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Because of the exponential form of Gaussian density,

working with the natural logarithm of decision function is

more convenient In other words, we can use the form:

And it infers

IV.Neural Networks

The approaches discussed in the preceding is based on the

use of sample patterns to estimate statistical parameters

The patterns used to estimate these parameters usually are

called training patterns, and a set of such patterns from

each class is called a training set.

The process by which a training set is used to obtain

decision functions is called learning or training.

The statistical properties of the pattern classes in a problem

often are unknown or cannot be estimated

In practice, such decision-theoretic problems are best

handled by methods that yield the required decision

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IV.Neural Networks

An approach manage to organize some nonlinear computing

elements (called neurons) as a networks to classify a input

pattern

The resulting models are referred to by various names:

neural networks, neurocomputers, parallel distributed

processing (PDP) modelsm neuromorphic systems, layered

self-adaptive networks, connectionist models

Here we use the name neural networks or neural nets We

use these networks as vehicles for adaptively developing the

coefficients of decision functions via successive

presentations of training sets of patterns

IV.1 Perceptron

The most simple of neural networks is perceptron In its most

basic form, the perceptron learns a linear decision function that

dichotomizes two linearly separable training sets

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IV.1 Perceptron

Fig 12.14 shows schematically the perceptron model for

two pattern classes

The response of this basic device is based on weighted

sum of its inputs; that is

which is a linear decision function with respect to the

components of the pattern vectors

The coefficients wi, i=1,2,…, n, n+1, called weights

The function that maps the output of the summing

junction into the final output of the device sometimes is

called the activation function.

When d(x)>0 the activation function causes the output of perceptron to

be +1, it indecates the pattern x was recognized as belonging to class

ω1 The reverse is true when d(x)<0.

When d(x)=0, x lies on the decision surface separating the two pattern

classes The decision boundary implemented by the perceptron is

obtained by set Eq 12.2-29 equal to zero:

which is the equation of a hyperplane in n-dimensional pattern space

Geometrically, the first n coefficients establish the orientation of the

hyperplane, whereas the last coefficient, wn+1, is proportional to the

perdendicular distance from the orgin to the hyperplane.

IV.1 Perceptron

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Denote yi=xi, i=1,2, , n, and yn+1=1, then 12.2-29

becomes:

where

y=(y1,y2, ,yn,1)Tis now an augmented pattern vector and

w=(w1,w2, ,wn,wn+1) is called the weight vector.

The problem is how to establish the weight vector ?

IV.1 Perceptron

Training algorithms

Linearly separable classes: A simple, iterative algorithm for

obtaining a solution weight vector for two linearly separable training sets

follows For two training sets of augmented pattern vectors belonging to

pattern classes ω1and ω2, respectively.

IV.1 Perceptron

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Training algorithms

The correction increment c is assumed to be positive and, for now, to be constant

This algorithm sometimes is referred to as the fixed increment correction rule

IV.1 Perceptron

Training algorithms

IV.1 Perceptron

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Training algorithms

IV.1 Perceptron

Training algorithms

Nonseparable classes: One of the early methods of training

perceptron is Widrow-Hoff, or Least-Mean-Square (LMS)

delta rule, the method minimizes the error between the

actual and desired response at any time training step.

IV.1 Perceptron

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Training algorithms

IV.1 Perceptron

Training algorithms

IV.1 Perceptron

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IV.2 Multilayer Neural Networks

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IV.2 Multilayer Neural Networks

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This method is used for the comparison of region boundaries that

are described in terms of shape numbers

The degree of similarity, k, between two region boundaries

(shapes) is defined as the largest order for which their shape

numbers still coincide For example, let a and b denote shape

numbers of closed boundaries represented by 4-directional chain

codes These two shapes have a degree of similarity k if:

V.1 Matching Shape Numbers

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V.1 Matching Shape Numbers

Suppose that two region boundaries, a and b, are

coded into string denoted a1a2…an and b1b2…bm

Let α represent the number of matches between the

two strings, where a match occurs in the kth position

if ak=bk The number symbols that do not match is

β=0 if and only if a1a2…an≡ b1b2…bm (n=n)

A simple measure of similarity between a and b is

the ratio

V.2 String Matching

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V.3 Syntactic Recognition of Strings

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Use of semantics

Automata as string Recognizers

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