CHAPTERS

FRACTAL TECHNIQUES IN IMAGE CLASSIFICATION

5.1 Introduction

Classification is necessary for complete understanding of objects and in the

design of a pattern recogniser. By classification, we mean putting together objects that

possess common features. Usually, discriminant features are used for classification

purpose. Classifier design consists of establishing a mathematical basis for classification

procedure. A classification task cannot be properly solved without solid knowledge

about the application area. The main two aspects of classification problem are (i) the

principal nature of classification problem deserves a careful analysis before classification

is applied. (ii) the classification itself needs to be performed carefully and correctly. The

latter is again subdivided into two problems. One is to select the proper type and number

of features and the other is to devise an efficient and accurate classification technique.

For a successful classification task, the relation between the image features and the object

classes sought must be investigated in as much detail as possible. From the multitude of

possible image features, an optimum set must be selected which distinguishes the

different object classes unambiguously and with as few errors as possible ( Jahne and

Haubecker, 2000).

The different tasks in the classification procedure are (i) determine whether the

problem requires classification (ii) determine the relation between the problem related

features and features extracted by image processing operators (iii) select the best features

(iv) decide whether unsupervised classification can be used or whether training is

required with known samples (Jahne and Haubecker, 2000).

Classification of images is done in two ways.

(i) Pixelwise classification

Here each pixel is classified to one of the classes. Usually, clustering is

performed for pixelwise classification.

(ii) Object based classification

Here the object in the image is identified and classified using any patternrecognition technique.

In this chapter, classification of galaxy images by statistical and neural network

method is discussed. The importance of fractals has been observed in the form of the

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fractal feature namely, the fractal dimension. In addition to this, the spectral flatness

measure is also used as another feature. Membership functions were found out for

these features and the classifier is designed using nearest neighbor and neural network

techniques. This scheme was improved further by designing a second classifier which

employs the fractal signature for a set of scales as the feature set and by using the same

techniques. Both the classifiers were found to give reasonable success rates.

5.2 Classification methods

The classification methods are broadly divided into two categories. They are

parametric and non parametric techniques. The parametric classification is based on

Bayes' rule which states that the likelihood ratio between two pattern classes i and j is

greater than the ratio of the probabilities of occurrence of the pattern classes j over i.

Also, it is based on priori knowledge of the pattern classes. Bayes' decision making

refers to choosing the most likely class, given the value of feature or features. The

probabilities of class membership are calculated from Bayes' theorem. If the feature

value is denoted by x and the desired class is C, then P (x) is the probability distribution

for feature x in the entire population. P(C) is the prior probability that a random sample is

a member of class C. P (x/C) is the conditional probability of obtaining feature value x

given that the sample is from class C, then Bayes' theorem is given by

P(C/x) = (P(C)P(x/C))/P(x) .................(5.1)

In the case of non-parametric technique, the type of density function is unknown.

One example of nonparametric technique is the histogram technique. To form a

histogram, the range of the feature variable x is divided into a finite number of adjacent

intervals that include all the data. These intervals are called cells. The number or fraction

of samples falling within each interval is then plotted as a function of x. If a sample falls

directly on the boundary between intervals, then it is put into the interval to its right.

Another non-parametric technique is based on nearest neighborhood. This

technique classifies an unknown sample as belonging to the same class as the most

similar or nearest sample point in the training set of data, which is called a reference set.

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Nearest can be taken as the smallest Euclidean distance in n dimensional feature space,

which is the distance between the points a = (aj,a2,a3,a4, ......an) and b =

(b1,b2,b3, •••••• ,bn) defined by

n

.............(5.2)

i=l

where n is the number of features. The most commonly used distance function is

Euclidean but another approach is to use the sum of the absolute differences in each

feature, rather than their square as the overall measure of dissimilarity. The distance

measure is computed as

n

...............(5.3)

i=l

where d is known as the city block distance. The advantage of nonparametric

technique over parametric is that in most real world problems, even the types of density

functions will be unknown. It becomes very difficult to understand the distribution from

the histogram of data. The Bayesian decision technique is optimal if the conditional

densities of the classes are known. If the densities are unknown, they must be estimated

non-parametrically. These work in a better manner when the number of observations is

large. Nearest neighbour techniques can approximate arbitrarily complicated regions, but

their error rates are usually larger than Bayesian rates. The non-parametric classification

method also uses discriminant functions in order to separate different pattern classes.

Another method of classification is based on neural networks, which attempts

to model the activity of biological brain (Gose et aI., 2000). The human brain performs

its task so efficiently since it uses parallel computation effectively. Thousands or even

millions of nerve cells called neurons are organized to work simultaneously on the same

problem. A neuron consists of a cell body, dendrites which receive input signals, and

axons which send output signals. Dendrites receive input from sensory organs such as the

eyes or the ears and from axons of other neurons. Axons send output to organs such as

muscles and to dendrites of other neurons. Tens of billions of neurons are present in the

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human brain. A neuron typically receives input from several thousand dendrites and

sends output through several hundred axonal branches. Because of the large number of

connections between neurons and the redundant connections between them, the

performance of the brain is relatively robust.

An early attempt to form an abstract mathematical model of a neuron was made

by McCulloch and Pitts ( McCulloch and Pitts, 1943 ). Their model does the following

operations

• Receives a finite number of inputs XI. ""XM

M

• Computes the weighted sum s =L Wi Xi using the weights WI, ••• ,WM

i=1

• Thresholds s and outputs 0 or 1 depending on whether the weighted sum is less

than or greater than a given threshold value T.

Node inputs with positive weights are called excitory and node inputs with

negative weights are called inhibitory. The action of the model neuron outputs 1 if

..............(5.4)

The output is zero otherwise. This can be rewritten as

where Wo =-T and Xo =1. Then output is as follows,

output = 1 if D > 0

..............(5.5)

..............(5.6)

output = 0 if D < 0 (5.7)

The weight Wo is called the bias weight. The difficult aspect of neural net is

training or to find weights that solve problems with acceptable performance. Algorithms

used for implementing neural network are back propagation algorithm, Hopfield net,

Radial basis function etc.

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Cluster analysis is an independent field in pattern recognition methodologies.

It is done when there is no a priori knowledge about the classes. The clustering analysis

algorithms are divided into agglomerative and divisive according to the initial

descriptions. Among agglomerative algorithms, hierarchical clustering algorithms and

minimal spanning trees are important. The classification methods are summarized in

Table 5.1 (Gose et aI., 2000 ; Jain, 2000).

Method Classification technique

Template matching Assign patterns to the most similartemplate.

Nearest Mean Classifier Assign patterns to the nearest class mean.Subspace Method Assign patterns to the nearest class

subspace.k-Nearest Neighbor Rule Assign patterns to the majority class

among k nearest neighbor using aperformance optimized value for k.

Bayes-plug in Assign pattern to the class which has themaximum estimated posterior probability.

Logistic Classifier Maximum likelihood rule forlogistic(sigmoidal) posterior probabilities.

Parzen Classifier Bayes plug-in rule for Parzen densityestimates with performance optimizedkernel.

Fisher Linear Discriminant Linear classification using MSEoptimization.

Binary Decision Tree Finds a set of thresholds for a pattern-dependent sequence of features.

Perceptron Iterative optimization of a linearclassifier.

Multi-layer perceptron Iterative MSE optimization of two or(Feed forward Neural more layers or perceptrons (neurons)network) using sigmoid transfer functions.Radial Basis Network Iterative MSE optimization of a feed

forward neural network with at least onelayer of neurons using Gaussian-liketransfer functions.

Support Vector classifier Maximizes the margin between theclasses by selecting a minimum numberof support vectors.

Table 5.1: Different classification methods and the techniques employed

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5.3 Classification of galaxy images

The classification of galaxies has been the subject of study for a number

of years. The morphological classification scheme of galaxies was introduced by Hubble

in early 1900s. Though as years passed by, several modifications were made to this

scheme, it is still considered as the basis for any classification procedure. The

morphological type describes the global appearance of a galaxy. Also, it provides

information about the evolution of galaxies, its structure and stellar content. In the

conventional approach, galaxies were classified by visual inspection (Burda and

Feitzinger, 1992). The drawbacks of this scheme are that they are slow, ambiguous and

could not be applied for distant galaxies.

The multivariate nature of galaxies makes classification of galaxies difficult

and less accurate. A highly automated classification system is required which can

separate galaxy properties better than the conventional systems. Different classification

procedures have been introduced by experts in the past few decades ( Odewahn,1995;

Nairn et al., 1995; Molinari, 1998). The techniques employed mainly are either statistical

or neural network based. In the statistical approach, the decision boundaries are

determined by the probability distribution of the patterns belonging to each class. A more

recent method is based on neural networks. Here the most commonly used type of

network is the feed forward network which includes multilayer perceptron and radial

basis function. Several modifications are available for this algorithm, of which the most

recent is the one developed by Philip et al. (2002), known as difference boosting neural

network for star-galaxy classification. If the features are distributed in a hierarchical

manner, syntactic approach can be made.

Different types of parameters are defined for the classification purpose. They

are galaxy axial ratio, surface brightness, radial velocity, flocculence etc (Burda and

Feitzinger, 1992). Classifications like morphological, structural or spectral are done

based on these parameters. Morphological classification of galaxies was investigated by

Nairn et al. (1995) using artificial neural networks with the parameters like galaxy axial

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ratio, surface brightness etc. Spectral classification has been found to correlate well with

morphological classification. In spectral classification, the galaxies are classified by

comparing the relative strengths of the components with those of galaxies of known

morphological type (Zaritsky, 1995). Andreon has classified galaxies structurally using

their structural components (Andreon, 1997). Classification of galaxies can also be done

based on luminosity functions ( Han, 1995). Molinari (1998 ) has used Kohonen Self

Organising Map to classify galaxies based on luminosity functions.

The two main steps that preceed classification of an image are feature extraction

and feature selection ( if the image is processed to reduce noise and enhance image

properties). Feature extraction involves deciding which properties of the object (size,

shape etc) best distinguish among the various classes of objects and this should be

computed. The feature extraction phase identifies the inherent characteristics of an image.

By using these characteristics, the subsequent task of classification is done. Features are

based on either the external properties of an object (boundary) or on its internal

properties. Here, classification based on fractal features is discussed.

The term feature selection refers to algorithms that select the best subset of

the input feature set. Algorithms which create new features based on either

transformations or combinations of the original feature set are called feature extraction

algorithms. However, the terms feature extraction and feature selection are used

interchangeably in the literature. Often feature extraction precedes feature selection.

5.4 Image Catalogue

The catalogue of 113 galaxies discussed In chapter 3 is considered for

classification purposes. These galaxies span morphological types from -5 to 12 (Frei et

aI., 1996). These are divided into four groups. The galaxies belonging to morphological

types

-5 and -4 belong to elliptical group

-3 to -1 belong to lenticulars group

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o to

10 to

9 belong to spiral group

12 belong to irregular group

In this work, two classification schemes are designed; the first one makes use of

fractal dimension and spectral flatness measure and the second one makes use of fractal

signature. For galaxy classification scheme using fractal dimension and spectral flatness

measure, the dataset was divided into two groups, namely ellipticals and spirals. The first

two group forms ellipticals and the other three groups form spirals. For fractal signature

study, galaxies belonging to first and third groups are considered. The first group is

labelled as elliptical and the third as spirals.

5.5 Preliminary investigation

Before proceeding to fractal features, like fractal dimension and fractal signature,

the geometrical features namely magnitude and diameter of the galaxy images are

considered in a preliminary study. The classifiers are designed using neural network

which employs backpropagation algorithm. The classifiers designed are two class

classifier and five class classifier and linear and nonlinear classifier. Comparitive

studies are done based on the performance between two class classifier and five class

classifier and linear and non linear classifiers. The galaxy types are followed as in T type

Revised Hubble System (Nairn et al., 1995).

(i) Two class classifier and five class classifier

Here supervised learning technique is done for classification of galaxies. In both

cases, the first half for the data set is used for training purpose, and the second half for

testing.

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Two class classifier

In this classifier, the input layer was presented with the different feature sets.

The first feature set consists of only magnitude, second one consists of only diameter

and the third feature set consists of magnitude and diameter. The output layer

corresponds to the two classes, namely ellipticals and spirals. The procedure was tested

for different sets of hidden nodes.

Five class classifier

The same feature sets as in the above case are considered here. The output

classes are ellipticals, SO/Sa, Sb, Sc and Irr. Galaxy types from -5 to -1 form the first

group, 0 to 2 form the second group, 3 and 4 form the third group, 5 and 6 form the

fourth group and from 7 to 12 form the fifth group. The groups SO/Sa and Sb are

early spirals and Sc are late spirals. Irr forms the irregular group.

The following tables ( Tables 5.2 and 5.3) show the results of comparison. N

denotes the number of hidden nodes. PI, P2 and P3 denote the feature set. The success

rate in each case is given in Table 5.2. It could be observed that the feature set as well

as the output classes will contribute to significantly to the success rate of the classifier

designed.

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Parameters Hidden N=2 N=5

nodes

Magnitude 3 65% 55 %

5 65 % 62%

7 68 % 66%

9 75 % 67%

Diameter 3 55 % 54%

5 60% 59%

7 65 % 61 %

9 67% 62%

Magnitude 3 68 % 63 %

& Diameter 5 69% 64%

7 69% 67%

9 70% 66%

Table 5.2: Comparison of success rates of two and five class classifiers, where N denotes

the number of classes.

It could be observed from Table 5.2 that success rate increases as the number of

hidden nodes increases. Among the feature sets, the third feature set namely, magnitude

and diameter proved to be better than the other two. When the number of hidden nodes

are more, magnitude alone outperforms the other two. In all cases, the five class

classifier is giving a lower success rate compared to that of the two class classifier. But if

the highest and second highest probabilities are considered, then the success rate of five

class classifier is above 90 %.

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Hidden PI PI P2 P2 P3 P3

Nodes (N=2) (N=5) (N=2) (N=5) (N=2) (N=5)

3 9160 13350 7960 11060 8060 15685

5 12310 16950 11310 13540 11460 19650

7 14450 18690 14260 16980 14460 22650

9 17320 19980 17168 18790 17670 24750

Table 5.3: Iterations required for convergence of two and five class classifier.

It is from Table 5.3 that observed that as the number of hidden nodes increases, the

number of iterations required to converge also increases. Also, the number of iterations

required for two class classifier is less compared to that of five class classifier. The

error tolerance set is 0.05.

(ii) Linear and nonlinear classifiers

A linear classifier employs a linear discriminant function whereas a

nonlinear classifier uses nonlinear functions of the input. In the case of parametric

decision making, the type of class density is either known or can be assumed.

Sometimes, the data may not well fit to anyone of the standard distributions. Here

nonparametric techniques like histograms can be employed, where an approximate

density function is obtained from the samples. Another approach is to assume the

functional form of the decision boundary of that form which separates the classes.

Here, adaptive. decision boundary algorithm is implemented for linear classification

and back propagation algorithm(explained in 5.5.1) is implemented for non linear

classification (Gose et al., 2000).

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Adaptive Decision Boundary algorithm .

Suppose the discriminant function having the form

D = Wo + WIXI + ... + 'WnXn •••••.••••.•.(5.8)

is used to classify samples containing M features into two classes.

D = 0 ........... ...(5.9)

is the equation of the decision boundary between the two classes. The weights

WO,Wj, .....wn are to be chosen in such a way that the classifier exhibits better

performance on the test set. A sample with feature vector x = (Xj,X2, .... ,Xn) is

classified into one of the two classes if D > 0 and in the other class if D <= O.

In the adaptive or training phase, samples are presented to

the current form of the classifier. Whenever a sample is correctly classified, no

change is made in the weights but when a sample is incorrectly classified, each

weight is changed in the direction of the correct output. The algorithm then

proceeds to the next sample and this process is repeated until all the samples are

considered for a number of times.

The adaptive decision boundary algorithm consists of the following steps.

(i) Initialise the weights Wo, WI. .....,Wn to small random values. The initial guess

will contribute to the speed of convergence of the classifier.

(ii) Choose the next sample x = (XI,X2, ...... ,Xn) from the training set. Let the true

class of D be d such that d = 1 or -1 represents the true class.

(iii) Compute D =Wo+ WIXI + .. , +'WnXn

(iv) If D is not equal to d, replace Wi by Wi + CdXi for i = 1, ....,M where c is a

positive constant which represents the step size.

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(v) Repeat steps 2 to 4 with each sample of the training set. When finished, run

through the entire data set and note the classification rate. If it is, satisfactory,

stop the iteration. Else, continue with the training process until the specified

tolerance is achieved.

The results obtained from ADB algorithm are given in Table 5.4 and 5.5. Here PI

denotes magnitude, P2 denotes diameter, P3 denotes magnitude and diameter.

Error Class Linear Non linear Linear Non linear Linear Non linear

tolerance PI PI P2 P2 P3 P3

0.05 E 57 % 57% 42% 42% 57% 66%

S 71 % 73 % 65 % 69 % 73 % 81 %

0.005 E 57% 71 % 42% 42% 57% 71 %

S 77% 79% 69% 75 % 77% 83 %

E 71 % 71 % 42% 57% 71 % 71 %

0.0005 S 77% 81 % 73 % 79% 79% 85 %

Table 5.4: Comparison of success rates of linear and nonlinear classifiers

It could be observed from Table 5.4 that as the error tolerance decreases, the success

rate increases. Also, the success rate of spirals is always higher that of ellipticals. This

may be due to the less number of ellipticals present in the entire group. The success rate

of a non linear classifier is always high compared to that of the linear classifier.

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Error Linear Non linear Linear Non linear Linear Non linear

tolerance PI PI P2 P2 P3 P3

0.05 1002 12310 2400 11310 2890 11460

0.005 1040 14650 2650 13680 3150 14670

0.0005 1200 16750 2800 15750 3368 16580

Table 5.5: Iterations required for convergence of linear and nonlinear classifer

It could be observed that as the error tolerance decreases, the number of iterations

required for convergence also increases. This can lead to better classification results.

Table 5.5 gives the iterations required for convergence when the number of hidden nodes

is five.

5.6 Design of classifiers

After initial studies, classifiers are designed using two fractal feature sets. The

first feature set consists of fractal dimension and spectral flatness measure. The second

feature set consists of the fractal signature values over a range of scales. The

classification techniques employed are nearest neighbor method and neural network

technique implemented using back propagation algorithm.

5.6.1 Galaxy classification using fractal dimension and spectral flatness measure

The features fractal dimension and spectral flatness measure are calculated based

on the discussion in chapters 2 and 3. A membership function is defined for these two

features and the grade of membership is used as the input feature set. Fuzzy concepts

are often used for classification purposes. Fuzzy models were introduced by Lotfi Zadeh

in late 60's to represent fuzziness or vagueness in our day-to-day life. The basic

structures underlying in fuzzy models are computational mathematics and models. The

membership function is the underlying power of every fuzzy model since it is capable of

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modeling the gradual transition from a less distinct region to another in a suitable way.

This characteristic of fuzzy model makes it suitable for pattern recognition applications.

Membership values determine how much fuzziness a fuzzy set contains. Its value

measure the degree to which objects satisfy imprecisely defined properties. Identifying

the membership function is an important task in fuzzy logic applications. Besides the

heuristic selection of membership functions, many other techniques have been proposed

to produce membership functions which reflect the actual data distribution by using

unsupervised or supervised learning algorithms. With heuristic selections, the choice of

membership function is usually problem dependent. Triangular, trapezoidal and bell

shaped functions are three commonly used membership functions. Mter choosing the

fuzzy membership function shape, the membership functions for all fuzzy subsets and for

each input and output are constructed ( Chi et al., 1996). Spiekermann (1992) has

developed a fully automated morphological classification system for faint galaxies using

fuzzy algebra and heuristic methods. The catalogue considered was ESO-Uppasala and

the parameters were extracted from digitized plates. Altogether 96 parameters were

defined which were statistical features. In the classifier, fuzzy theory was used to

determine the membership function and heuristic methods were applied to the grades

obtained from the membership functions. For each galaxy, there are two outputs, grades

of membership fuctions representing the two classes. So, the sample is represented in a

196 dimensional feature space. Later the dimension is reduced to two and block metric is

applied to the feature space which cuts the spectrum into boxes, which represent Hubble

equivalent types.

In this case, the membership functions follow the model of a parabola with

shoulders(figure 5.1). There are two output classes for the classifier, A and B, which

denote ellipticals and spirals respectively. The fuzzy set is a normal one with

membership function denoted by ~. The peak point is defined as the value :J!eak from the

domain X such that /-lA(:x?eak) =1 is 0.5 .The value is the same with other class too. The

. . h al crosscross pomt IS t e v ue X in X where /-lA (xcross) = /-lB (xcrOSS) > O. A cross point level

can then be defined as the membership grade of xcross in either fuzzy subset A or fuzzy

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subset B. Here there are 2 cross points, one at 0.235 and the other at 0.765 and the cross

point level is O.5.The input feature, fractal dimension is normalized in the range (0, 1 ].

Here a = 0 and b = 1. c and d are defined as (a+b) /2 and (a+b) /8 respectively. The

membership functions for the two classes, J..lA and J..lB are computed as given in

equations 5.9 and 5.10. Here g is a constant whose value equals 0.0351. This value is

computed by assuming that the parabola passes through ( 0.5, 0 ) and ( 0.875, 1 ) in the

case of spirals (computed in equation 5.9 ) and through ( 0.5, 1 ) and ( 0.125, 0 ) in the

case of ellipticals (computed in equation 5.10 ).

The membership functions for the classes A and B are as follows

IlA = {

IlB = {

a1- (x-c)"2/(4g)

a

1(x-c)"2/(4g)

1

x<=dd < x <=l-dx> 1-d

x<=dd < x <=l-dx> 1-d

...........(5.10)

..........(5.11)

In both cases, the value of g is found to be 0.0351. This is obtained since the parabola

passes in the range 0 through 1, considering the y-axis.

The membership function is plotted below with x-axis denoting the normalized value of

the feature and y-axis denoting the grade of the membership function. The parabola with

the general form x2 =4ay denotes the spiral galaxy and i = - 4ay denotes the elliptical

galaxy.

100

0.80.60.40.2

1

0.8

0.6

0.4

0.2

O+--L.-,.__--~'"'--~,.__-__,~~__,

o

Figure 5.1: Membership function for fractal dimension

So if the fractal dimension of the image is known, the grade of membership

function for both the classes can be found out. A threshold is applied to the grade which

would roughly classify the samples into two groups as spirals and ellipticals. The

percentage of samples correctly classified has been found to be 73.45 %. The

classification rate of ellipticals is less compared to that of spirals. Now, another

parameter is added to improve the success rate of the classifier. The parameter defined is

spectral flatness measure (sfm) (discussed in section 3.5). As in the case of fractal

dimension, a similar membership function is defined for sfm ( the parabola is a little

more stretched ; by 0.0625 in both directions) and the grade of membership function of

sfm for the two classes is found out. The object is then represented in a four-dimensional

feature space. A classifier based on back propagation algorithm is designed using these

features as input parameters. The neural net architecture is given in Figure 5.2. The

architecture of the network is a multilayered one where the nodes in a layer are fully

connected to the nodes in the next layer. The input layer contains nodes representing the

four features and the hidden layer contains four nodes. The output layer consists of the

output node. A threshold is applied to the value of the output node to determine the class.

Also, a bias node whose value equals one is added to input layer and hidden layer.

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input layer hidden layer output layer

Figure 5.2: Neural network architecture

The back propagation algorithm consists of two steps (Gose et al., 2000)

(i) A feed forward step in which the outputs of the nodes comprising the hidden

layers and the output layer are computed. The output values are calculated as a linear

combination of the weight and the node value of the previous layer, which is then

presented to the sigmoid function. Output value of a computed node is

X. (k+l)J = R (L Wij (k+l) Xj (k») .............(5.12)

Here Xj (k) is the value of the jth node in kth layer and Wij (k+I) is the weight of the link

connecting ith node in kth layer to jth node in (k+1) sl layer.

(ii) A back propagation step where the weights are updated backwards from the

output layer to one or more hidden layers. The back propagation step uses the steepest

descent method to update the weights so that the error function

E = .............(5.13)

is minimized where dj is the desired output class. The network architecture is as follows.

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The input layer consists of four feature values and the hidden layer consists

of four hidden nodes excluding the bias node. The network is trained for different sets of

iterations (n) and different hidden nodes to test the perfonnance of the classifier. The

average success rate in each case is given below (Figures 5.3 - 5.7 ).

Performance when n=10,000

~ 100

~ 75lZ 508 25i]l 0-+--+-+-+--+--+---1,-----+--+--1

1 2 3 4 5 6 7 8 9 10

hidden nodes -->

Figure 5.3:Variation of success rate with hidden nodes

Performance when n=15,OOO

100

A75I

I

Ql

~ 50IIIIIIQ)

8 25::JIII

0

1 2 3 4 5 6 7 8 9 10

hidden nodes ••>

Figure 5.4 : Variation of success rate with hidden nodes

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n=18,OOO

r~t: :: ==: :,1 2 3 4 5 6 7 8 9 10 11

hidden nodes

Figure 5.5 : Variation of success rate with hidden nodes

n=20,OOO

r~t ,::::: ,1 2 3 4 5 6 7 8 9 10 11

hidden nodes

Figure 5.6: Variation of success rate with hidden nodes

n=25,OOO

(1) 1::t=:~l/) ,:::l/)(1)00::J I Il/)

1 2 3 4 5 6 7 8 9 10 11

hidden nodes

Figure 5.7 : Variation of success rate with hidden nodes

It could be seen that the average success rate is maximum (80.53%) for n =

25,000 and number of hidden nodes= 5.

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5.6.2 Galaxy classification using fractal signature

One of the important properties of fractal objects is the surface area. For images,

the change of gray level surface has been measured at different scales. The change in

measured area with changing scale is used as the fractal signature and these can be

compared for classification. Peleg et al. (1984), introduced a texture analysis method that

measures the area of the gray level surface at varying resolutions.

Computation of fractal signature

For a pure fractal gray level image the surface area, A(e) is computed as

A (e) = F e2-D

, (5.14)

where e is the resolution of the gray levels in the image, D is fractal dimension and F is

a constant. The change in measured area with changing scale is used as the fractal

signature of the texture.

The surface area of the galaxy image is computed by the method suggested by

Mandelbrot for curve measurement (Peleg et al.,1984)

Initially, for e = 0,

=

=

g (i, j)

g(i, j)

.................(5.15)

.................(5.16)

where g(i, j) represents the gray level function.

From e == 1 onwards,

ue (i, j) = max{ut;.,(i, j)+1, max ut;.\(m, n)}

l(m,n)-(i,j)I<=l

Ie (i, j) = min {le-li, j)-l, min le-im, n)}

l(m,n)-(i,j)I<=l

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.................(5.17)

.................(5.18)

For computing Ue at different points, the four immediate neighbours are considered.

Following this, the volume is computed by

Ve = ~ (ue(i, j) -Ie (i, j)).

ij

The surface area is computed as

A (e) = (Ve- Ve-l ) / 2e

. (5.19)

..................(5.20)

Ifthe image is a fractal, the plot of A (e) verses e on a log -log scale is a straight

line. Typical variations of A (e) with e for spiral and elliptical galaxies are given in

Figure 5.8. It could be observed that the plots of spirals tend to be a straight line which

substantiates that images of spiral galaxies are more fractal prone compared to that of

ellipticals. The slope S (e) of A (e) is the fractal signature. The fractal signature for

e =2,3, ...... , 48 were computed for elliptical and spiral galaxies of 113 nearby galaxy

catalogue. It could be observed that fractal signatures of ellipticals resemble each other.

(Figures 5.9 and 5.10 ). The fractal signatures of spirals become more prominent with

respect to their morphological type.

106

t4

3m

16

(Jl

a; 10 15 Zl 25 3l :J; <10 0 2 4 0 I~>4

~1CJliE)->

1211

4lZ!<lIB

4lB

11

o 2 4

I~

o 2 4

Ie:¢}>

o 2 4Ie:¢}>

Figure 5.8 : Variation of A(E:) with E: on log-log scale. Spirals (top) and

ellipticals (bottom)

107

1.04406

0.5

f 0.0~

~'0.5

-to

-1.5

-2.0 ';;-"7;.:--;!';;:'""-::"-:-'''-:''~'''''''~-'-~''''''1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2

A(e) -->

0.5

1 0.0..,en ·0.5

-1.0

-1.5

-2.0 ~n?'1~;;:;-;;~"-:"'7'""'""-'-"""''"'''-'2.0 2.1 2.2 2.3 2.4 2.5 2.6

A(e) -->

0.5 1.0

4486 4564

~0.0 ~.

,~.. -0.5 ~.

U; $-to en ....

-1.5 .l~

-2.0 .1.0

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 1.6 1.6 2.0 2.2 2.4 2.6A(e) --> A(e) -->

1.04621 1.0

0.50.5 4636

f0.0

0.0

i ~ -0.5-0.5

,.-.$

-1.0en -1.0

-1.5 ·1.5

-2.0 -2.01.8 2.0 2.2 2.4 2.6 2.0 2.2 2.4 2.6 2.8 3.0

A(e) --> A(e) -->

A(e) -->

Figure 5.9 : Variation ofis given on topright.

1.0

0.5

~ 0.0

~-O.5II)

-1.0

-1.5

-2.0

2.0 2.2 2.4

5322 1.05813

0.5

~0.0

$ -0.5en

·1.0

-1.5

-2.0

u u ~ u u U U M UA(e) -->

slope, S (e) with area, A (e) for elliptical galaxies. Galaxy id

108

·0.4

0

27153079

·0.6·1

1 ·0.8

.;.·2I

~ -1.0$

CIJ

UJ ·3

·1.2

·1.4

·4

·1.6·5

-1.81.5 2.0 2.5 3.0 3.5 4.0

.614.5

A(e) -->

2 3 4 5 6

A(e) -->

0.0 0.5

·0.53198 0.0 3486

1-;;;- '1.0

1 ·0.5

Cil-;;;-

'1.5

Cil -1.0

-2.0

·1.5

-2.0

·2.5 ·2.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 2 3 4 5 6

A(e) -->A(e) -->

0.0

0.0

35963631

1-0.5

·0.5 .;.~

$

, ·1.0

UJ ·1.0 if·1.5

·1.5 ·2.0

-2.0 ·2.5

2.0 2.5 3.0 3.5 4.0 4.52 3 4 5 6

A(e) -->A(e) --->

'2·~.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

ACe) -->

-1.5

36720.0 3726

1 ·0.5

$UJ -1.0

.2.5 ";;""'~;-----::-='--'-:':'-=-"'-::''''''''''~~....L.._J-,..o1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

A(e) -->

-0.5

1 -1.0~

$UJ

·1.5

-2.0

F~gure 5.10: Variation of slope, S (€:) with area, A (€:) for spiral galaxies. Galaxy id is

gIVen on topright.

109

5.6.2.1 Classification methods

(i)Nearest Distance approach

Galaxy images were classified into ellipticals and spirals by comparing the distances

between their fractal signatures. For two images hand iz, whose signatures are 81 and 82,

the distance is defined as,

D (iI, iz) = L ( 81 (e) - 82 (e) )210g ((e+Yz) / (e-Yz» (5.21)

The first three images of elliptical galaxies were used to define the elliptical cluster and

the first ten images of spiral galaxies were used to define spiral cluster. The clusters were

defined by taking the average of the fractal signature for the various scales defined.

Out of 14 ellipticals, only one was misclassified and out of 90 spirals, only seven were

misclassified.

ii) Neural Network Approach

As in the above case, a neural net is implemented using back propagation algorithm. The

input layer consists of A (E), E = 2, ..47. Altogether, it took 15,000 iterations for the

algorithm to converge. Out of 14 ellipticals, only one was misclassified and out of 90

spirals, only four were misclassified.

The results of the two techniques are depicted in Table 5.6.

Gal Dist Dist O/p Gal Dist Dist O/pid toEC toSC DVC value DVC DSC id toEC toSC DVC value DVC DSC

2768 0.09 0.32 E 0.01 E E 5055 0.72 0.07 8 0.02 E 83377 0.02 0.56 E 0.007 E E 5248 4.33 2.10 8 0.99 8 83379 0.04 0.61 E 0.005 E E 5364 2.07 0.90 8 0.99 8 84125 0.06 0.46 E 0.47 E E 5371 3.09 1.37 8 0.99 8 84365 1.33 0.52 8 0.99 8 E 6384 2.00 0.66 8 0.99 8 84374 0.20 0.29 E 0.34 E E 2715 3.07 1.91 8 0.99 8 84406 0.13 0.25 E 0.42 E E 2976 2.82 1.53 8 0.99 8 84472 0.33 1.07 E 0.29 E E 3079 6.06 4.52 S 0.99 S S4486 0.37 0.96 E 0.06 E E 3198 2.92 1.14 S 0.003 E S4621 0.22 1.19 E 0.004 E E 3486 3.76 1.72 S 0.99 S S4636 0.06 0.60 E 0.044 E E 3596 1.28 0.27 S 0.99 S S5322 0.08 0.84 E 0.005 E E 3631 3.45 1.80 S 0.99 S S5813 0.09 0.43 E 0.01 E E 3672 3.09 1.80 S 0.99 S S4564 0.77 2.13 E 0.004 E E 3726 1.86 0.64 S 0.99 S S

110

3166 0.30 0.27 S 0.99 S S 3810 4.88 2.86 S 0.99 S S

5701 1.20 0.22 S 0.99 S S 3877 1.93 1.07 S 0.99 S S3623 1.38 0.91 S 0.99 S S 3893 2.68 1.52 S 0.99 S S4594 4.86 2.95 S 0.99 S S 3938 3.79 1.91 S 0.99 S S5377 0.39 0.37 S 0.99 S S 4123 1.38 0.57 S 0.99 S S2775 0.30 0.11 S 0.99 S 'S 4136 2.56 0.91 S 0.99 S S2985 0.98 0.24 S 0.99 S S 4157 1.30 1.07 S 0.99 S S3031 0.46 0.93 E 0.99 S S 4254 7.84 4.58 S 0.99 S S3368 0.57 0.15 S 0.99 S S 4414 2.26 0.87 S 0.99 S S4192 0.67 0.36 S 0.99 S S 4535 1.88 0.56 S 0.99 S S4450 0.38 0.04 S 0.99 S S 4651 2.17 0.83 S 0.99 S S4569 0.43 0.32 S 0.99 S S 5033 0.55 0.22 S 0.99 S S4725 0.46 0.28 S 0.99 S S 5334 19.47 14.95 S 0.99 S S4826 0.41 0.45 E 0.99 S S 2403 2.52 0.97 S 0.99 S S2683 2.43 1.17 S 0.99 S S 2541 9.89 6.60 S 0.003 E S3351 0.56 0.32 S 0.99 S S 3184 2.43 1.18 S 0.99 S S3675 1.28 0.44 S 0.99 S S 3319 9.60 6.42 S 0.99 S S4013 0.78 0.32 S 0.99 S S 3556 4.46 2.28 S 0.99 S S4216 0.13 0.40 E 0.01 E S 4144 2.92 2.00 S 0.99 S S4394 1.93 0.89 S 0.99 S S 4189 1.98 0.71 S 0.99 S S4501 0.70 0.17 S 0.99 S S 4487 1.79 0.57 S 0.99 S S4548 0.16 0.55 E 0.96 S S 4559 1.98 0.70 S 0.99 S S4579 0.05 0.40 E 0.91 S S 4654 2.69 1.00 S 0.99 S S4593 0.34 0.06 S 0.98 S S 4731 4.12 1.92 S 0.99 S S5746 1.94 1.44 S 0.99 S S 5669 2.35 0.80 S 0.99 S S5792 1.47 0.64 S 0.99 S S 6015 4.28 2.17 S 0.99 S S5850 1.24 0.54 S 0.99 S S 6118 1.42 0.76 S 0.99 S S5985 1.38 0.57 S 0.99 S S 6503 5.17 3.01 S 0.99 S S2903 0.69 0.24 S 0.99 S S 4498 4.98 2.90 S 0.99 S S3147 1.33 0.64 S 0.99 S S 4571 5.08 2.94 S 0.99 S S3344 2.20 0.87 S 0.99 S S 5585 1.97 0.66 S 0.99 S S3953 1.06 0.26 S 0.99 S S 4178 3.66 1.72 S 0.99 S S4030 3.53 1.72 S 0.99 S S 4242 24.50 18.56 S 0.99 S S4088 6.44 3.89 S 0.99 S S 4861 4.46 2.28 S 0.99 S S4258 6.44 3.89 S 0.99 S S 5204 7.49 4.54 S 0.99 S S4303 3.76 1.66 S 0.99 S S 4449 6.70 4.03 S 0.99 S S4321 0.87 0.27 S 0.99 S S4527 0.19 0.26 E 0.99 S S4689 3.01 1.43 S 0.99 S S5005 0.59 0.73 E 0.02 E S

Table 5.6 : Classification results: SC:Spiral cluster EC:Elliptical clusterDVC:Derived class,DSC:Desired class

111

The classwise performance of the two techniques is given in Figure 5.11. The first bar

graph gives the success rate for nearest distance and the second one gives the success rate

using neural network.

100

~

i :: +-_----..........12 3 4 5

Figure 5.11 : Classwise performance of the two techniques

5.7Conclusion

In this chapter, classification of galaxy images using fractal features is discussed.

The features considered are fractal dimension and spectral flatness measure.A

membership function is defined for these two features and the grade of membership is

used as the input feature set to the classifier. The success rate of the classifier is 80%.

Another fractal property, the fractal signature is computed for the two types of nearby

galaxies . It could be observed that the fractal signature of spirals and ellipticals vary

between the groups, though they are similar within the group. Moreover, taking the

fractal signature as input feature set, the neural net could give a classification rate of

95%. The less number of ellipticals when compared to spirals gives rise to 95% success

rate, though only one from elliptical group was misclassified.

112