CHAPTERS FRACTAL TECHNIQUES IN IMAGE...
Transcript of CHAPTERS FRACTAL TECHNIQUES IN IMAGE...
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.
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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