Multi-level Fractal Decomposition Based Feature Extraction

Advanced Studies in Theoretical Physics
Vol. 8, 2014, no. 20, 849 - 856
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/astp.2014.49124
Multi-level Fractal Decomposition Based
Feature Extraction Using Two Dimensional
Discrete Wavelet Transforms
Amitabh Wahi
Department of Information Technology
Bannari Amman Institute of Technology
Sathyamangalam – 638401
Erode District, Tamil Nadu, India
S. Sundaramurthy
Department of Information Technology
Bannari Amman Institute of Technology
Sathyamangalam – 638401
Erode District, Tamil Nadu, India
Copyright © 2014 Amitabh Wahi and S. Sundaramurthy. This is an open access article
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, the multifractal scheme provides a richer framework to extract the
fractal components using 2D discrete wavelet transform than that of the
conventional methods. In general, most of the signals and images are complex
objects and possess a high degree of redundant information. The statistical
properties of signals and natural images reveal that natural images can be viewed
through different segments, which are most probably fractal components in nature.
Such Fractal Components are very informative about the geometry of the images
from which they were extracted. Those Components are usually equal to edges
present in the Image. The discrete wavelet transform analysis is used for
extracting the most revealing parts of the images using such as Haar and
Daubechies wavelets. The key advantage of Discrete Wavelet transform over
Fourier transforms is temporal resolution characteristic. The Wavelet transform
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Amitabh Wahi and S. Sundaramurthy
captures both location and frequency information. The statistical properties of the
extracted fractal components can be used for signal analysis, image reconstruction
and object recognition in the multi environment digital images.
Keywords: Fractal, Decomposition, Edges, Wavelets, Discrete Wavelet
Transforms
1 Introduction
The statistical analysis of signals and images implies to detect the origin of
redundancies like that of the power spectrum. In the last years a new statistical
study has arisen, that of multifractals in natural images [1][2]. The multifractal
framework provides an efficient framework than the conventional method of
detecting the edges. This method makes possible to split any image into a
collection of Fractal sets, from which one of them is supposed to be the most
informative, that set is usually edge-like [1,2].
In most of the fields like object recognition, medical image analysis and Signal
processing, Extracting features from the images and signals are essential to
provide exact results. There are so many approaches widely used for extracting
the features from the input sources for improving the recognition rate [10, 11, and
15]. For optimal thresholding of medical images, Multi Objective Particle Swarm
Optimization technique is introduced by optimizing the interclass variance and the
Shannon entropy [8].
In the field of Fractal geometry, Fractals and Fractal dimension play important
roles to estimate the features in signals and images. To reduce the computation
complexity, the Fractal dimension is computed using fuzzy and escape time
dimension in [9]. Edge detection is the process of detecting meaningful transitions
in an image. The purpose of detecting edges in an image is to identify areas of an
image where a large change in intensity takes place. An edge is a local concept
where as a region boundary is a more global idea according to its definition. An
ideal edge is an asset of connected pixels, each of which is located at an
orthogonal step transition in gray level [1, 2, and 15].
A reasonable definition of edge requires the ability to measure gray-level
transitions in a meaningful way. In the Conventional edge detection methods, the
presence of an edge at a point in an image can be determined by using first and
second derivative of the gray level profile. However, providing a reasonable,
non-conventional definition of “edge” is more controversial [1, 5, 6, 12, 14]. The
wavelet based feature extraction is used in the classification of war scenes, Fractal
surfaces and EEG data [3, 6, 14, and 13]. The ANN is used for the classification
of different Images by taking the extracted features as input to the neurons [4, 6,
and 7].
In this paper, the multi fractal framework is used to obtain the most informative
component using the discrete wavelet transform. Section 2 introduces the
multifractal framework and its significance in the field signal and image Analysis.
Multi-level fractal decomposition
851
Section 3 shows the extraction of fractal components from the given sample.
Section 4 deals with the experimental implementation and results.
2 Multifractal Framework
The multifractal framework a new scheme allows introducing a natural
classification technique, which gives information about the edges present in the
scene as well as textures. This framework takes advantage of multiscaling
properties of images to decompose them as a collection of different fractal
components, each one associated to a fractal singularity component. The
singularity exponent provides the information about the changes in the intensity
level in an image. Among all the fractal components , some of the components,
characterized by the least possible exponent, provides the most important parts of
the whole image and those components seems to be like edges present in the
Scene [1, 2, 3].
The multifractal framework is especially well adapted to certain types of
images, but a great variety of real world images seem to be well described in
this framework. The fractal components are structures of great geometrical
relevance, determined by the statistical properties of natural images. Such Fractal
components are also called as Most Singular Manifold [1, 2], which is strongly
related to the edges or contours of the objects. For the certain type of multifractals
detected in the real world images, the existence of this kernel stresses the
importance of the multifractal classification and extracting the MSM is a good
technique to compress and code images. The Fractal figures generally share the
following features in common, No characteristic length, Self-similarity and
Non-integer dimension called as Fractal dimension.
3 Extraction of Fractal Components of Images using 2DWT
A wavelet is a waveform of effectively limited duration that has an average
value of zero. Wavelet analysis represents the new step, called a windowing
technique with different size of regions. Wavelet analysis allows the use of long
time intervals where more precise low-frequency information is wanted, and
shorter regions where high-frequency information is wanted.
The wavelet transform has the ability to provide the solution for the
multi-resolution problem. Wavelet analysis is capable of revealing aspects of data
that other signal analysis techniques miss aspects like self-similarity,
discontinuities in higher derivatives, breakdown points etc.
In this paper, the 2D discrete wavelet transform (DWT)is used to detect the most
singular fractal exponents from the image using wavelets like Haar and
Daubechies. Compared with Fourier transforms, The important advantage of
DWT is temporal resolution. The wavelet captures both location and frequency
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Amitabh Wahi and S. Sundaramurthy
information. The computational complexity of discrete wavelet transform is also
less compared with continuous wavelet transform [15].
The two dimensional sequence d (k,l) is commonly referred to as the discrete
wavelet transform of f(t). The value d(k,l) is computed by using the equations 3.1
and 3.2.
1
M
W ( j0 , k ) 
W ( j , k ) 
1
M
 f ( x) jo, k ( x)
(3.1)
x
 f ( x) j ,k ( x)
(3.2)
x
Fractal Dimension is used to measure the complexity of an object.The Fractal
dimensions D(h) , which can be predicted from statistical properties of the images .
The fractal dimension is given in the equation 3.3 .
D(h)  D 
 h  h 
h  h 


1

log

 
 d  D 
(3.3)
where,
   log 1  h d  D 
(3.4)
The Fig. 1 shows the block diagram of the fractal decomposition of the images
using Wavelet Transforms.
Fig. 1 Block diagram of the Fractal Decomposition
4 Experiment implementation and Results
The proposed Fractal decomposition method is tested using Matlab software on
INTEL XEON E5506 QUAD 2.13 GHZ core processor machine with Windows
XP. The proposed procedure is tested on three classes of image datasets. The class
Multi-level fractal decomposition
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1 dataset contains the sample images taken by the Nikon camera. The class2
dataset contains image samples from the MIT data base. The class 3 contains
standard images like Lena and mandrill. It was assumed that the images were free
from noises hence no noise removal method was used to filter the noises. The
sample images are resized to 512x512 for the feature extraction process. The
following are the wavelets used for the determination of Fractal exponents. The
Fig 2. Shows the class1, class2 and class3 sample images with reduced size.
The following procedure is used to extract the Fractal components.
1. Resize the input image to 512 X 512 Image.
2. Compute the gradient value for each pixel in the given image.
3. Compute the wavelet coefficients for each pixel in the image using 2D DWT
Haar and Daubechies(db2) wavelets.
4. Determine the threshold using quantile function.
5. Extract the fractal components or coefficients using the threshold value.
Table I shows the number of fractal components determined from the
three-sample image classes using the wavelets Haar and Db2 for the level1 and
level2 decomposition. The number of fractal exponents is more in class2
compared to the remaining two classes in the level1and level2 decomposition.
In addition, the fractal components extracted are more in Haar over Db2. Table II
shows the execution time for the multilevel decomposition using Haar and Db2.
The comparison results on number of Fractals and execution time are depicted in
the graph 1 and 2 respectively.
Fig 2. Class1, Class2 and Class3 Image samples
Table I Number of Fractal components
# of fractal exponents
Image
Level 1
Level2
Sample
Haar
db2
Haar
db2
Class1
3253
3184
2102
2049
Class2
15701
15331
7055
7006
Class3
10573
10384
4356
4216
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Amitabh Wahi and S. Sundaramurthy
Table II Execution Time in Seconds
Execution time in seconds
Image
Level 1
Level2
Sample
Haar
db2
Haar
db2
Class1
0.297
0.313
0.274
0.289
Class2
0.301
0.314
0.273
0.282
Class3
0.296
0.311
0.271
0.280
Level 1
Level2
Graph 1. Number of fractals
Level 1
Graph 2.
db2
Class3
Haar db2 Haar db2
Class2
Haar
Class2
Class1
db2
Class1
0,32
0,3
0,28
0,26
0,24
Haar
20000
15000
10000
5000
0
Class3
Level2
Execution Time
5 Conclusion
In this paper, the proposed method uses multifractal framework to
decompose the images into Fractal components using 2D wavelet transforms.
The key advantage of wavelet over Fourier is temporal resolution. The discrete
wavelet transform captures both frequency components and location information.
It is shown that there exists a large family of wavelets which can be used to detect
and isolate the fractal components . Haar and Daubechies wavelets are used for
multi-level decomposition to extract the Fractal components. The method
proposed here could be used to determine which properties are important to keep
in order using the extracted components. From the extracted Fractal components,
the various statistical properties can be obtained for image analysis, object
recognition and signal analysis in various domains using artificial neural
networks.
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Received: September 1, 2014

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