PDE BASED FEATURES FOR TEXTURE ANALYSIS USING WAVELET TRANSFORM

International Journal on Cybernetics & Informatics (IJCI) Vol. 5, No. 1, February 2016
DOI: 10.5121/ijci.2016.5114 143
PDE BASED FEATURES FOR TEXTURE ANALYSIS
USING WAVELET TRANSFORM
P. S. Hiremath1
and Rohini A. Bhusnurmath2
1
Department of Computer Science (MCA), KLE Technological University,
BVBCET Campus, Hubli-580031, Karnataka, India.
2
Department of P.G. Studies and Research in Computer Science,
Gulbarga University, Gulbarga-585106, Karnataka, India.
ABSTRACT
In the present paper, a novel method of partial differential equation (PDE) based features for texture
analysis using wavelet transform is proposed. The aim of the proposed method is to investigate texture
descriptors that perform better with low computational cost. Wavelet transform is applied to obtain
directional information from the image. Anisotropic diffusion is used to find texture approximation from
directional information. Further, texture approximation is used to compute various statistical features.
LDA is employed to enhance the class separability. The k-NN classifier with tenfold experimentation is
used for classification. The proposed method is evaluated on Brodatz dataset. The experimental results
demonstrate the effectiveness of the method as compared to the other methods in the literature.
KEYWORDS
Anisotropic Diffusion, Wavelet Transform, Texture Approximation, Partial Differential Equation (PDE).
1. INTRODUCTION
Texture is a recurring pattern in image intensity. Texture can be defined as a similarity grouping
in an image [1]. Texture analysis represents mathematical approach that captures gray level
variations in an image. Texture perception is a function of radiometric and spatial scales. It is
used in computer vision and image analysis. An image is the matrix of pixels and a texture in an
image can be considered as descriptors of interrelated pixels. Tuceryan and Jain [2] categorized
texture analysis approaches into statistical, signal processing, geometrical, and model based
methods. The study of human vision system reveals that spatial or frequency representation
preserving local and global information is suitable for quasi periodic signal. Haralick et al. [3]
proposed gray level co-occurrence matrix (GLCM) for texture feature extraction. This approach
explored spatial gray level dependence of texture. Tamura et al. [4] proposed a computation of six
visual properties like contrast, coarseness, likeliness, regularity, directionality and roughness. In
[5] wavelet transform is used to obtain features for texture classification. Smith and Chang [6, 7]
used the statistical features such as mean and variance from wavelet sub-bands as texture
measures. The local directional binary patterns (LDBP) and non-subsampled contourlet transform
(NSCT) based texture classification using k-NN classifier is introduced in [8]. The effective
LDBP’s are investigated which characterize local image texture [9]. In [10], support vector
machine is used for rotation and shift invariant texture classification. Thyagarajan et al. [11]
proposed wavelet transform with co-occurrence matrix for texture analysis. The commonly used
texture descriptors that have been used successfully to real-world textures are the Laws’ texture
energy measures [12] and Fourier power spectrum [13]. The fractal concept developed by
Mandelbrot [14] provides an excellent representation of the roughness of natural textural surfaces.
The partial differential equation (PDE) based anisotropic diffusion filter is used for image
processing. The PDE techniques are widely used in image and signal processing [15]. Most

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methods based on PDE are used to smooth the image while preserving the edges [16]. An
effective method for texture classification using local directional binary pattern co-occurrence
matrix and anisotropic diffusion is presented in [17]. Anisotropic diffusion for color texture
image classification is explored on RGB color space [18]. Extraction of LDBP features using
diffusion approach on different datasets is presented in [19]. Randen and Husoy [20] concluded
in a survey of filtering methods that future direction of research is extraction and classification of
powerful texture features with low computational cost. This observation has inspired to develop
proposed method.
The combination of transform based method and statistical method for feature extraction is
proposed in the paper. The objective of this paper is to obtain better classification accuracy with
reduced computational cost. Wavelet transform is used to obtain directional information.
Anisotropic diffusion is applied on directional information to find texture approximation. Further,
various statistical features are computed from texture approximation. Linear discriminant analysis
(LDA) enhances the class separability. The k-NN classifier is used for classification. The method
is evaluated on sixteen texture classes from Brodatz image dataset. The average classification
accuracies obtained by using various statistical feature sets are compared with other methods in
the literature.
2. PROPOSED METHOD
The proposed method comprises the following steps:
i. Apply Haar wavelet transform on input image to yield H, V and D subbands.
ii. Apply anisotropic diffusion on H, V and D components up to n diffusion steps and obtain
texture approximation.
iii. Extract statistical features from texture approximation image and use LDA on feature sets
for optimization
iv. Use k-NN classifier to classify the feature set of textural image.
These methods are described briefly as given below.
2.1. Wavelet Transform
During the past decades wavelet analysis has become a powerful tool for multi-resolution
analysis. Intuitively, multiscale wavelet analysis is an ideal approach to analyze texture because it
is well recognized that scale is one of the most important aspects of texture information. The
wavelet based methods are computationally effective over other methods for the texture
classification. The different wavelet transform functions filter out different range of frequencies
(i.e., subbands). Thus, wavelet is a powerful tool, which decomposes the image into low
frequency and high frequency subband images. The implementation and theoretical aspects of
wavelet based algorithms are discussed in [21, 22, 23]. Time dependent signal analysis at
different resolution is achieved by wavelet transform. The wavelet transform has several
advantages making it attractive for texture analysis. That include –textures are represented at the
suitable scale in varying the spatial resolution. Wavelet functions are available in wide range of
choices, so that wavelet best suited for texture analysis can be chosen in a specific application. In
an image, the wavelet decomposition is obtained with separable filtering along the columns and
the rows and of an image [23]. The decomposition of a signal into a set of detail coefficients (H,
V, D) and an approximation coefficient (A) is obtained in discrete wavelet transform. The D
subband represents diagonal details (high frequencies in both directions – the corners), H gives
horizontal high frequencies, V gives vertical high frequencies, and the image A corresponds to
the lowest frequencies. The Figure 1 represents the level 1 (1-scale) image decomposition. The
original Brodatz texture image D51 is shown in the Figure 1(b) and its 1-level Haar wavelet
transform is shown in the Figure 1(c).

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A H
V D
(a) (b) (c)
Figure 1. Wavelet transform (a) Subbands of level 1 decomposition (b) Brodatz texture image D51 (c) 1-
level Haar wavelet transform of the image in (b).
2.2. Anisotropic Diffusion
Anisotropic diffusion filter proposed by Perona - Malik [15] is used to smooth an image without
altering important parts of the image such as edges. The edge detection is done using scale space
approach. The family of parameterized images is produced in combination of original image and
image dependent filter. This makes anisotropic diffusion a space variant and nonlinear
transformation of the image. The relation between filtered image F(x,y,t) and the original image
F0(x,y) is represented by F(x,y,t) = F0(x,y) * GK(x,y,t), where GK(x,y,t) is a Gaussian kernel
having variable scale parameter t. The filtering is the result of iterative heat diffusion. The initial
condition is given by the equation F(x,y,0) = F(x,y) , where t represents time. The diffusion
process smoothes the image and is stopped at the edges and boundaries. The method is described
in brief in Appendix.
2.3. Statistical Features
Statistical methods are used to compute local features by considering gray values in an image
[24]. Statistical methods are classified into first order (one pixel), second order (two pixels) and
higher order (more than two pixels) statistics [24]. The first-order statistics captures properties
(e.g. average and variance) considering individual pixel values, whereas second and higher order
statistics represent the spatial dependency between two or more pixel values relative to each
other. Various feature vectors considered for experimentation are described below.
2.3.1. First order statistics for texture analysis (F1)
First order texture features are calculated from the gray values of original image. These statistics
do not consider relationships between neighboring pixels. Five features, namely, mean, median,
standard deviation, skewness and kurtosis are considered for experimentation. Texture analysis
based on these statistical features lacks the information about the relationship between
neighboring pixels.
2.3.2. Haralick features (F2)
Haralick et al. [3] suggested second order statistics of textures that considers relationship between
pairs of pixels. They used gray level co-occurrence matrices (GLCM) which have become well
known and widely used textural features. GLCM shows a function of joint probability
distributions of pairs of gray level pixels. The most commonly used seven textural features,
namely, contrast, energy, entropy, homogeneity, maximum probability, cluster shade and cluster
prominence over angles 0, 45, 90 and 135 degrees (7 features x 4 angles = 28 number) are
considered for the experimentation.

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2.3.3. Gray level difference statistics (F3)
This method is described in [25]. Five features, namely, homogeneity, contrast, energy, entropy
and mean are calculated from a single grey level difference probability distribution vector. These
features are obtained from sum of four vectors for 0, 45, 90 and 135 degrees. Methods based on
second-order statistics have higher discrimination capability than the structural and power
spectrum (transform based) methods [25].
2.3.4. Neighborhood gray tone difference matrix (F4)
Image texture can be represented as a function of the structural and tonal relationships between
the pixels. Tone is based mainly on pixel intensity (gray values), while the structure is the spatial
(location) relationship between pixels [26]. A Grey-Tone Difference Matrix (GTDM) was
proposed in [27] to correlate texture measures with human perception of textures. A GTD matrix
is a vector containing G elements. Its elements are computed as the difference between gray value
of pixel and average of gray values over a square by sliding the window centered at the pixel.
Features, namely, coarseness, contrast, busyness, complexity, and texture strength are considered.
2.3.5. Statistical feature matrix (F5)
This method of feature extraction was introduced in [28]. Four features, namely, coarseness,
contrast, period and roughness are considered.
2.3.6. Law’s texture energy measures (F6)
This method is described in [29, 30]. Six texture energy measures are given by standard deviation
of convolved image over entire region of interest. The measures are derived from three simple
vectors. L3 = (1, 2, 3) which represents averaging; E3 = (-1, 0, 1) calculating first difference
(edges); and S3 = (-1, 2, -1) corresponding to the second difference (spots). Mutually multiplying
these vectors by considering the first term as a column vector and the second term as row vector,
results in 3 X 3 matrix known as Law’s masks. The Law’s mask is convolved over texture image
and six energy statistics are calculated, which are used for texture description.
2.3.7. Fractal dimension texture analysis (F7)
Texture analysis based on fractal was introduced in [14]. The geometric complexity of spatial
pattern of textures is represented using fractals [14]. The word `fractal' refers to complex patterns
that re-occur at various scales and are independent of scales. Fractal metrics provide features
patterns of self similarity at different scales. The fractal dimension gives a global descriptor of
complexity or irregularity of a geometric object. It has been found that fractal dimension encode
textures in nature, which represents the irregularity of textures. The fractal dimension is used to
calculate roughness of a surface. The larger the fractal dimension, the rougher is the texture.
2.3.8. Fourier power spectrum (F8)
Texture descriptors are scale dependant. A texture is described in multiple resolutions to decrease
the scale sensitivity. An appropriate scale to achieve the maximum texture discrimination may be
chosen. For calculating multiscale feature, time-frequency method known as Fourier spectral
method [31] is adopted. It is an image in a space whose co-ordinate system has an interpretation
that is closely related to the characteristics of a texture (such as size or frequency). Two features,
namely, radial sum and angular sum features are computed.

147
2.3.9. Shape (F9)
Five shape features: size (x, y), area, perimeter and perimeter^2 /area are considered for
experimentation. Texture size measure may be based on co-occurrences of edge pairs with
opposite edge directions at constant distance in a direction perpendicular to the edge directions.
These statistics are derived from the second order statistics.
3. FEATURE EXTRACTION AND TEXTURE CLASSIFICATION
Feature extraction is the important stage of texture analysis. Features obtained from this stage are
used for texture classification.
3.1. Feature extraction
The steps of the proposed method are given in the Algorithm 1.
Algorithm 1: Training Algorithm
Step 1 : Read the training image block I.
Step 2 : Decompose the image, using Haar wavelet transform, into horizontal (H),
vertical (V) and diagonal (D) components.
Step 3 : Subject the H, V and D components to anisotropic diffusion.
Step 4 : Obtain texture approximation IHtxr, IVtxr, IDtxr for H, V and D components,
respectively.
Step 5 : Compute statistical features (as in section 2.3) for images IHtxr, IVtxr, IDtxr obtained
in Step 4, which form feature vector F. The F is stored in the feature database,
with class label.
Step 6 : The Steps 1 – 5 are repeated for all the training blocks of all the texture class
images and the training feature set (TF) is obtained.
Step 7 : LDA is applied on training feature set (TF) of Step 6. The discriminant feature
set (TFLDA) is obtained, which is then used for texture classification.
Step 8 : Stop.
The Algorithm 1 is executed up to different numbers of diffusion steps (t) and extracted various
features as discussed in section 2.3.
3.2. Texture classification
The texture classification is performed using k-NN classifier with ten-fold experimentation, based
on Euclidean distance [32]. The steps of testing algorithm are given in the Algorithm 2.
Algorithm 2 : Testing Algorithm (Classification of test images)
Step 1 : Read the test image block Itest.
Step 2 : Decompose the image Itest, using Haar wavelet transform, into horizontal (H),
vertical (V) and diagonal (D) components.
Step 3 : Subject the H, V and D components to anisotropic diffusion.
Step 4 : Obtain texture approximation ItestHtxr, ItestVtxr, ItestDtxr for H, V and D components,
respectively.
Step 5 : Compute statistical features (as in section 2.3) for images ItestHtxr, ItestVtxr, ItestDtxr
obtained in the Step 4, to form feature vector Ftest.
Step 6 : Project Ftest on LDA components stored in TFLDA and obtain the weights which
constitute test image feature vector FtestLDA.

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Step 7 : (Classification) The k-NN classifier (k = 3) based on Euclidean distance is
employed to classify the test image Itest using FtestLDA vectors.
Step 8 : Stop.
4. EXPERIMENTAL RESULTS AND DISCUSSION
4.1. Dataset
The sixteen texture images from Brodatz album [33] are considered for experimentation and these
are shown in the Figure 2. Each Brodatz texture sample represents one class. Each texture image
is of 256x256 pixels with 256 gray levels. The experimental dataset includes regular textures,
namely, D4, D3, D36, D75, D16, D21, D68, D24, nonregular textures D82, D11, D29, D104,
D71 and highly regular textures D6, D52, D51. Each texture image is subdivided into 16 non
overlapping blocks of 64x64 pixels. Thus, totally 256 blocks are considered. The texture images
in the dataset have different gray scale properties. The performance of unbiased texture
classification is estimated by performing ten-fold experiments. The average of ten experimental
results is reported in the paper. The images are randomly separated into distinct training and
testing sets. The 50% of the sub images are taken as training set and remaining 50% sub images
are used for testing.
Figure 2. Texture images from Brodatz album from left to right and top to bottom: D3, D4, D6, D11, D16,
D21, D24, D29, D36, D51, D52, D68, D71, D75, D82, D104.
4.2. Experimental results
The experimentation of the proposed method is carried out on Intel® Core™ i3-2330M @
2.20GHz with 4 GB RAM using MATLAB 7.9 software. The Haar wavelet transform is
employed to decompose the image, resulting in average (A), horizontal (H), vertical (V) and
diagonal (D) components. The H, V and D components of the image are then subjected to
anisotropic diffusion to find texture approximation. Further, different statistical features
(discussed in section 2.3) are computed from the texture approximations. The LDA is used to

149
enhance the class separability. The k-NN classifier is used for classification. The values for
optimization of parameters of anisotropic diffusion are: lambda = 0.25 and conduction coefficient
= 60. The experimentation for each feature set is executed up to 10 diffusion steps. The average
classification accuracy is computed for the sixteen class problem, where 16 texture classes are
considered for the experimentation from Brodatz texture dataset [33].
The Table 1 shows the comparison of average classification accuracy, average training time and
average testing time for various feature sets using the optimal number of diffusion steps. Totally,
thirty feature sets (including combinations of feature sets in section 2.3) are taken for
experimentation. The optimal number of diffusion step is the diffusion step at which the best
classification results are obtained for a given feature set. The corresponding training and testing
time are recorded.
Table 1. Comparison of average classification accuracy (ACA), average training time (ATrTm)
and average testing time (ATsTm) for the various feature sets based on wavelet transform and
anisotropic diffusion using the optimal number of diffusion step (DS) for Brodatz dataset.
Sl.
No.
Feature set DS ACA
(%)
ATrTm
(sec.)
ATsTm
(sec.)
No. of
features
1 F1 9 79.30 3.71 0.24 15
2 F2 8 92.89 35.92 2.25 84
3 F3 7 84.69 3.06 0.19 15
4 F4 1 86.17 4.28 0.27 15
5 F5 1 73.44 3.39 0.22 12
6 F6 10 92.11 4.20 0.27 18
7 F7 9 42.19 4.01 0.25 12
8 F8 1 39.84 1.54 0.10 6
9 F9 1 19.84 1.37 0.09 15
10 F1+F3 9 93.20 4.61 0.29 30
11 F1+F3+F4 7 95.78 6.32 0.40 45
12 F1+F3+F4+F5 10 98.75 8.19 0.51 54
13 F1+F3+F4+F5+F6 9 97.58 8.20 0.52 72
14 F1+F3+F4+F5+F6+F7 9 96.41 8.61 0.54 84
15 F1+F3+F4+F5+F6+F7+F8 9 94.84 8.83 0.55 90
16 F1+F3+F4+F5+F6+F7+F8+F9 4 36.09 7.48 0.47 105
17 F1+F3+F5+F6 9 98.36 5.45 0.34 57
18 F3+F4+F5+F6 1 97.73 6.80 0.43 57
19 F1+F4+F5+F6 10 98.28 8.64 0.54 57
20 F6+F7+F8+f9 4 35.78 3.90 0.25 51
21 F1+F3+F5 10 98.36 6.33 0.40 39
22 F1+F3+F6 9 97.42 5.10 0.32 48
23 F1+F4+F5 6 97.81 6.91 0.44 39
24 F3+F4+F5 10 98.05 8.79 0.55 39
25 F6+F7 10 94.06 5.19 0.33 30
26 F4+F5 6 97.66 7.47 0.47 24
27 F3+F4 1 93.20 4.14 0.26 30
28 F5+F6 10 97.34 6.27 0.40 27
29 F8+F9 1 22.58 1.42 0.09 21
30 F2+F4 9 86.88 41.73 2.61 99
It is observed from the Table 1 that the single feature set (F1 to F9) gives poor classification
accuracy except F2 and F6. Even though the feature set F2 based on Haralick features gives
92.89% of classification accuracy, it is computationally expensive. The same observation is

150
verified for the combination of feature sets F2+F4. Hence, F2 is not considered with other
combinations of feature set. The combination of feature set gives improved classification
accuracy compared to the feature set considered alone. Hence, the accuracy of classification
depends on the type of features used.
It is also observed from the Table 1 that the feature sets F1, F3, F4, F5 and F6 are dominant
feature sets. These feature sets give better classification accuracy in combinations. The optimal
average classification accuracy of 98.75% is observed for F1+F3+F4+F5 feature set (Sl. No. 12)
at the tenth diffusion step with average training time of 8.19 sec. and average testing time of 0.51
sec.
The proposed method is experimented on same dataset [33] as used in [8] and [19] so that, results
can be compared. The methods NSCT and LDBP used in [8] and [19] are briefly described in
Appendix. The Table 2 shows comparison of average classification accuracy attained by the
proposed method and other methods in the literature on Brodatz dataset.
Table 2. Comparison of average classification accuracy (%) obtained by the proposed method and other
methods in the literature [8, 19] on Brodatz dataset.
Image no.
Average classification accuracy (%)
Method in [8]
(NSCT & LDBP)
Method in [19]
(PDE & LDBP)
Proposed method
( WT & PDE)
D104 100 100 100
D11 100 96.25 96.25
D16 100 100 100
D21 100 100 100
D24 100 98.75 98.75
D29 100 100 100
D3 100 95 100
D36 87.5 98.75 95
D4 100 100 97.5
D51 100 97.5 95
D52 100 100 97.5
D6 100 100 100
D68 100 100 100
D71 100 97.5 100
D75 100 100 100
D82 87.5 100 100
Average classification
accuracy (%)
98.43 98.98 98.75
Average training
time (sec.)
278.50 14.92 8.18
Average testing
time (sec.)
12.42 0.93 0.51
It is observed from the Table 2 that the average classification accuracy of the proposed method is
improved as compared to the method based on NSCT and LDBP [8]. It is marginally less
compared to the method based on PDE and LDBP [19]. The training time and testing time of the
proposed method is considerably reduced. Thus, the proposed method is effective and
computationally less expensive.

151
5. CONCLUSIONS
In this paper, a novel method of PDE based statistical features for texture analysis using wavelet
transform is proposed and is implemented on Brodatz dataset. The proposed method is compared
with the other methods in the literature [8, 19]. Following conclusions can be made from the
experimentation:
 Better classification accuracy using simple statistical descriptors is achieved.
 Combination of feature sets improves the average classification accuracy.
 Feature set F2 (that computes the Haralick features) and other combination of feature
sets containing F2 (F2+F4) is observed to be computationally expensive.
 Most feature sets are computationally inexpensive making it suitable for real time
applications.
 The classification accuracy can be increased with more sophisticated classifier.
 The computational cost is reduced significantly as compared to other methods in the
literature, while better classification accuracy is attained.

ACKNOWLEDGEMENTS
The authors are grateful to the reviewers for their valuable comments and suggestion, which
substantially improved the quality of the paper.
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New York.
[34] Weickert, J., (1998) Anisotropic Diffusion in Image Processing. ECMI Series, Teubner –Verlag.
[35] Coifman, R. R., & Donoho, D. L., (1995) “Translation Invariant Denoising in Wavelets and
Statistics”, Lecture Notes in Statistics, pp 125-150.
[36] Cunha, A. L., Zhou, J., & Do, M. N., (2006) “The Nonsubsampled Contourlet Transform: Theory,
Design and Applications”, IEEE Trans. Image Processing, Vol. 15, No.10, pp 3089-3101.
APPENDIX
1. Anisotropic diffusion
The Partial differential equation (PDE) [19] techniques have been widely used for enhancing
images. The diffusion process is equivalent to a smoothing process with a Gaussian kernel (linear
diffusion). A major drawback of the linear diffusion is its uniform filtering of local signal features
and noise. This problem was addressed by Perona and Malik [15], who proposed a nonlinear
diffusion process, where diffusion can take place with a variable diffusion in order to control the
smoothing effects [34] is represented by Eq. (1):
(1)
e
KI
Ip
2
)/||(||
||)(||



153
where  Ip  is the conduction function, I is the gradient magnitude and the parameter K controls
the conduction.
It was demonstrated [15] that anisotropic diffusion clearly outperforms the canny edge detector,
making image boundaries sharp. The Figure A1 represents the anisotropic diffusion on Brodatz
texture image D104 of size 64 x 64. The row 1 represents the effect of anisotropic diffusion
resulting in smoothing of image (cartoon approximation). The row 3 is the texture component of
the corresponding images in the row 1, obtained by the difference between original image and it’s
cartoon approximation. Row 2 and 4 depict the contour representations of images in row 1 and
row 3, respectively.
(a) I (b) t1 (c) t2 (d) t3 (e) t4 (f) t5
Figure A1. Effect of anisotropic diffusion on Brodatz texture image D104 of size 64 x 64. Row 1 represents
the smoothed images [(b) to (f)] of the original image I in (a) at successive diffusion steps t1 through t5.
Row 3 represents the texture component of the image in row 1. Row 2 and 4 represent the contour plot of
the corresponding images in row 1 and row 3, respectively [19].
2. Local directional binary patterns
The most important property of local directional binary patterns (LDBP) [8] is computational
simplicity. The basic idea of LDBP is that, 3x3 kernel of image can be treated as basic texture
region. The gray value of central pixel is compared with the gray values of eight pixels around it.
The central gray pixel value is the threshold value. If the gray value of surrounding pixel is larger
than gray value of central pixel, the surrounding pixel is marked as one otherwise zero. The
binary values of all surrounding pixels can be obtained. All surrounding pixels are given different
metrics. The metrics is multiplied with a binary value of surrounding pixels as shown in the
Figure A2. Further, the sum of product of binary value and metrics of all surrounding pixels is set
as the value of local directional binary pattern of central pixel. The value of local directional
binary patterns of all pixels in an image can be obtained through such calculation neglecting the
pixels of edges. The LDBP weight fb of the central pixel (xc, yc) can be calculated using the Eq. (2)
and Eq. (3).
     

7
0
45*cos,
j
cjccb jffvyxf (2)
where
 
,0
.0
1
0






 x
x
xv (3)

154
fc and fj are the pixel values at the center and jth
neighbor in 3x3 kernel of image. Finally, a matrix
of LDBP weights is obtained.
Figure A2. Transformation of neighborhood pixels to calculate central pixel weight in LDBP. (a) A sample
neighborhood, (b) Resulting binary thresholded result, (c) LDBP mask, (d) Resultant weights after
multiplying corresponding elements of (b) and (c) [18].
3. Nonsubsampled contourlet transform
An important feature of nonsubsampled contourlet transform (NSCT) is its stability with respect
to shifts of the input signals [35]. The lack of shift invariance during image processing will cause
pseudo Gibbs phenomena around singularities. In order to enhance directional selectivity and
shift invariance and to get rid of the frequency aliasing of contourlet, Cunha et al. [36] presented a
shift invariant version of the contourlet transform namely, NSCT. To obtain a shift invariant,
directional multiresolution image representation, the NSCT is built upon iterated nonsubsampled
filter banks. The NSCT combines nonsubsampled pyramids to provide multi scale decomposition
and nonsubsampled directional filter bank (DFB) to provide directional decomposition. The two
level NSCT decomposition is shown in the Figure A3.
Figure A3. Nonsubsampled contourlet transform (a) Nonsubsampled filter bank (b) Idealized frequency
partitioning [36].
The building block of the nonsubsampled pyramid is shown in the Figure A3(a). It is a two
channel nonsubsampled filter bank which has no downsampling or upsampling and therefore is
shift invariant. The ideal frequency response of a nonsubsampled DFB is shown in the Figure
A3(b). The building block of a nonsubsampled DFB is a two channel nonsubsampled filter bank

155
AUTHORS
Dr. P.S. Hiremath Professor, Department of Computer Science (MCA), KLE
Technological University, BVBCET Campus, Hubli- 580031, Karnataka, India. His
research areas of interest are Image Processing, Pattern Recognition, Computational
Fluid Dynamics, Optimization Techniques, Data Mining, Web Mining and Knowledge
Discovery Techniques. He has published 190 research papers in peer reviewed
International Journals.
Rohini A. Bhusnurmath is currently pursuing her Ph.D. in Computer Science in the
Department of P.G. Studies and Research in Computer Science at Gulbarga University,
Gulbarga- 585106, Karnataka, India. Presently she is working as a Lecturer in the
department of Computer Science, at Govt. P.U. College for Girls, Bijapur, Karnataka,
India. Her research areas of interest are Image Processing and Pattern Recognition.

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