International Journal of Advancements in Research & Technology, Volume 2, Issue4, April-2013
ISSN 2278-7763
249
Texture Classification Using Curvelet Transform
1
1
S. Prabha, 2Dr. M. Sasikala
Department of Electronics and Instrumentation Engg, Anand Institute of Higher Technology, Chennai
2
Department of Electronics and Communication Engg, C.E.G Campus, Anna University, Chennai.
Abstrat-Brain tumors are due to abnormal growths of tissue in
the brain. The most common group is gliomas, followed by
meningiomas. Magnetic resonance imaging (MRI) is currently
an indispensable diagnostic imaging technique for the early
detection of any abnormal changes in tissues and organs. It
possesses fairly good contrast resolution for different tissues. It is
therefore widely used to provide images which distinguish brain
tumours from normal tissues. Although MRI can clearly supply
the location and size of tumours, it is unable to classify tumour
types, determination of which usually requires a biopsy.
However a biopsy is a painful process for patients, and in some
cases such as brain stem gliomas, may be too hazardous. These
limitations necessative development of new analysis techniques
that will improve diagnostic ability. One promising technique is
texture analysis, which characterizes tissues to determine
changes in functional characteristics of organs at the onset of
disease. In this work texture classification based on curvelet
transform has been performed. A curvelet based texture feature
set is extracted from the region of interest. Texture features set
consists of entropy and energy. Fuzzy-c-means algorithm is used
as a classifier to classify two sets of brain images, benign tumour
and malignant tumour.
Index Terms: Active contours; Curvelets; Texture classification;
Magnetic resonance image.
1. INTRODUCTION
The analysis of texture in images provides an important cue to
the recognition of objects. It has been recently observed that
different image objects are best characterized by different
texture methods (Kaplan 1999). Successful applications of
texture analysis methods have been widely found in
industrial, biomedical, remote sensing areas and target
recognition
Texture methods used can be categorized as
statistical, geometrical, model based and signal processing
which is proposed by Tuceryan and Jain (1998). Some
statistical methods used are co-occurrence matrix features and
auto correlation function. Tuceryan and Jain (1990) proposed
geometrical methods in which textures are considered to be
composed of texture primitives and are extracted and
analyzed. Several stochastic models have been proposed for
texture modeling and classification such as Gaussian Markov
random fields and spatial auto correlation function model
proposed by Patrizio, Alessandro and Gaetano (2002).
The signal processing techniques are mainly based
on texture filtering for analyzing the frequency contents either
in spatial domain or in frequency domain. Filter bank instead
of a single filter has been proposed, giving raise to several
Copyright © 2013 SciResPub.
multi channel texture analysis systems such as gabor filters
and wavelet transforms proposed by Unser (1989). The major
disadvantage of gabor transform is that its output are not
mutually orthogonal which may result in a significant
correlation between texture features.
In the last decade, Arivazhagan and Ganesan(2003)
performed texture classification method based on wavelet
transform. The success of wavelets is mainly due to the good
performance for piecewise smooth functions in one
dimension. Unfortunately, such is not the case in two
dimensions.Wavelets in two dimensions are obtained by a
tensor product of one dimensional wavelet and they are thus
good at isolating the discontinuity across an edge, but will not
see the smoothness along the edge (Minh and Martin 2003).
To overcome the weakness of wavelets in higher
dimensions, Candes and Donoho(1998) pioneered a new
system of representations named ridgelets which deal
effectively with line singularities in two dimensions. The idea
is to map a line singularity into point singularity using the
radon transform. So ridgelet transform allows representing
edges and other singularities in a more efficient way than
wavelet transform which is proposed by Patrizio
Campisi,Alessandro and Gaetano(2002).
In image processing, edges are typically curved
rather than straight and ridgelets alone cannot yield efficient
representations. However at sufficiently fine scales, curved
edges are almost straight and so to capture curved edges, one
ought to be able to deploy ridgelets in a localized manner, at
sufficiently fine scales. Candes and Donoho(2002) proposed
another multiscale transform called curvelet transform which
is designed to handle curve discontinuities well. Here the idea
is to partition the curves into collection of the ridge fragments
and then handle each fragment using the ridglet transform.
Lucia Dettori and Lindsay Semler(2007) proposed automated
imaging system for classification of tissues in medical images
obtained from CT scan It is found that curvelet transform
outperforms all other multi-resolution techniques yielding
high accuracy rates.
The texture classification algorithm proposed in this article
consists of four main steps: segmentation of region of interest
from MRI scans, application of the discrete curvelet transform
on the region of interest, extraction of the most discriminative
texture features from the curvelet coefficients and creation of
a classifier(fuzzy-c-means) that identifies the various tissues.
The general algorithm is summarized in the methodology
diagram below (Fig. 1)
International Journal of Advancements in Research & Technology, Volume 2, Issue4, April-2013
ISSN 2278-7763
Brain
Image
Segmentation of
region of inter est
250
Where C is any other variable curve, and the constants c1 and
Curvelet
Transfor m
c2 depending on C , are the averages of u0 inside C and
respectively outside C . In this simple case, it is obvious that
C0 , the boundary of the object, is the minimizer of the fitting
term
Benign
/Malignant
tumor
Textur e
Descriptor s
Cla ssifier
inf
C
Fig.1 Block Diagram of Texture Classification
The paper is structured as follows: Section 2
describes segmentation of region of interest using active
contour model. Section 3 discusses the discrete curvelet
transform and feature extraction. Section 4 presents texture
classification methods.
2. SEGMENTATION
In this work, segmentation of region of interest is performed
using active contour model without edges (Tony 2001). The
model can detect objects whose boundaries are not necessarily
defined by gradient. Energy is minimized which can be seen
as a particular case of the minimal partition problem. In this
work, the stopping term does not depend on the gradient of
the image, as in the classical active contour models, but is
instead related to a particular segmentation of the image. The
initial curve can be anywhere in the image, and interior
contours are automatically detected. The model is trying to
separate the image into regions based on intensities.
2.1 DESCRIPTION OF THE MODEL
{F1 (C )  F2 (C )}  0  F1 (C0 )  F2 (C0 )
This can be seen easily. For instance, if the curve C is outside
the object, then F1(C)>0 and F2(C)≈ 0. If the curve C is
inside the object, then F1(C)≈ 0 but F2(C)>0 If the curve C is
both inside and outside the object, then F1(C)>0 and F2(C)>0.
Finally, the fitting energy is minimized if C=C0 , i.e., if the
curve C is on the boundary of the object. In our active contour
model minimize the above fitting term and add some
regularizing terms, like the length of the curve C, and (or) the
area of the region inside C. Therefore, we introduce the
energy functional F(c1,c2, C) , defined by
F (c1 , c2 , C )  .Length(C )  v. Area(inside(C ))
+
1

2
u0 ( x, y )  c1 dxdy
inside ( C )
+ 2

2
u0 ( x, y )  c2 dxdy
outside (C )
1 , 2 > 0 are fixed parameters. In
almost all our numerical calculations fix the value of 1 =
2 =1 and v = 0.
Where μ ≥ 0, v ≥ 0,
Therefore consider the minimization problem as
The evolving curve C is defined in Ω, as the boundary of an
open subset ω of Ω (i.e. ω  Ω , and C=∂ω). In this work,
inside(C) denotes the region ω , and outside(C) denotes the
region Ω\ ω . Our method is the minimization of an energy
based-segmentation. Assume that the image u0 is formed by
two regions of approximatively piecewise-constant intensities,
i
o
of distinct values u0 and u0 . Assume further that the object
i
to be detected is represented by the region with the value u0 .
i
Let denote its boundary by C0. Then u0 ≈ u0 inside the object
o
[or inside (C0)], and u0 ≈ u0 outside the object [or outside
(C0)]. Consider the following “fitting” term:
F1(C)+F2(C)=
2

u0 ( x, y)  c1 dxdy
+

inside ( C )
2
u0 ( x, y )  c2 dxdy
outside (C )
Copyright © 2013 SciResPub.
inf
c1 , c2 , C
F (c1 , c2 , C )
Benign and malignant tumor images are given as input to the
active contour model which gives segmented image as shown
in fig.2.
International Journal of Advancements in Research & Technology, Volume 2, Issue4, April-2013
ISSN 2278-7763
Fig.2 Input image and segmented image using active contour
model
3. Discrete Curvelet transform
Curvelets are effective at detecting image activity along
curves. The discrete version implemented in this work uses a
‘USFFT’ (Unequally spaced Fast Fourier transform)
algorithm. The fast discrete curvelet transform is simpler,
faster and less redundant. This approach uses a decimated
rectangular grid tilted along the main direction of each
curvelet.
The discrete curvlet transform is implemented via USFFT in
four steps.
1. Apply the 2D FFT(fast fourier transform) and obtain
Fourier samples f[n1, n2], −n/2 <= n1, n2 < n/2.
2. For each scale/angle pair (j, l), resample (or interpolate)
f[n1, n2] to obtain sampled values f[n1, n2 − n1 tanөl] for (n1,
n2) Є Pj .
3. Multiply the interpolated (or sheared) object f with the
parabolic window Uj , effectively localizing f near the
parallelogram with orientation өl, and obtain
f [n1 , n2 ]  f [n1 , n2  n1 tan l ]U j [n1 , n2 ]
4. Apply the inverse 2D FFT to each fj,l, hence collecting the
discrete coefficients cD(j,l,k).
The approximate scales and orientations can be seen in fig 3.
Fig. 3 Finite tiling by the polar ‘wedges’
Copyright © 2013 SciResPub.
251
Fig. 3 illustrates the basic digital tiling. The windows
Uj,l, smoothly localize the fourier transform near the sheared
wedges obeying the parabolic scaling. The shaded region
represents one such typical wedge.
The inputs to the curvelet transform are x and
nscales. x is N by N pixel array of an image. nscales defines
number of scales including the coarsest wavelet level. The
output of curvelet transform represents cell array of curvelet
coefficients in which integer scale varies from coarsest to
finest scale. Two parameters involved in the digital
implementation of the curvelet transform are number of
resolution and number of angles at the coarsest level. The
parameters are bound by the two constraints: the maximum
number of resolutions depends on the original image size and
the number of angles at the second coarsest level must be at
least eight and multiple of four. Several features were
calculated on the curvelet coefficient. The following four
feature vectors were investigated: entropy and energy. Since
the region of interest is 16x16 pixels, the maximum possible
resolution extraction is two levels of resolutions.
4. Texture classification method
The original image is decomposed using discrete curvelet
transform. Texture features are calculated from each Curvelet
sub-band. In order to improve the classification gain, cooccurrence matrix is formed for each sub-band of discrete
curvelet transform, which gives the information about the
spatial distribution of gray scale values. From the cooccurrence matrix, the features such as energy and entropy are
calculated.
4.1 Feature Extraction
Texture classification is based on the texture features
extracted from the curvelet co-efficient. A gray tone spatial
dependence matrix approach introduced by Haralick (1979),
which is well known statistical method for extracting second
order texture information from images is used. Twodimensional co-occurrence (gray-level dependence) matrices
are generally used in texture analysis because they are able to
capture the spatial dependence of gray-level values within an
image. The gray-level co-occurrence matrix can reveal certain
properties about the spatial distribution of the gray levels in
the texture image. A 2D co-occurrence matrix, P is an n x n
matrix, where n is the number of gray-levels within an image.
A pixel with the gray level intensity value i occurs in a
specific spatial relationship to a pixel with the value j in graylevel co-occurrence matrix (GLCM). The matrix acts as an
accumulator so that P[i , j] counts the number of pixel pairs
having the intensities i and j. Pixel pairs are defined by a
distance and direction which can be represented by a
displacement vector d =(dx,dy), where dx represents the
number of pixels moved along the x-axis, and dy represents
the number of pixels moved along the y-axis of an image . In
order to quantify this spatial dependence of gray level values,
International Journal of Advancements in Research & Technology, Volume 2, Issue4, April-2013
ISSN 2278-7763
various textural features including Entropy, Energy (Angular
Second Moment), are calculated.
This method is based on the estimation of the second
order joint conditional probability density function P(i,j/ d,ө)
where ө= 0,45,90 and 135 degrees. Each P(i,j / d,ө) is the
probability going from gray level i to gray level j, given that
the inter sample spacing is d and the direction is given by
angle ө (offset degree). This is also referred to as cooccurrence matrix. The co-occurrence matrix is calculated for
the source images for ө = 0 degrees and distance d = 1 (offset
distance). Four texture features are calculated from the cooccurrence matrix.
Let P(i,j) denote the co-occurrence matrix and N the
number of distinct gray level in the quantized image. The
following four texture features are calculated from cooccurrence matrix.
4.2 Entropy
252
5. RESULTS AND DISCUSSION
Twenty benign tumor images and sixteen malignant tumor
images are used for the study. The tumor regions in the MRI
brain images are segmented using active contour model.
Benign tumor images used in this work are shown in fig.4.
Malignant tumor images are shown in fig.5.
Curvelet transform is applied to the region of
interest. For region of interest of 16x16 pixels, finest scale
curvelet subband matrix is obtained. Texture features such as
entropy and energy are extracted from each curvelet sub band.
Then feature values are given as input to fuzzy-c-means
algorithm, all benign tumor images are correctly classified,
only one malignant tumor image is not correctly classified.
Curvelet based feature extraction yielded higher accuracy
rates than wavelet and ridgelet transform.Table1 shows tumor
classification of benign and malignant images. The
classification accuracy obtained for curvelet transform is
97.22%.
Entropy measures the randomness of a gray-level distribution.
The entropy is expected to be high if the gray levels are
distributed randomly throughout the image
M
Entropy
N
 
i 1

P [ i , j ] log P [ i , j ]
j 1
Where P[i,j] are the pixel values at the (i,j)
coordinates of the image. The size of the image is M*N.
4.3 Energy (Angular second moment)
Fig.4 benign tumor images
Energy measures the number of repeated pairs. The energy is
expected to be high if the occurrence of repeated pixel pairs is
high
M
Energy

N

i 1
P 2 [i, j ]
j 1
Where P[i,j] are the pixel values at the (i,j)
coordinates of the image. The size of the image is M*N.
Fig.5 Malignant tumor images
4.4 Fuzzy c-means Algorithm
Unsupervised clustering method is used as a classifier for
detecting brain tumor images. In the unsupervised clustering
method, fuzzy-c-means algorithm is used for classification.
Fuzzy c-means (FCM) is a data clustering technique in which
a dataset is grouped into number of clusters with every data
point in the dataset belonging to every cluster to a certain
degree. For example, a certain data point that lies close to the
center of a cluster will have a high degree of belonging or
membership to that cluster and another data point that lies far
away from the center of a cluster will have a low degree of
belonging or membership to that cluster. In this paper the
extracted feature value of energy and entropy from the
curvelet transform are given as input to FCM. FCM
automatically classifies the benign and malignant tumors with
high classification accuracy..
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Table 1
Tumor Classification
Tumor
Correct
Classification
MisClassification
Benign(20)
20
0
Malignant(16)
15
1
Total(36)
35
1
International Journal of Advancements in Research & Technology, Volume 2, Issue4, April-2013
ISSN 2278-7763
253
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