Curvelet based Image Compression using Support Vector Machine and Core Vector Machine – A Review

Abstract

Images are very important documents nowadays. To work with them in some applications they need to be compressed, more or less depending on the purpose of the application. To reduce transmission cost and storage requirements, competent image compression schemes without humiliation of image quality are required. Several image coding techniques were developed so far for both lossless and lossy image compression. Extensions of 1-D transforms such as wavelet transform have limitations of capturing geometry of image edges. Functions that have discontinuities along straight lines cannot be effectively represented by normal wavelet transforms But natural images have geographic lines such as edges, textures which cannot be well reconstructed if compression is done by 1-D Transforms. Nowadays image coding is done, using Curvelet Transform since it supports different orientations of image textures. An investigation is done on various types of image coding techniques based on Curvelet Transform that exist. This paper deals with study of image compression techniques using Curvelet Transform based on Support vector machine and Core vector machine with their performance results.

Full text

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
675
Curvelet based Image Compression using Support Vector Machine and Core
Vector Machine – A Review
P.Anandan1, R.S.Sabeenian2
Abstract
Images are very important documents nowadays. To
work with them in some applications they need to be
compressed, more or less depending on the purpose
of the application. To reduce transmission cost and
storage requirements, competent image compression
schemes without humiliation of image quality are
required. Several image coding techniques were
developed so far for both lossless and lossy image
compression. Extensions of 1-D transforms such as
wavelet transform have limitations of capturing
geometry of image edges. Functions that have
discontinuities along straight lines cannot be
effectively represented by normal wavelet
transforms But natural images have geographic
lines such as edges, textures which cannot be well
reconstructed if compression is done by 1-D
Transforms. Nowadays image coding is done, using
Curvelet Transform since it supports different
orientations of image textures. An investigation is
done on various types of image coding techniques
based on Curvelet Transform that exist. This paper
deals with study of image compression techniques
using Curvelet Transform based on Support vector
machine and Core vector machine with their
performance results.
Keywords
Curvelet Transform, Support vector machine, Core vector
machine, Image Compression, Image Coding.
1. Introduction
The technique used to decrease data storage
requirements and communication costs, is data
compression, which reduces redundancies in data
representations. Due to the pervasive distribution
Manu script received June, 2014.
P.Anandan, Department of Electronics and Communication
Engineering, R.M.D. Engineering College, Kavaraipettai-601206,
Thiruvallur District, Tamilnadu, India
Dr.R.S.Sabeenian, Department of Electronics and
Communication Engineering, Sona College of Technology, Salem-
63600, Tamilnadu, India
digital image contents, compression of images or data
outcomes inconsiderable reduction in the storage
capacity of the memory devices. Transferring images
without compression over digital networks needs
very high bandwidth. Due to that the need for
efficient storage and transferring medical images is
noticeably increasing, image compression is
essential. The heart of any image processing tasks is
an efficient representation of visual information lies
in the image. The medical data is articulated as
images or other types of digital signals, such as
Magnetic Resonance Imaging, Computer
Tomography, Ultrasound, Positron Emission
Tomography etc.
Transferring image information into transform
domain will be more competent rather the image
itself. Image transformations can be done using
various transforms such as Discrete Cosine
Transform, Discrete Fourier Transform, Wavelet
Transform, Contourlet Transform etc. Compression
can be achieved by transferring data into transform
domain, quantizing the transformed coefficients and
encoding the quantized coefficients.
To avoid redundancy, the transform must be atleast
biorthogonal, and to save CPU time, the transform‟s
algorithm must be fast. Various image compression
techniques are exists so far. Even so observations
have noted that wavelets may not be best choice
presenting natural images. This is because the
tendency of wavelets to ignore to smoothness along
the edges (cannot provide „sparse‟ representation).
This property has been taken advantage by some
novel transforms like ridge let and curvelet
transform. These can be formed into elements which
are anisotropic and exhibit high directional
sensitivity.
The curvelet discrete transform are theoretically
simpler, faster and less redundant than earlier
implementations. In this paper, we have reviewed
some Curvelet based image compression techniques
with Support vector machine and Core vector
machine which approximates the curvelet coefficients
using a fewer support vectors and weights.

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
676
2. The Curvelet Transform
To overcome the drawbacks of wavelet transform, A
new multi-resolution transform was developed by
Candés and Donoho in 1999. The transform that is a
two-dimensional anisotropic extension of wavelet,
originally designed to represent edges and other
singularities beside curves better than wavelet
transforms[5]. Although curvelets is an extension of
wavelets, there exists a association between curvelet
and wavelet subbands.
In wavelet transform, the elements have only location
and scale parameters. In curvelt transform, the
elements have location, scale and orientation
parameters. The fundamental defect reside in wavelet
transform is that, unable to represent edge
discontinuities along curves. In compression process,
limited number of coefficients are required but in
case of reconstructing the edges properly along
curves several wavelet coefficients are employed.
This is mainly because of the reason that, in case of
mapping large wavelet coefficients, edges are
repeated at scale after scale [12]. It required a
transform that can handle 2D singularities along the
curves sparsely distributed. As a result new multi
resolution wavelet transform is produced[13].
Curvelet basis elements possess wavelet basis
function qualities but these also oriented at various
directions as a result of it edge discontinuities and
other singularities are well defined by it when
compared to wavelet transform [14]. Curvelet
transform comes under multiscale geometric
transform. One of the special member in it wavelet
transform. This transform has multiscale pyramid
with many directions at each length and scale.
The superiority of curvelets over wavelets in cases
such as,
i. Optimally sparse representation of objects
with edges.
ii. Optimal image reconstruction in several ill-
posed problems.
iii. Optimal sparse representation of wave
propagators.
3. The Second Generation of Curvelet
Transform
The Curvelet transform has been taken into two
major revisions. At begining the curvelet transform
(“curvelet 99”transform now) used a computer series
of steps involving the ridgelet analysis of the radon
transform of the image [candes & Donoho, 2000]. It
was found that it performance was slowly exceeding.
Soon after they introduced, researchers developed
numerical algorithms for their implementation
[Donoho Duncan, 2000] and reported on a series of
practical successes[Starck, Murtagh, Candes
&Donoho, 2003].
In order to make use of curvelets and easy
understanding curvelets are redesigned. In this new
method, the use of the ridgelet transform was
eliminated, as a result redundancy is reduced and
speed is increased. It is faster, simpler and less
redundant than “curvelet 99” transform [2] and [10].
3.1 Continuous-Time Curvelet Transforms
Authors have worked throughout in 2D(i.e., R2) with
spatial variables ‟x‟, with „w‟ a frequency domain
variable, and with r and θ polar coordinates in the
frequency domain. They start with a pair of windows
w(r) called as “radial window” and v(t) called as
“angular window”[7]. These windows are smooth,
non-negative and real-valued, with W taking positive
real arguments and supported on r ϵ (1/2,2) and „v‟
taking real arguments and supported on t ϵ [-1,1]. For
each j ≥ j0, the frequency window Uj is defined in the
fourier domain as,
According to equation 1, Uj is a polar „„wedge”
window, as show in Fig. 1.
Figure1. Continuous curvelet support in the
frequency domain
To define the waveform φj(x) by means of its Fourier
transform, φj(x) is a „„mother” curvelet in the sense
that all curvelets at scale 2 - j can be obtained by
rotations and translations of φj(x). Introduce the
equispaced sequence of rotation angles θ l=2π.2-[j/2].l
with l = 0, 1, . such that 0≤ θ l<2 π, and the sequence
of translation parameters k = (k1, k2) ϵ Z2.
Uj(r,θ) = 2-3j/4W(2-jr)V(2[j/2] θ/2π) ----------(1)
̴2j
̴2j/2

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
677
With these notations, curvelets are defined by
where R0 is the rotation by θ radians.
A curvelet coefficient is then simply the inner
product between an element ƒ ϵ L2(R2) and a
curvelet φ j,l,k,
Reconstruction formula is
3.2 Digital Curvelet Transform
The window Uj smoothly extracts frequencies near
the dyadie corona and near the angle in continuous
time. For cartesian array, the corona and rotations are
not especially adapted. It is convenient to replace
them by Cartesian equivalent. It is done based on
concentric squares and shears [1] and [8]. The
“Cartesian window “ can be defined as
Figure2. Digital curvelet tiling of space and
frequency
Wj (ω) is a window of the form
where φ is defined as the product of low-pass one-
dimensional windows
The function φ obeys 0≤ φ ≤1, might be equal to 1
on [-1/2,1/2], and vanishes outside of [-2, 2] [11].
4. Curvelet Transform and Support
Vector Machine for Image
Compression
4.1. SVM Regression for Image Compression
Because of good generalization ability the SVM has
been widely used. At first, it is designed to solve
pattern recognition problem. Regression is an
extension use of classification. It is a non-seperable
classification that each data point can be tought of
being as its own class.
In regression process, a set of training points are
given, the real function is approximated with in a
predefined error Ɛ by choosing the minimum
number of training points. There is a corresponding
weight for each training point chosen by the SVM
(support vector). Number of Vectors and Weights is
less than training points, which is that SVM
regression can carry out data compression.
The regression problem can be formulated as follows:
SVM attempts to learn the input - output relationship
from the given training points (x1,y1),
(x2,y2)………(xl,yl) where xi ϵ Rn and yi ϵ R. In
the case of regression, vapnik‟s linear loss function is
used with insensibility zones as a measure of the
error between f(x) and y.
4.2. Compression of Curvelet Coefficients
using SVM
Initially, the original image is decomposed into
number of sub-bands. These sub-band are the
representation of image in different frequency range
and has different importance. Most of the image
energy is resides in lowest sub-band, in the
reconstruction of image it plays a vital role because
low frequency information is highly sensible to
human eyes. So the authors have used different
compression technique for different subbands to
import information with given bit rate. The lowest
sub-band is encoded by DPCM, which is nearly
lossless. The SVM regression compressed the finer
scale sub-bands which approximates the curvelet
coefficients using a fewer support vectors and
weights. In addition, some of the finer scale sub-
bands are discarded directly due to the reason that
φj,l,k(x) = φ(Rθl(x-xkj,l)) ------------------(2)
ƒ= ∑ ‹ ƒ, φj,l,k › φj,l,k -----------------(4)
j,l,k
( , , ) : , ( ) ( )
R
c j l k f f x x dxj, l, k j, l,k
²
φφ
---(3)
Ũj(ω) := W j (ω)Vj (ω) ---------------
(5)
Wj (ω) = ( φ2j+1(ω) - φ2j(ω) )1/2 , j ≥ 0 ---------(6)
φj(ω1, ω2) = φ(2-j ω1) φ(2-j ω2) --------------(7)
N
ƒ(x,w) = ∑wiφi(x) -------------(8)
i=1
Error = ǀƒ(x,w) – y ǀ = 0 if ǀ y - ƒ(x,w) ǀ ≤ Ɛ
=ǀy-ƒ(x,w)ǀ-Ɛ if ǀ y - ƒ(x,w) ǀ > Ɛ
---(9)
0
50
100
150
200
-50
-100
-150
-200
0
50
100
150
200
-50
-100
-150

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
678
they only contain a little amount of energy and have
little noticeable effect on image quality.
Figure3. Compression Scheme
Figure4. Decompression Scheme
4.3. Curvelet Coefficients Organization
By means of fast discrete curvelet transform the
original image was decomposed into frequency
domain. Based on decomposition rules, we can get
the scale number (nscales=log2n-3, where [m,n] =
size(image)). In this paper, nscales equals 6, because
the size of image they used was 512 pixels ×
512pixels.
After the process of decomposition the original
image was divided into three levels: coarse Detail
and fine. The low frequency coefficients were
assigned to coarse (inner level). The high frequency
coefficients were assigned to fine (outer mostlevel).
The middle frequency coefficients were assigned to
Detail. The detailed structure of the curvelet
coefficients are shown in Table 1.
The characteristics of curvelet coefficients are given
below:
(1) Most of the image energy is compressed
into the lowest sub-band. The rest energy is
spread over other sub-bands, reducing from
low frequency to high frequency.
(2) The highest value of coefficients focused on
the first level.
(3) The lowest value of coefficients focused on
the final level.
(4) With the enhanced number of scale,
coefficients includes more zero.
4.4. Curvelet Coefficients Normalization
The important step of image compression is
normalization. Normalizing curvelet coefficients is
used in SVM regression method. It will produce the
weight that are lower in maginitude but having
similar value and also makes the weight more
compressible. It is mainly used to overcome the
coefficients of different subbands variation.
Coefficients normalization can be done using the
relation
where cmin and cmax are the minimal and the maximal
curvelet coefficients in the sub-band, respectively, c
is the coefficient to be normalized and cꞌ is the value
after normalization.
4.5. Curvelet Coefficient Encoding
We can get the support vectors and weights after
SVM regression and they should be encoded in the
coding bit stream. In decoding process, to produce
the original curvelet coefficients by the regression
modes, the sum of regression modes is used with the
support vectors and weight [9].
The weight in the SVM regression provides the
support vector one by one. If any input training point
is choosen as the support vector, it should have some
corresponding weight. The support vector has two
meanings in the SVM regression method: One is
input and second one is the position of input. Finally
the support vectors and weight are combined and
encoded together.
5. Curvelet Transform and Core
Vector Machine for Image
Compression
5.1. Core Vector Machine for Image
Compression
SVM has been widely used in many areas because of
its good generalization ablity. In SVM
implementations, the training time complexity is
scales between O(m) and O(m2.3) and it can be
further driven down to O(m) alongwith the use of
parallel mixture[15]. By generalizing the underlying
minimum enclosing ball problem by CVM algorithm,
it can be used with any linear or non linear kernals
and also it obtains optimal solutions appropriately
provable. In the number of training patterns „m‟,the
asymptotic time complexity of CVR is linear, while
the space complexity of CVR is independent of „m‟.
c - cmin
cmax - cmin
C ' =
-----------(10)
Coarse
Detail
SVM
regression
Fine
Curvelet
Transform
Original
image
Discard
Compressed
data
Curvelet
coefficient
Encoded by
DPCM
Inverse
Curvelet
Transform
Detail
Curvelet
Coefficient
Coarse
Restoration
by SVM
Fine
Reconstructed
image
Added in
coefficient
Compressed
data
Decoded

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
679
Authors s hows that CVR is independent of „m‟.
Authors have shown that CVR has comparable
performance with SVR, the CVR method is much
faster and it produces fewer core vectors on very
large data sets. So the CVR method can be inserted
into the image compression algorithm to gain much
improvement of compression ratio[1].
Table 1: Structure of the Curvelet Coefficients
5.2. The CVM Algorithm
For the Core-set „St‟, the ball‟s center „ct‟and radius
„Rt‟ at the tth iteration, the CVM algorithm follows as
given below:
i. Initialize S0, c0 and R0.
ii. Stop if there is no training point z such that
φ(z) falls outside the (1+ε)-ball
B(ct,(1+ε)Rt).
iii. Find z (core vector) such that φ(z) is
furthest away from ct. Set St+1=Stᴗ{z}.
This can be made more competent by using
the probabilistic speedup method that finds a
„z‟ which is only approximately the
outermost.
iv. Find new MEB(St+1) and set ct+1 =
cMEB(St+1) and Rt+1 = rMEB(St+1).
v. Increment t by step1 and go back to Step2.
5.3. Compression achieved by CVM
Corresponding to the orientation characteristics of
each subband, the proper scan order is used to map
from two dimension block into one dimension vector.
It is called as „y‟ for convenient. To form the vector
„x‟ by the pos itions of the elements in „y‟. The input
is „x‟ and the output „y‟ in the CVM regression
model. In training process, moreover the Ɛ, kernel
type and kernel parameters affects the compression
efficiency in the model. Different types of data is
suited by different kernels. Due to that the coefficient
from one block is almost considered as Gaussian
function, the Gaussian function is choosed as the
regression kernel[1].
6. CVR versus SVR
CVR algorithm can be used with any linear non
linear kernals and also it obtains optimal solutions
appropriately provable. In the number of training
patterns „m‟, the as ymptotic time complexity of
CVR is linear, while the space complexity of CVR is
independent of „m‟. The CVR method is much fas ter
and it produces fewer vectors on very large data sets.
The performance of CVR is better than the SVR
implementation when the data set is large, and it
produces fewer support vectors. More over all the
core vectors are useful support vectors. On the very
large data set ,the time required to find theoretical is
constant with respect to the training set.
7. Performance Parameters
Performance of any image compression can be
obtained by PSNR (Peak Signal-to-Noise Ratio) and
CR (Compression Ratio) parameters.
where, N denotes the total number of pixels
f(i,j) denotes the pixel value in the original image.
F(i,j) denotes the pixel value in the reconstructed
image.
Increase in Bit Rate improves the quality of the
reconstructed image. Some of the readings were
taken from the above literature to represent the
performance parameter.
Levels
Scales
Orientations
Matrix form
Coarse
Cell[1]
1
32 x 32
Detail
Cell[2]
32 (4 x 8)
16X12
12x16
16x12
12x16
Cell[3]
32 (4 x 8)
32x22
22x32
32x22
22x32
Cell[4]
64(4 x 16)
64x22
22x64
64x22
22x64
Cell[5]
64(4 x 16)
128x44
44x128
128 x 44
44x128
Fine
Cell[6]
1
512 x 512
PSNR(dB) = 20xlog10(Maximumpixel value)
(MSE)1/2 ---(11)
where, MSE represents the mean squared error
CR(bpp) = Size of the compressed image
Total number of pixels -------(13)
MSE = 1/N x ∑i∑j (f(i,j)-F(i,j))2 ----(12)

International Journal of Advanced Computer Research (ISSN (print): 2249-7277 ISSN (online): 2277-7970)
Volume-4 Number-2 Issue-15 June-2014
680
Table 2. Comparison of image compression
techniques based on Curvelet Transform
Figure5. Graphical Representation of PSNR from
Table 2
8. Conclusion
Image quality of the image after compression is the
main criteria that all the compression techniques
should hold. Here we discussed some existing image
compression techniques based on Curvelet Transform
with their performance results. In section 4, Curvelet
Transform and support vector machine for
Image compression is discussed. In section 5,
Curvelet Transform and core vector machine for
Image compression is discussed. Experimental results
Figure6. Graphical Representation of CPU time
from Table2
from both the papers shows that the curvelet
transform along with core vector machine gains
better compression performance than that of curvelet
transform along with support vector machine both in
PSNR and CPU time. At the same time, the algorithm
works fairly well for declining block effect at higher
compression ratios. The results are only a preliminary
investigation of compressing curvelet coefficient
using CVM regression, and there is much can be
done to improve the performance. For example, the
method of encoding curvelet sub-bands should be
more flexible, which makes CVM learn data
dependency more efficiently.
For real-time image transmission or storage process,
all the compression techniques are useful. Each one
of the coding techniques is different from the other.
The selection of high PSNR value will lead to
maintain the quality of the image and success in
compression process.
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Refer
ence
Technique used
PSNR
/dB
CR
%
CPU
Time/s
[1]
Curvelet Transform
and Core Vector
Machine
27
22
30.2
[2]
Curvelet Transform
and Support Vector
Machine
26.93
22
34.4
28
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CVM SVM
C
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i
m
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Fig 6. Graphical Representation of CPU time from Table 2.

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P.Anandan received B.E Degree in
Electronics and Communication
Engineering from Anna University,
Chennai and M .E degree in Applied
Electronics from Anna University,
Coimbatore, Tamilnadu, India in 2005
and 2010. Currently he is working as an
Assistant Professor in Electronics and
Communication Engineering Department in R.M.D.
Engineering College, Kavaraipettai-601206, Thiruvallur
District, Tamilnadu, India. He is a Research Scholar in
Anna University Chennai, in the area of image processing.
His research interests include Transform based Image
Compression.
Dr.R.S.Sabeenian is currently
working as a Professor in ECE
Department in Sona College of
Technology, Salem, Tamil Nadu,
India.He received his Bachelors in
Engineering from Madras University
and his M asters in Engineering in
Communication Systems from Madurai
Kamaraj University. He received his Ph.D. Degree from
Anna University, Chennai in the year 2009 in the area of
Digital Image processing. He is currently heading the
research group named Sona SIPRO (SONA Signal and
Image PROcessing Research Centre) centre located at the
Advanced Research Centre in Sona College of Technology,
Salem.He has published more than 65 research papers in
various International, National Journals and Conferences.
He has also published around seven books. He is a reviewer
for the journals of IET,UK and ACTA Press Singapore.He
received the “Best Facul ty Award” among Tamil Nadu,
Karnataka and Kerala states for the year 2009 given by the
Nehru Group of Institutions, Coimbatore and the “ Best
Innovative Project Award” from the Indian National
Academy of Engineering, New Delhi for the year 2009 and
“ISTE RajarambapuPatil National Award” for
Promising Engineering Teacher for Creative Work done in
Technical Education for the year 2010 from ISTE.
He has also received a Project Grant from the All India
Council for Technical Education and Tamil Nadu State
Council for Science and Technology, for carrying out
research. He received two “Best Research Paper Awards”
from Springer International Conference and IEEE
International Conference in the year 2010.He was also
awarded the IETE BimanBehariSen Memorial National
Award for outstanding contributions in the emerging areas
of Electronics and Telecommunication with emphasis on
R&D for the year 2011.The Award was given by Institution
of Electronics and Telecommunication Engineers (IETE),
New Delhi. He is the Editor of 6 International Research
Journals International Journal of Information Technology,
Asian Journal of Scientific Research, Journal of
ArtificialIntelligence, Singapore Journal of Scientific
Research, International Journal of M anufacturing Systems
and ICTACT Journal of Image Processing. He is also
associated with the Image Processing Payload of the
PESIT Pico Satellite Project which is to be launched by
the end of December, 2013.He is the External Expert
Member for Board of Studies of Adhiyaman College of
Engineering,Hosur and M.Kumarasamy College of
Enineering,Karur.He is the Honoaray Treasurer of IETE
Salem Sub Centre from 2010 onwards. He is the Co-
ordinator for AICTE-INAE DVP Scheme.His areas of
interest include texture analysis, texture classification and
pattern recognition.He delivered more than 50 guest
lectures and chaired more than 25 national and international
conferences.Recently he received “ISTE PeriyarAward
for Best Enginee ring Coll ege Teacher” for the year 2012.