100 CHAPTER 5 IMAGE COMPRESSION TECHNIQUE USING NONSUBSAMPLED CONTOURLET TRANSFORM 5.1 INTRODUCTION Image compression is currently a well-known topic for researchers. Due to fast growth of digital media and the following need for reduced storage and to convey the image in an efficient way, better image compression approach is required. Among the rapid growth of digital technology in consumer electronics, their requirement to protect raw image data for future editing or repeated compression goes on increasing. Thus, image compression has become a dynamic area of research in the field of image processing to lessen file size. Wavelet and curvelet transformations are widely used transformation techniques to carry out compression. But both have their own limitations which affects overall performance of the compression process. This research work focuses on presenting a non-linear image compression technique that compresses image both radically and angularly. Nonsubsampled Contourlet Transformation (NSCT) has the potential to approximate the natural images comprising contours and oscillatory patterns. In addition to this transformation, vector quantization technique is used to remove the redundancies in the images. Finally, this technique uses Hybrid Set Partitioning in Hierarchical Trees (HSPIHT) with Huffman for
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100
CHAPTER 5
IMAGE COMPRESSION TECHNIQUE USING
NONSUBSAMPLED CONTOURLET TRANSFORM
5.1 INTRODUCTION
Image compression is currently a well-known topic for
researchers. Due to fast growth of digital media and the following need for
reduced storage and to convey the image in an efficient way, better image
compression approach is required.
Among the rapid growth of digital technology in consumer
electronics, their requirement to protect raw image data for future editing
or repeated compression goes on increasing. Thus, image compression has
become a dynamic area of research in the field of image processing to
lessen file size. Wavelet and curvelet transformations are widely used
transformation techniques to carry out compression. But both have their
own limitations which affects overall performance of the compression
process. This research work focuses on presenting a non-linear image
compression technique that compresses image both radically and angularly.
Nonsubsampled Contourlet Transformation (NSCT) has the potential to
approximate the natural images comprising contours and oscillatory
patterns. In addition to this transformation, vector quantization technique is
used to remove the redundancies in the images. Finally, this technique uses
Hybrid Set Partitioning in Hierarchical Trees (HSPIHT) with Huffman for
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efficient encoding process. The experimental results of Non-sub sampled
contourlet transformation, with vector quantization and HSPIHT are better
when compared with existing transformations techniques.
5.2 PROBLEMS IN WAVELET BASED CONTOURLET
TRANSFORM
A non-redundant transform is needed for image coding. Certain
types of non-redundant geometrical image transforms that depends on the
DFB have been presented by the researchers. The octave-band directional
filter banks which are introduced by Paul S Hong & Mark JT Smith (2002)
correspond to a new family of the DFB that attains radial decomposition as
well. The CRISP-contourlet is another transform proposed by Lu & Do
(2003), which is implemented based on the contourlet formation, other than
nonseparable filter banks are merely used. Truong & Oraintara (2005)
presented a Non-uniform DFB which is a modified form of CRISP-
contourlets and is the other non-redundant directional transform, which
also gives multiresolution.
Neither of the above non-redundant directional schemes have
been used in a practical image processing application. Ramin Eslami &
Hayder Radha (2008) introduced a Wavelet-Based Contourlet Transform
(WBCT), in which DFB is applied to all the feature subbands of wavelets
in an analogous way that one constructs contourlets. The major difference
is that, wavelets are used instead of the Laplacian pyramids in contourlets.
As a result, the WBCT is non-redundant and can be adapted for some
resourceful wavelet-based image coding methods.
The main drawbacks of the WBCT and other contourlet-based
transforms contain artifacts that are occurred by setting some transform
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coefficients to zero for nonlinear approximation and also owing to
quantizing of the coefficients for coding.
The image focusing multiple objects contains more information
than those which just focus one object. This type of image is broadly used
in various areas such as remote sensing, medical imaging, computer vision
and robotics. Actually, all cameras used in computer visualization systems
are not pin-hole devices but consist of convex lenses. Therefore, they
endure from the problem of partial depth of field (Chunming Li et al 2004).
It is often not possible to get an image which contains all relevant objects
in focus. The objects in front of or behind the focus plane would be
blurred. To overcome this problem, image fusion is introduced by the
author in several images with different focal points are shared together to
create an image with all objects fully focused (Pan et al 1999). Over past
years, numbers of techniques were proposed by the researchers for
multifocus image fusion. The easiest multifocus image fusion technique in
spatial domain is to take the average of the source images pixel by pixel,
would lead to numerous unpredicted side effects like reduced difference
Yuancheng & Yang (2008). Various methods depends on multiscale
decomposition (MSD) as presented by Pajares & Cruz (2004).
Wavelet transform for 1-D piecewise smooth signals are better,
though it has severe limitations with high dimensional signals. 2D split
wavelet performs better at isolating the discontinuities at object edges, but
cannot effectively signify the line and the curve discontinuities. In contrast,
it can only capture restricted directional information. Accordingly, WT
based method cannot conserve the relevant features in source images well
and will possibly introduce some artifacts and variation in the combined
results (Yuancheng & Yang 2008). Minh N Do & Vetterli (2005) presented
a Contourlet Transform (CT) is a two dimensional image representation
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method. It is attained by combining the directional filter bank (DFB) as
discussed by Burt & Adelson (1983); Bamberger & Smith (1992).
Compared with the traditional DWT, the CT is not only with multi-scale
and localization, but also with mulitdirection and anisotropy.
Consequently, the CT can symbolize edges and further singularities along
curves. On the other hand, CT possesses shift-invariance and results in
artifacts along the edges to some level. Arthur et al (2006) proposed a
transform known as nonsubsampled contourlet transform (NSCT).
The NSCT come into the perfect properties of the CT, and in the meantime
consist of shift invariance. In image fusion, the NSCT has more
information fusion together and is obtained and the influence of
misregistration on the fused information which can also be reduced. As a
result, the NSCT is more appropriate for image fusion and also in image
compression etc.
5.3 NONSUBSAMPLED CONTOURLET TRANSFORM:
FILTER DESIGN AND APPLICATIONS
Several image processing tasks are executed in a particular
domain other than the pixel domain, frequently using invertible linear
transformation. This linear transformation can be superfluous or not, based
on whether the set of basic functions is linear independent. By means of
redundancy, it is feasible to improve the set of fundamental functions so
that the representation is more resourceful in capturing several signal
behavior. Applications of imaging are of edge detection, contour detection,
denoising and image restoration that can significantly assistance from
redundant representations. In the situation of multiscale expansions
employed with filter banks, dropping the fundamental requirement
provides the chance of an expansion that is shift-invariant, a vital property
in a number of applications. For example in image denoising ,by means of
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thresholding in the wavelet domain, the lack of shift-invariance reasons
pseudo-Gibbs phenomena in the region of singularities (Coifman &
Donoho 1995). Thus, most of the modern wavelet denoising routines
employ an expansion with low shift sensitivity than the standard maximally
decimated wavelet decomposition as explained in Chang et al (2000);
Sendur & Selesnick (2002) and Minh Do & Vetterli (2005) proposed a
contourlet transform which is a directional multiscale transform that is built
by combining the Laplacian pyramid (LP) and the directional filter bank
(DFB). Due to downsamplers and upsamplers present in both the LP and
DFB, the contourlet transform is not shift-invariant. The nonsubsampled
contourlet transform (NSCT) is obtained by coupling a nonsubsampled
pyramid structure with the nonsubsampled DFB.
Figure 5.1 Idealized frequency partitioning obtained with the NSCT
5.3.1 Nonsubsampled Contourlet Transform
The idea behind a fully shift invariant multiscale directional
expansion similar to contourlets is to obtain the frequency partitioning of
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Figure 5.1 without resorting to critically sampled structures that have
periodically time-varying units such as downsamplers and upsamplers. The
NSCT construction can thus be divided into two parts: (1) A
nonsubsampled pyramid structure which ensures the multiscale property
and (2) A nonsubsampled DFB structure which gives directionality.
5.3.2 Nonsubsampled Pyramid
The shift sensitivity of the LP can be remedied by replacing it
with a 2-channel nonsubsampled 2-D filter bank structure. Such expansion
is similar to the 1-D a trous wavelet expansion Shensa (1993) and has a
redundancy of J + 1 when J is the number of decomposition stages.
The ideal frequency support of the low-pass filter at the j-th stage is the
region jjjj 2,
2x
2,
2. Accordingly, the support of the high-pass
filter is the complement of the low-pass support region on the
1j1j1j1j 2,
2x
2,
2square. The proposed structure is thus different
from the tensor product, a trous algorithm. It has j+1 redundancy.
By contrast, the 2-D a trous algorithm has 3j+1 redundancy.
5.3.3 Nonsubsampled Directional Filter Bank
The directional filter bank is constructed by combining critically
sampled fan filter banks and pre/post re-sampling operations. The result is
a tree-structured filter bank which splits the frequency plane into
directional wedges.
A fully shift-invariant directional expansion is obtained by
simply switching off the downsamplers and upsamplers in the DFB
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equivalent filter bank. Due to multirate identities, this is equivalent to
switching off each of the downsamplers in the tree structure, while still
keeping the re-sampling operations that can be absorbed by the filters. This
results in a tree structure composed of two-channel nonsubsampled filter
banks. The NSCT is obtained by carefully combining the 2-D
nonsubsampled pyramid and the nonsubsampled DFB (NSDFB) Arthur et
al (2006).
The resulting filtering structure approximates the ideal partition
of the frequency plane displayed in Figure 5.2. It must be noted that,
different from the contourlet expansion, the NSCT has a redundancy given
by J0j j
l2R where jl2 is the number of directions at scale j.
Figure 5.2 Two kinds of desired responses (a) The pyramid desired
response (b) The fan desired response
The directional filter bank of Bamberger & Smith (1992) is
constructed by combining critically-sampled two-channel fan filter banks
and resampling operations. The result is a tree-structured filter bank that
splits the frequency plane in the directional wedges shown in Figure 5.3(a).
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Figure 5.3 The directional filter bank (a) Ideal partitioning of the 2-
D frequency plane into 23 = 8 wedges (b) Equivalent
multi-channel filter bank structure. The number of
channels is L = 2l, where l is the number of stages in the
tree structure
Using multirate identities Bamberger & Smith (1992), the tree-
structured DFB can be put into the equivalent form shown in Figure 5.3
(b), where the downsampling/upsampling matrices Sk for 0 2l-1 are
given by Equation (5.1)
;12k2for;12k0for
)2,2(diag)2,2(diag
S 1l1l
1l
1l
1l
k (5.1)
with ‘l’ denoting the number of stages in the tree structure. It is
clear from the above that the DFB is not shift-invariant. A shift-invariant
directional expansion is obtained with a nonsubsampled DFB (NSDFB).
The NSDFB is constructed by eliminating the downsamplers and
upsamplers as in Figure 5.3(b). This is equivalent to switching off the
downsamplers in each two-channel filter bank in the DFB tree structure
and upsampling the filters accordingly. This results in a tree consisting of
two-channel nonsubsampled filter banks. The equivalent filter in the kth
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channel of the analysis NSDFB tree is the filter )z(Ueqk as in Figure 5.3 (b).
Thus, each filter )z(Ueqk is a product of l simpler filters. Just like the DFB,
all NSFB’s in the NSDFB tree structure are obtained from a single NSFB
with fan filters (Figure 5.4b) which illustrates a four channel
decomposition. Note that in the second level, the upsampled fan filters
1,0i),z(U Qi have checker-board frequency support, and when combined
with the filters in the first level, gives the four directional frequency
decomposition that is shown in Figure 5.4. The synthesis filter bank is
obtained similarly. Just like the NSP case, each filter bank in the NSDFB
tree has the same computational complexity as that of the prototype NSFB.
Figure 5.4 A four-channel nonsubsampled directional filter bank
constructed with two-channel fan filter banks
(a) Filtering structure (b) Corresponding frequency
decomposition
Figure 5.4(a) shows a four-channel filtering structure the
equivalent filter in each channel is given by )z(U)z(U)z(U Qji
eqk .
Figure 5.4(b) shows the corresponding frequency decomposition.
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5.4 APPLICATIONS OF NON-SUBSAMPLED
CONTOURLET TRANSFORM
5.4.1 Image Denoising
In order to illustrate the potential of the NSCT designed using
the techniques previously discussed an Additive White Gaussian Noise
(AWGN) removal is implemented from the images by means of
thresholding estimators, and tested the NSCT fewer than two denoising
schemes described below:
1) Hard Threshold
For the hard threshold estimator, a global threshold
ijnj,i KT for each directional subband is chosen. This has been termed K-
sigma thresholding (Starck et al 2002). Set K = 4 for the finer scale and
K = 3 for the remaining ones. This method known as NSWT-HT when
applied to NSWT coefficients and NSCT-HT when applied to NSCT
coefficients. Five scales of decomposition are used for both NSCT and
NSWT. For the NSCT 4, 8, 8, 16, 16 directions in the scales are used from
coarser to finer respectively.
2) Local Adaptive Shrinkage
Perform soft thresholding (shrinkage) independently in each
subband. The threshold (Grace Chang et al 2000) is chosen by
Equation (5.2).
n,j,i
ijN2
j,iT (5.2)
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where n,j,i denotes the variance of the n-th coefficient at the ith
directional subband of the jth scale, and ijN2 is the noise variance at scale j
and direction i. It is shown in Grace Chang et al et al (2000) that shrinkage
estimation with x
2T , and assuming X generalized Gaussian distributed,
yields a risk within 5% of the optimal Bayes risk. Po & Do (2006) studied
that contourlet coefficients are well modelled by generalized Gaussian
distributions. The signal variances are estimated locally using the
neighbouring coefficients contained in a square window within each
subband and a maximum likelihood estimator. The noise variance in each
subband is inferred using a Monte-Carlo technique where the variances are
computed for a few normalized noise images and then averaged to stabilize
the results. This method is known as local adaptive shrinkage (LAS).
Effectively, LAS method is a simplified version of the denoising method
proposed in Grace Chang et al (2000) that works in the NSCT or NSWT
domain.
The other applications of NSCT is image enhancement, image
fusion, image compression, applications of medical image analysis, video
compression etc.
5.4.2 Advantages of NSCT
The nonsubsampled contourlet transform (NSCT) not only has
multiresolution but has multidirectional properties
By using non-subsampled contourlet transform ,coefficients in
small-scale and gradient information of each colour channel is
identified efficiently
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The NSCT is a fully shift-invariant, multi scale, and multi
direction expansion that have a fast implementation. Here,
filters are designed with better frequency selectivity, thereby
achieving better sub band decomposition
5.5 PROPOSED METHOD USING NON-SUBSAMPLED
CONTOURLET TRANSFORM
The basic steps of the proposed image compression algorithm
are shown in Figure 5.5. All these steps are invertible, therefore lossless,
except for the Quantize step. Quantizing is the process of reduction of the
precision of the floating point values of the WBC transforms.
Figure 5.5 Process of the proposed approach
5.5.1 Nonsubsampled Contourlet Transform
First, contourlet transform down samplers and up samplers are
formed in both the laplacian pyramid and the Directional Filter Bank
(DFB). Thus, it is not shift-invariant, which causes pseudo-Gibbs
phenomena around singularities. NSCT is an improved form of contourlet
transform. It is employed in some applications, in which redundancy is not
a major issue, i.e. image fusion. In contrast with contourlet transform, non-
subsampled pyramid structure and non-subsampled directional filter banks
are employed in NSCT. The non-subsampled pyramid structure is achieved
Non sub sampled
Contourlet Coefficient
Dead zone Quantization
HSPIHT Encoding
with Huffman
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by using two-channel nonsubsampled 2-D filter banks. The DFB is
achieved by switching off the down samplers/up samplers in each two-
channel filter bank in the DFB tree structure and up sampling the filters
accordingly. As a result, NSCT is shift-invariant and leads to better
frequency selectivity and regularity than contourlet transform. Figure.5.6
shows the decomposition framework of contourlet transform and NSCT