IMAGE COMPRESSION USING CONTOURLET TRANSFORM AND MULTISTAGE VECTOR by hhn16881

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									                                  GVIP Journal, Volume 6, Issue 1, July 2006




  IMAGE COMPRESSION USING CONTOURLET TRANSFORM AND
           MULTISTAGE VECTOR QUANTIZATION
                      S.Esakkirajan1, T.Veerakumar2, V. Senthil Murugan3, R.Sudhakar4
             1,3
                Department of Electrical and Electronics Engineering, PSG College of Technology
          2,4
              Department of Electronics and Communication Engineering, PSG College of Technology
                               Peelamedu, Coimbatore-641 004, Tamilnadu,India
            rajanesakki@yahoo.com, tveerakumar@yahoo.co.in, senthilmurugan_eee@yahoo.co.in ,
                                        sudha_radha2000@yahoo.co.in

Abstract                                                       applications such as TV transmission, video
This paper presents a new coding technique based on            conferencing, facsimile transmission of printed
contourlet transform and multistage vector                     material, graphics images, fingerprints and drawings.
quantization. Wavelet based Algorithms for image               Compression can be achieved by transforming the
compression results in high compression ratios                 data, projecting it on a basis of functions, and then
compared to other compression techniques. Wavelets             encoding this transform. In this paper, we examine the
have shown their ability in representing natural               design of image coder by integrating contourlet
images that contain smooth areas separated with                transform [2] with Multistage Vector Quantization
edges. However, wavelets cannot efficiently take               (MSVQ) [3]. Vector quantization (VQ) is a
advantage of the fact that the edges usually found in          quantization technique [4] applied to an ordered set of
natural images are smooth curves. This issue is                symbols. The superiority of VQ lies in the block
addressed by directional transforms, known as                  coding gain, the flexibility in partitioning the vector
contourlets, which have the property of preserving             space, and the ability to exploit intra-vector
edges. The contourlet transform is a new extension to          correlations. Multistage VQ divides the encoding task
the wavelet transform in two dimensions using                  into several stages. The first stage performs a
nonseparable and directional filter banks. The                 relatively crude encoding of the input vector using a
computation and storage requirements are the major             small codebook. Then, the second stage quantizer
difficulty in implementing a vector quantizer. In the          operates on the error vector between the original
full-search algorithm, the computation and storage             vector and the quantized first stage output. The
complexity is an exponential function of the number            quantized error vector provides a refinement to the
of bits used in quantizing each frame of spectral              first approximation. The indices obtained by
information. The storage requirement in multistage             multistage vector quantizer are then encoded using
vector quantization is less when compared to full              Huffman coding. Contourlets have the property of
search vector quantization. The coefficients of                preserving edges and fine details in the image; the
contourlet transform are quantized by multistage               encoding complexity in multistage vector quantization
vector quantization. The quantized coefficients are            is less when compared to tree structured vector
encoded by Huffman coding to get better quality i.e.,          quantization. This motivates us to develop a new
high peak signal to noise ratio (PSNR). The results            coding scheme by integrating contourlet transform
obtained are tabulated and compared with the existing          with multistage vector quantization.
wavelet based ones.
                                                                  The remainder of the paper is organized as follows:
                                                               Section 2 focuses on contourlet transform, Section 3
Keywords: Contourlet Transform, Directional Filter             emphasizes on multistage vector quantization, Section
bank, Laplacian Pyramid, Multistage Vector                     4 deals with the proposed image compression scheme
Quantization.                                                  and finally conclusions are drawn in Section 5.

1. Introduction                                                2. Contourlet Transform
     A fundamental goal of image compression [1] is                 The Contourlet Transform is a directional
to reduce the bit rate for transmission or data storage        transform, which is capable of capturing contours and
while maintaining an acceptable fidelity or image              fine details in images. The contourlet expansion is
quality. Image compression is essential for                    composed of basis function oriented at various


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                                       GVIP Journal, Volume 6, Issue 1, July 2006


directions in multiple scales, with flexible aspect                  In general, the contourlet construction allows for
ratios. With this rich set of basis functions, the              any number of DFB decomposition levels ‘lj’ to be
contourlet transform effectively capture smooth                 applied at each LP level ‘j’. For the contourlet
contours that are the dominant feature in natural               transform to satisfy the anisotropy scaling relation,
images. In contourlet transform, the Laplacian                  one simply needs to impose that in the PDFB, the
pyramid does the decomposition of images into                   number of directions is doubled at every other finer
subbands and then the directional filter banks analyze          scale of the pyramid. Fig. 2(b) graphically depicts the
each detail image as illustrated in Fig. 1.                     supports of the basis functions generated by such a
     The pyramidal directional filter bank (PDFB) [5],          PDFB.
was proposed by MinhDo and Vetterli, which                           As can be seen from the two shown pyramidal
overcomes the block-based approach of curvelet                  levels, the support size of the LP is reduced by four
transform by a directional filter bank, applied on the          times while the number of directions of the DFB is
whole scale also known as contourlet transform (CT).            doubled. Combine these two steps, the support size of
The grouping of wavelet coefficients suggests that              the PDFB basis functions are changed from one level
one can obtain a sparse image expansion by first                to next in accordance with the curve scaling relation.
applying a multi-scale transform and then applying a            In this contourlet scheme, each generation doubles the
local directional transform to gather the nearby basis          spatial resolution as well as the angular resolution.
functions at the same scale into linear structures. In          The PDFB provides a frame expansion for images
essence, first a wavelet-like transform is used for edge        with frame elements like contour segments, and thus
(points)                                                        is also called the contourlet transform.
detection, and then a local directional transform for
contour segments detection. With this insight, one can
construct a double filter bank structure (Fig.2 (a)) in
which at first the Laplacian pyramid (LP) is used to
capture the point discontinuities, and followed by a
directional filter bank (DFB) to link point
discontinuities into linear structures [6]. The overall
result is an image expansion with basis images as
contour segments, and thus it is named the contourlet
transform. The combination of this double filter bank
is named pyramidal directional filter bank (PDFB).




                                                                    Fig 2. (a) Block diagram of a PDFB, and (b) Supports for
        Fig 1 A flow graph of the Contourlet Transform
Fig. 2(a) shows the block diagram of a PDFB. First a                                      Contourlets
standard multi-scale decomposition into octave bands
is computed, where the low pass channel is sub-                 A. Laplacian Pyramid
sampled while the high pass is not. Then a directional               One way of achieving a multiscale decomposition
decomposition with a DFB is applied to each high                is to use a Laplacian pyramid (LP), introduced by
pass channel. Fig. 2(b) shows the support shapes for            Burt and Adelson [7].
contourlets implemented by a PDFB that satisfies the                 The LP decomposition at each level generates a
anisotropy scaling relation. From the upper line to the         down sampled lowpass version of the original and the
lower line, the scale is reduced by four while the              difference between the original and the prediction,
number of directions is doubled. PDFB allows for                resulting in a bandpass image as shown in Fig. 3(a). In
different number of directions at each scale/resolution         this figure, ‘H’ and ‘G’ are called analysis and
to nearly achieve critical sampling. As DFB is                  synthesis filters and ‘M’ is the sampling matrix. The
designed to capture high frequency components                   process can be iterated on the coarse version. In Fig.
(representing directionality), the LP part of the PDFB          3(a) the outputs are a coarse approximation ‘a’
permits subband decomposition to avoid “leaking” of
low frequencies into several directional subbands,
thus directional information can be captured
efficiently.



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                                        GVIP Journal, Volume 6, Issue 1, July 2006




      Fig 3. Laplacian pyramid scheme (a) analysis, and (b)
                         reconstruction.

and a difference ‘b’ between the original signal and
the prediction. The process can be iterated by
decomposing the coarse version repeatedly. The
original image is convolved with a Gaussian kernel
[8]. The resulting image is a low pass filtered version                         Fig 4. Laplacian pyramid structure.
of the original image. The Laplacian is then computed              wedge-shaped frequency partition as shown in Fig. 5.
as the difference between the original image and the               The original construction of the DFB in [9] involves
low pass filtered image. This process is continued to              modulating the input signal and using diamond-
obtain a set of band-pass filtered images (since each              shaped filters. Furthermore, to obtain the desired
one is the difference between two levels of the                    frequency partition, an involved tree expanding rule
Gaussian pyramid). Thus the Laplacian pyramid is a                 has to be followed. As a result, the frequency regions
set of band pass filters. By repeating these steps                 for the resulting subbands do not follow a simple
several times a sequence of images, are obtained. If               ordering as shown in Fig. 4 based on the channel
these images are stacked one above another, the result             indices. The DFB is designed to capture the high
is a tapering pyramid data structure, as shown in Fig.             frequency components (representing directionality) of
4 and hence the name. The Laplacian pyramid can                    images [1]. Therefore, low frequency components are
thus be used to represent images as a series of band-              handled poorly by the DFB. In fact, with the
pass filtered images, each sampled at successively                 frequency partition shown in Fig. 5, low frequencies
sparser densities. It is frequently used in image                  would leak into several directional subbands, hence
processing and pattern recognition tasks because of its            DFB does not provide a sparse representation for
ease of computation. A drawback of the LP is the                   images. To improve the situation, low frequencies
implicit oversampling. However, in contrast to the                 should be removed before the DFB. This provides
critically sampled wavelet scheme, the LP has the                  another reason to combine the DFB with a
distinguishing feature that each pyramid level                     multiresolution scheme. Therefore, the LP permits
generates only one bandpass image (even for multi-                 further subband decomposition to be applied on its
dimensional cases), which does not have “scrambled”                bandpass images. Those bandpass images can be fed
frequencies. This frequency scrambling happens in                  into a DFB so that directional information can be
the wavelet filter bank when a highpass channel, after             captured efficiently. The scheme can be iterated
downsampling, is folded back into the low frequency                repeatedly on the coarse image. The end result is a
band, and thus its spectrum is reflected. In the LP, this          double iterated filter bank structure, named pyramidal
effect is avoided by downsampling the lowpass                      directional filter bank (PDFB), which decomposes
channel only.                                                      images into directional subbands at multiple scales.
                                                                   The scheme is flexible since it allows for a different
B. Directional Filter Bank                                         number of directions at each scale. Fig. 6, 7 and 8
     In 1992, Bamberger and Smith [9] introduced a                 shows the contourlet transform of the images Lena,
2-D directional filter bank (DFB) that can be                      Fingerprint and Barbara respectively. For the visual
maximally decimated while achieving perfect                        clarity, only two-scale decompositions are shown.
reconstruction. The directional filter bank is a                   Each image is decomposed into a lowpass subband
critically sampled filter bank that can decompose                  and several bandpass directional subbands.
images into any power of two’s number of directions.
The DFB is efficiently implemented via a l-level
treestructured decomposition that leads to ‘2l’
subbands with




                                                                                      Fig 5. DFB frequency partitioning


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                                       GVIP Journal, Volume 6, Issue 1, July 2006


                                                                 codebook with no imposed constraints in its structure.
                                                                 The resulting encoding and storage complexity, of the
                                                                 order of 2kr, may be prohibitive for many applications.
                                                                 A structured VQ scheme which can achieve very low
                                                                 encoding and storage complexity is multistage VQ
                                                                 (MSVQ). In MSVQ, the kr bits are divided between L
                                                                 stages with bi bits for stage ‘i’. The storage
                                                                                                 L
                                                                 complexity of MSVQ is         ∑ 2b i       vectors, which can
         Fig 6. Contourlet Transform of “Lena” image.                                          i =1
                                                                 be     much      less    than        the     complexity    of
                                                                  L
                                                                 ∏ 2bi   = 2 kr vectors for unstructured VQ. MSVQ
                                                                 i =1
                                                                 [10] is a sequential quantization operation where each
                                                                 stage quantizes the residual of the previous stage.




      Fig 7. Contourlet Transform of “Fingerprint” image.




                                                                               Fig 9. Encoder block diagram of MSVQ

                                                                     The structure of MSVQ encoder [11] consists of
                                                                 a cascade of VQ stages as shown in Fig. 9. For an L-
                                                                 stage MSVQ, an l th –stage quantizer Ql ,
       Fig 8. Contourlet Transform of “Barbara” image.
                                                                 l =0,1,2… L − 1 is associated with a stage codebook
It can be seen that only contourlets that match with             Cl contains K l stage code vectors. The set of stage
both location and direction of image contours produce
significant coefficients. Thus, the contourlet transform         quantizers {Q0,Q1,.......,QL −1} are equivalent to a
effectively explores the fact, that the edges in images          single quantizer Q , which is referred to as the direct-
are localized in both location and direction. One can            sum vector quantizer.
decompose each scale into any arbitrary power of
two’s number of directions, and different scales can             MSVQ Encoder
be decomposed into different numbers of directions.                    In the MSVQ encoder as shown in Fig.9, the
This feature makes contourlets a unique transform                input vector ‘X’ is quantized with the first stage
that can achieve a high level of flexibility in                  codebook producing the first stage code vector Q0(X),
decomposition while being close to critically sampled.           a residual vector y0 is formed by subtracting Q0(X)
Other multiscale directional transforms have either a            from ‘X’. Then y0 is quantized using the second stage
fixed number of directions or are significantly over             codebook, with exactly the same procedure as in the
complete.                                                        first stage, but with ‘y0’ instead of ‘X’ as the input to
                                                                 be quantized. Thus, in each stage except the last stage,
                                                                 a residual vector is generated and passed to the next
3. Multistage Vector Quantization                                stage to be quantized independently of the other
    In vector quantization, an input vector of signal            stages.
samples is quantized by selecting the best matching                     MSVQ is an error refinement scheme, inputs to a
representation from a codebook of ‘2kr’ stored code              stage are residual vectors from previous stage and
vectors of dimension k. VQ is an optimal coding                  they tend to be less and less correlated as the process
technique in the sense that all other methods of coding          proceeds.
a random vector in ‘k’ dimensions with a specific
number b=kr of bits are equivalent to special cases of           MSVQ Decoder
VQ with generally suboptimal codebooks. However,                      The decoder as shown in Fig. 10 receives for
optimal VQ assumes single and possibly very large                each stage an index identifying the stage code vector
                                                                 selected and forms the reproduction X by summing

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                                    GVIP Journal, Volume 6, Issue 1, July 2006


the identified vectors. The overall quantization error is                In this work we do not design a different
equal to the quantization residual from the last stage.                  codebook for each individual layer. The
Sequential searching of the stage codebooks renders                      same codebook is applied to all layers.
the encoding complexity to the storage complexity                    6. The indices obtained from Multistage Vector
 L                                                                       Quantization are encoded by Huffman
∑ 2bi                                                                    coding.
i =1                                                                 The proposed scheme uses static Huffman coding
                                                                 where the same Huffman table is used for different
                                                                 images. This way overhead of sending Huffman tables
                                                                 along with coded data is eliminated

                                                                 5. Results and Discussion
                                                                 We present the encoding results of 256 x 256, 8 bit
                                                                 resolution       ‘LENA’,         ‘BARBARA’           and
                                                                 ‘FINGERPRINT' images. We have tested our
                                                                 algorithm for the class of natural image that do not
                                                                 contain large amounts of high frequency or oscillating
                                                                 patterns which is nothing but Lena image. The same
               Fig 10. Decoder block diagram of MSVQ
                                                                 algorithm is applied to the test image that exhibit
                                                                 large amounts of high frequency and oscillating
                                                                 patterns, which is Barbara image. Other than low and
4. Proposed Scheme                                               high frequency image, the algorithm is also applied to
The proposed algorithm is summarized below.                      the image, which has both high, and low frequency
    1. To decorrelate the relationship between the               part, which is fingerprint [13]. For simplicity, we have
        pixels, contourlet transform is applied first to         considered only two stages in the multistage vector
        all the test images taken. Different                     quantization. The same algorithm can be extended to
        directional and pyramidal filter banks are               many stages. As a trial, we have incorporated three
        considered for decomposition. This is the                stages in MSVQ for Lena image and found that the
        initialization stage in the proposed algorithm.          quality of the reconstructed image is good, but the
    2. Group neighboring contourlet coefficients                 execution time is more when compared to two stages
        into one vector.                                         in MSVQ. During transmission of images, the impact
        2 X 2 contourlet coefficients are grouped into           of different types of noises in the test image should be
    a vector.                                                    taken into account. In our work, the prime motive is
    3. Take the absolute values of all vector                    compression and not transmission hence the impact of
        components since signs and absolute values               noise is not taken into account. The codes are run on a
        of vector components are encoded separately              Pentium IV PC with 256Mb RAM.
        in our algorithm, we consider only the
        magnitude of each vector component in the                      Table I gives the result of the proposed scheme
        refinement process.                                      against wavelet based multistage vector quantization
    4.    Find the training vectors for the first layer          for Fingerprint image and the corresponding plot is
        codebook.                                                shown in Fig. 11. Table II and III gives the result of
        This can be done by two different ways. One              PSNR values for different pyramidal and directional
        is to include all training vectors of the first          filters when applied to the fingerprint image and the
        layer, i.e., symbols with norms larger than              corresponding plots are shown in Fig. 12 and 13
        the first threshold T1. Another is to                    respectively.
        manipulate the components of vectors, e.g.
        multiplied by 2 or 4, so that all the vectors             Table-I PSNR values for Wavelet and Contourlet
        fall in the subspace of the first layer. The                      Transform of Fingerprint image
        latter approach contains a much larger                   Bits per    ‘Haar’     P-filter:   P-filter:   P-filter:
        training set and richer patterns than the                Dimens-     Wavelet    ’Haar’      ’Haar’      ’Haar’
        former one. We choose the second method in                 ions                 D-filter:   D-filter:   D-filter:
        our coding scheme.                                         (bpd)                 ’9-7’       ’pkva’      ’5-3’
    5. Perform multistage codebook training.                      0.125      25.8820    27.7315     27.7584     26.4390
                                                                   0.25      38.7565    40.1037     40.0999     39.3212
        The codebook training includes: find the
                                                                    0.5      50.5928    51.9884     51.9807     51.3174
        centroids of the training set, and the residual
                                                                   0.75      54.3465    55.5676     55.5614     54.8794
        codewords of the first stage, second stage,                 1.0      59.9010    62.6486     62.6456     62.1502
        and etc. The training method is Lloyd-Max
        iteration, which is often referred to as Linde,
                                                                 From the Fig. 20 we can infer that the proposed
        Buzo and Gray (LBG) [12].
                                                                 scheme outperforms the wavelet based multistage


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                                                                        GVIP Journal, Volume 6, Issue 1, July 2006


vector quantization. In the case of fingerprint image,                                                    Table IV gives the result of the proposed scheme
the ‘Haar’ and ‘9-7’ as the pyramidal and directional                                                     against wavelet based multistage vector quantization
filter combination gives better PSNR result when                                                          for Lena image and the corresponding plot is shown
compared to other pyramidal and directional filter                                                        in Fig. 14. Table V and VI gives the result of PSNR
combinations.                                                                                             values for different pyramidal and directional filters
                           65
                                        Wavelet Vs Contourlet for Fingerprint image                       when applied to the Lena image and the
                                        Wavelet=Haar                                                      corresponding plots are shown in Fig. 15 and 16
                           60           Contourlet=Haar + 9-7
                                        Contourlet=Haar + 5-3                                             respectively.
                           55

                                                                                                                                        Contourlet for Fingerprint image
                           50                                                                                             45
                PSNR--->




                                                                                                                                   P:Filter=9-7,D:Filter=5-3
                           45                                                                                                      P:Filter=PKVA,D:Filter=5-3
                                                                                                                                   P:Filter=5-3,D:Filter=PKVA
                                                                                                                          40
                           40


                           35
                                                                                                                          35




                                                                                                               PSNR--->
                           30


                           25
                            0.1    0.2      0.3     0.4   0.5    0.6   0.7     0.8    0.9    1                            30
                                                 Bits per dimensions------>

                                                                                                                          25
           Fig 11. Plot of PSNR Vs bit rate for fingerprint image

                            Table-II PSNR values for Contourlet                                                           20
                                                                                                                           0.1   0.2   0.3     0.4    0.5   0.6   0.7     0.8   0.9     1
                             Transform of Fingerprint image                                                                                  Bits per dimensions------>
    Bits per                                P-filter:             P-filter:             P-filter:         Fig 13. Comparison of bit rate vs. PSNR between different pyramid
    Dimens-                                   ’9-7’                ’pkva’                ’5-3’             and directional filters using Contourlet transform with Multistage
      ions                                  D-filter:             D-filter:             D-filter:                      Vector quantization for Fingerprint image
     (bpd)                                   ’pkva’                 ’9-7’                ’9-7’
     0.125                                  21.6754               21.1154               22.8744
     0.25                                   28.4383               27.8619               30.1356
      0.5                                   35.1472               34.3806               31.9153
     0.75                                   38.7381               37.8090               35.1001                            Table-IV PSNR values for Wavelet and
      1.0                                   41.2961               40.1303               37.3613                                        Contourlet
                                                                                                                                Transform of Lena image
                                             Contourlet for Fingerprint image                               Bits per               ‘Haar’            P-filter:     P-filter:          P-filter:
                  45
                                    P:Filters=9-7,D:Filters=PKVA                                            Dimens                 Wavelet           ’Haar’        ’Haar’             ’Haar’
                                    P:Filters=PKVA,D:Filters=9-7                                             -ions                                   D-filter:     D-filter:          D-filter:
                                    P:Filters=5-3,D:Filters=9-7
                  40                                                                                         (bpd)                                    ’9-7’         ’pkva’             ’5-3’

                  35
     PSNR--->




                                                                                                             0.125                 25.9767           27. 4774       27.4773           26.7345
                  30
                                                                                                              0.25                 38.8644           40.3459       40. 3470           39.5909
                  25
                                                                                                              0.5                  50.6572           52. 3395       52.3274           51.6232

                  20
                   0.1            0.2      0.3      0.4   0.5    0.6   0.7      0.8    0.9       1            0.75                 54.2564           55.7400        55.7370           55.0487
                                                  Bits per dimensions------>                                   1.0                 59.7543           62. 3710       62.3666           61.9104
Fig 12. Comparison of bit rate vs. PSNR between different pyramid
 and directional filters using Contourlet transform with Multistage
             Vector quantization for Fingerprint image

                           Table-III PSNR values for Contourlet
                            Transform of Fingerprint image
     Bits per                                P-filter:           P-filter:             P-filter:
     Dimens-                                  ’9-7’               ’pkva’                 ’5-3’
       ions                                  D-filter:           D-filter:             D-filter:
      (bpd)                                   ’5-3’                ’5-3’                ’pkva’
      0.125                                  21.4573             21.1001               22.8840
      0.25                                   28.3826             27.8567               30.1386
       0.5                                   35.1316             34.3795               31.9156
      0.75                                   38.7230             37.8072               35.1000
       1.0                                   41.2917             40.1301               37.3612



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                                                                       GVIP Journal, Volume 6, Issue 1, July 2006


                                         Wavelet Vs Contourlet for Lena image                              Table VII gives the result of the proposed scheme
                          65
                                       Wavelet=Haar                                                        against wavelet based multistage vector quantization
                                       Contourlet=Haar + 9-7
                          60
                                       Contourlet=Haar + 5-3
                                                                                                           for Barbara image and the corresponding plot is
                          55                                                                               shown in Fig. 17. Table VIII and IX gives the result
                                                                                                           of PSNR values for different pyramidal and
                          50
                                                                                                           directional filters when applied to the Barbara image
               PSNR--->




                          45                                                                               and the corresponding plots are shown in Fig. 18 and
                          40                                                                               19 respectively.
                                                                                                            From the Fig. 22 we can infer that the proposed
                          35
                                                                                                           scheme outperforms the wavelet based multistage
                          30                                                                               vector quantization. In the case of Barbara image, the
                          25
                                                                                                           ‘5-3’ and ‘pkva’ as the pyramidal and directional filter
                           0.1    0.2      0.3     0.4   0.5    0.6   0.7     0.8   0.9     1
                                                 Bits per dimensions------>                                combination gives better PSNR result when compared
                           Fig 14. Plot of PSNR Vs bit rate for Lena Image                                 to other pyramidal and directional filter combinations.

                               Table-V PSNR values for Contourlet
                                   Transform of Lena image                                                                                         Contourlet for Lena image
                                                                                                                             70
   Bits per                                P-filter:             P-filter:            P-filter:                              65
                                                                                                                                          P:Filter=9-7,D:Filter=5-3
   Dimens-                                   ’9-7’                ’pkva’               ’5-3’                                              P:Filter=PKVA,D:Filter=5-3
                                                                                                                             60
     ions                                  D-filter:             D-filter:            D-filter:                                           P:Filter=5-3,D:Filter=PKVA
                                                                                                                             55
    (bpd)                                   ’pkva’                 ’9-7’               ’9-7’
    0.125                                  25.7677               27.5363              28.8109                                50
                                                                                                                 PSNR--->
    0.25                                   38.8576               39.4803              42.0035                                45

     0.5                                   51.2431               51.5084              54.0942                                40

    0.75                                   54.7321               54.9806              57.5803                                35
     1.0                                   63.2737               63.5641              66.1124                                30
From the Fig. 21 we can infer that the proposed                                                                              25
                                                                                                                              0.1   0.2      0.3      0.4    0.5   0.6   0.7     0.8   0.9     1
scheme outperforms the wavelet based multistage                                                                                                     Bits per dimensions------>
vector quantization.
In the case of Lena image, the ‘5-3’and ‘pkva’ as the                                                      Fig 16. Comparison of bit rate vs. PSNR between different pyramid
pyramidal and directional filter combination gives                                                          and directional filters using Contourlet transform with Multistage
                                                                                                                           Vector quantization for Lena image
better PSNR result when compared to other pyramidal
                                                                                                                            Table-VII PSNR values for Wavelet and
and directional filter combinations.
                                                                                                                                         Contourlet
                                             Contourlet for Lena image
                                                                                                                                 Transform of Barbara image
                    70                                                                                        Bits                   ‘Haar’                 P-filter:     P-filter:          P-filter:
                                       P:Filter=9-7,D:Filter=PKVA                                             per                    Wavelet                ’Haar’        ’Haar’             ’Haar’
                    65
                                       P:Filter=PKVA,D:Filter=9-7
                    60                                                                                       Dimen                                          D-filter:     D-filter:          D-filter:
                                       P:Filter=5-3,D:Filter=9-7
                    55
                                                                                                             s-ions                                          ’9-7’         ’pkva’             ’5-3’
                                                                                                             (bpd)
    PSNR--->




                    50
                                                                                                             0.125                    25.9767               27.6510       27.6584            26.6174
                    45
                                                                                                             0.25                     38.8644               40.1235       40.1111            39.3832
                    40                                                                                        0.5                     50.6572               52.0361       52.0314            51.3565
                    35                                                                                       0.75                     54.2564               55.6162       55.6130            54.9234
                    30                                                                                        1.0                     59.7543               62.6668       62.6654            62.1643
                    25
                     0.1         0.2      0.3     0.4    0.5    0.6    0.7    0.8   0.9         1
                                                                                                                                          Wavelet Vs Contourlet for Barbara image
                                             Bits per dimensions------>                                                     65
Fig 15. Comparison of bit rate vs. PSNR between different pyramid                                                                         Wavelet=Haar
                                                                                                                            60
 and directional filters using Contourlet transform with Multistage                                                                       Contourlet=Haar + 9-7
                Vector quantization for Lena image                                                                          55
                                                                                                                                          Contourlet=Haar + 5-3

                                                                                                                            50
                               Table-VI PSNR values for Contourlet
                                                                                                               PSNR--->




                                                                                                                            45
                                   Transform of Lena image
  Bits per                                P-filter:              P-filter:                P-filter:                         40

  Dimens-                                   ’9-7’                 ’pkva’                    ’5-3’                           35
    ions                                  D-filter:              D-filter:                D-filter:
                                                                                                                            30
   (bpd)                                    ’5-3’                  ’5-3’                   ’pkva’
   0.125                                  26. 3517               27.9687                  28.8971                           25
                                                                                                                             0.1    0.2      0.3      0.4    0.5   0.6   0.7     0.8   0.9         1
    0.25                                  39.3317                39.7848                  42.0332                                                   Bits per dimensions------>
    0.5                                   51. 5548               51.8074                  54.1043
    0.75                                  55. 0525               55.2784                  57.5952                           Fig 17. Plot of PSNR Vs bit rate for Barbara Image
    1.0                                   63. 5971               63.8706                  66.1352



                                                                                                      25
                                                                        GVIP Journal, Volume 6, Issue 1, July 2006


                 Table-VIII PSNR values for Contourlet                                                          filters chosen are ‘5-3’ and ‘pkva’ respectively. From
                      Transform of Barbara image                                                                the figures, it is obvious that as the bit rate increases,
   Bits per                                P-filter:              P-filter:                   P-filter:         the visual quality of the reconstructed image increases
   Dimens-                                  ’9-7’                  ’pkva’                       ’5-3’           which is in accordance with Rate-Distortion theory.
     ions                                  D-filter:              D-filter:                   D-filter:
    (bpd)                                   ’5-3’                   ’5-3’                      ’pkva’                      Original image               Reconstructed image
    0.125                                  26.6045                 27.2880                     29.1200
    0.25                                   39.4620                 39.4224                     42.0637
     0.5                                   51.5451                 51.4194                     54.1801
    0.75                                   55.0620                 54.9239                     57.6369
     1.0                                   63.6032                 63.4831                     66.2182


                                                Contourlet for Barbara image
               70
                                       P:Filter=9-7,D:Filter=5-3
               65
                                       P:Filter=PKVA,D:Filter=5-3
                                                                                                                              (a)                              (b)
               60                      P:Filter=5-3,D:Filter=PKVA
                                                                                                                       Reconstructed image              Reconstructed image
               55
    PSNR--->




               50

               45

               40

               35

               30

               25
                0.1              0.2      0.3       0.4    0.5   0.6    0.7      0.8         0.9       1
                                                  Bits per dimensions------>


Fig 18. Comparison of bit rate vs. PSNR between different pyramid                                                             (c)                              (d)
 and directional filters using Contourlet transform with Multistage
              Vector quantization for Barbara image                                                               Fig 20. Original and decoded 256 x 256 Finger print image (a)
                                                                                                                  Original image (b) bpd=0.125,(c) bpd=0.25, (d) bpd=1.0 using
                                                                                                                                        P-filter = ‘5-3’ and
                                                                                                                                         D-filter = ‘pkva’
                Table-IX PSNR values for Contourlet                                                                        Original image               Reconstructed image
                   Transform of Barbara image
          Bits per     P-filter:  P-filter:   P-filter:
          Dimens-        ’9-7’     ’pkva’       ’5-3’
            ions       D-filter:  D-filter:   D-filter:
           (bpd)        ’pkva’      ’9-7’       ’9-7’
           0.125       27.6801    27.3461      27.2880
           0.25        40.2529    39.4977      40.5984
            0.5        52.2609    51.4890      52.8883
           0.75        55.7804    55.0027      56.3485
            1.0        64.3248    63.5480      64.9423
                                                                                                                              (a)                              (b)
                                                                                                                       Reconstructed image              Reconstructed image
                                            Contourlet for Barbara image
                          65
                                         P:Filter=9-7,D:Filter=PKVA
                          60
                                         P:Filter=PKVA,D:Filter=9-7
                          55             P:Filter=5-3,D:Filter=9-7

                          50
               PSNR--->




                          45

                          40

                          35

                          30

                          25
                           0.1     0.2      0.3      0.4   0.5   0.6   0.7     0.8     0.9         1                          (c)                              (d)
                                                 Bits per dimensions------>
                                                                                                                      Fig 21. Original and decoded 256 x 256 Lena image
                                                                                                                  (a) Original image (b) bpd=0.125, (c) bpd=0.25, (d) bpd=1.0
Fig 19. Comparison of bit rate vs. PSNR between different pyramid
                                                                                                                       using          P-filter= ‘5-3’ and D-filter= ‘pkva’
 and directional filters using Contourlet transform with Multistage
              Vector quantization for Barbara image

Fig. 20, 21 and 22 shows the original and
reconstructed images of fingerprint, Lena and Barbara
at different bit rates. The pyramidal and directional

                                                                                                           26
                                          GVIP Journal, Volume 6, Issue 1, July 2006


             Original image               Reconstructed image




                                                                                         (a)                           (b)
                 (a)                              (b)
          Reconstructed image              Reconstructed image




                                                                                          (c)                          (d)
                                                                              Fig 24. Original and decoded 256 x 256 Lena image, bpd
               (c)                               (d)                                                    at 0.25
 Fig 22. . Original and decoded 256 x 256 Barbara image                       (a) Original image (b) Single stage MSVQ, (c) Two Stage
      (a) Original image (b) bpd=0.125, (c) bpd=0.25, (d)                      MSVQ (d) Three Stage MSVQ using P-filter = ‘5-3’ and
             bpd=1.0 using          P-filter= ‘5-3’ and D-filter =                                 D-filter = ‘pkva’
             ‘pkva’
                                                                            Fig. 24 shows the reconstructed images of ‘Lena’ for
                                                                            different stages of MSVQ.

                                                                            We have compared the execution time and the quality
                                                                            of the reconstructed image by incorporating three
                                                                            stages in multistage vector quantization. The results
                                                                            are shown in Table X. From the table it is clear that
                                                                            as the number of stages in multistage vector
                                                                            quantization increases, the quality of the reconstructed
                                                                            image also increases at the expense of execution time.
                                                                            This is evident from the plot, shown in Fig. 23. In
                                                                            Table X, ‘bpd’ stands for bits per dimension.




    Fig 23. Plot of PSNR Vs Execution time for Lena Image


Table X Contourlet transform with Different stages in MSVQ for Lena image P: Filter = ‘5-3’ and D:
Filter = ‘pkva’


   bpd            Single Stage VQ                                     Two Stage VQ                           Three Stage VQ
                    PSNR in dB      Execut-ion time         PSNR in dB          Execut-ion time         PSNR           Execut-ion time
                                      in seconds                                  in seconds            in dB            in seconds
  0.125                15.6237           4.5320                  28.8971             8.2180            42.0332               12.4540

   0.25                22.2691           4.7810                  42.0332             9.1880            60.1697               12.1870

    0.5                28.8972           4.8130                  54.1043             9.3750            78.0523               12.7500

   0.75                32.7703           4.9060                  57.5952             9.4060            81.6577               13.3900

    1.0                35.4600           5.3600                  66.1352            10.1560            90.7476               13.4690




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                                  GVIP Journal, Volume 6, Issue 1, July 2006


6. Conclusion                                               [8] M. N. Do, “Directional Multiresolution Image
      In this paper, compression of images using            Representations,” Ph.D.Thesis, EPFL, Lausanne,
contourlet transform and multistage vector                  Switzerland, Dec. 2001.
quantization has been presented. An extensive               [9] R. H. Bamberger and M. J. T. Smith, “A filter
result has been taken on different images. It can be        bank for the Directional decomposition of images:
seen that the PSNR obtained by contourlet                   theory and design,” IEEE Trans. on Signal
transform is higher than that of wavelet transform.         Processing, vol. 40, no. 4, pp. 882-893, Apr. 1992.
Hence, a better image reconstruction is possible            [10] Jayshree Karlekar, P.G. Poonacha and U.B.
with less number of bits, by using contourlet               Desai, “Image Compression using Zerotree and
transform. Here, only four filter combinations are          Multistage Vector Quantization”, ICIP, Vol.2,
considered. We are currently pursuing with other            pp.610, 1997
filter combinations. The experimental results reveal        [11] Hosam Khalil, Kenneth Rose, “Multistage
the fact that MSVQ is suitable for low bit rate             vector quantizer optimization for packet networks,”
image coding. The proposed scheme shows output              IEEE Trans. Signal Proc. Vol. 51, No.7, pp.1870-
of good quality around 0.5 bits per dimension (bpd)         1879, July 2003.
and very good results at around 1 bpd. This scheme          [12] Y. Linde, A. Buzo and R.M.Gray, “An
can easily be extended to include more stages in            algorithm for vector quantizer design,” IEEE
MSVQ to improve the output image quality.                   Trans. Commun., vol.28, pp.84-95, Jan.1980
                                                            [13] R. Sudhakar, R. Karthiga and S. Jayaraman,
                                                            “Fingerprint compression using Contourlet
7. Acknowledgements                                         Transform with Modified SPIHT algorithm”,
          The authors wish to thank their teachers          Iranian Journal of Electrical and Computer
Dr. S. Jayaraman, Dr. N. Malmurugan for their               Engineering (IJECE), vol.5, No.1, pp.3-10, Winter-
continued support. They also thank their present            Spring 2006.
institution where they are working, for the
encouragement.

8. References
[1] M.Antonini, M.Barlaud, P. Mathieu, and
I.Daubechies, “Image coding using wavelet
transform”, IEEE Trans. Image Proc., 205-220,
Apr.1992.
[2] M. N. Do and M. Vetterli, “The contourlet
transform: an efficient directional multiresolution
image representation,” IEEE Trans. Of Image
Processing, vol.14, no.12, pp. 2091-2106, Dec.
2004.
[3] B.H.Juang and A.H.Gray, “Multiple stage
vector quantization for speech coding”, Proc. IEEE
Int.Conf.Acoust., Speech, Signal Processing (Paris,
France), pp. 597-600, Apr.1982.
[4] A.Gersho and R.M. Gray, Vector Quantization
and Signal Compression. Boston, MA: Kluwer,
1992.



[5] M. N. Do and M.Vetterli, “Pyramidal
directional filter banks and curvelets,” in Proc. Of
IEEE Int. Conf. on Image Proc., vol.3, pp.158-161,
Thessaloniki, Greece, Oct.2001.
[6] D.D. Y. Po and M. N. Do, “Directional
multiscale modeling of images using the contourlet
transform,” IEEE Trans. on Image Processing, to
appear, Jun. 2006.
[7] P. J. Burt and E. H. Adelson, “The Laplacian
pyramid as a compact image code,” IEEE Trans.
on Commun., vol. 31, no. 4, pp. 532-540, 1983.




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