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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 19 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. 20 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 21 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 22 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 23 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 24 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 27 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. 28