Color Image Compression using Set Partitioning in Hierarchical Trees Algorithm G. RAMESH 1, V.S.R.K SHARMA 2

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3 Color Image Compression using Set Partitioning in Hierarchical Trees Algorithm Fig.2. Data dependency diagram showing lifting scheme Implementation. Lifting scheme is derived from a poly phase matrix representation of the wavelet filters, a representation that is distinguishing between even and odd samples. Using the algorithm of filter factoring, we split the original filter into a series of shorter filters (typically Laurent polynomials of first degree). Those filters are designed as lifting steps; each step one group of coefficients are lifted (altered) with the help of the other one (classical dyadic transform always leads to two groups of coefficients, low-pass and high-pass). Data flow in the final algorithm is presented in Fig. 3. Data dependency diagram showing the data flow according to CDF9/7 lifting scheme implementation. Source signal is defined as x, the result is taken from HP and LP branches for high-pass and low-pass coefficients, respectively; a,b,c,d and K are constants computed by filter factorization process. Picture on the right is demonstrating boundary handling. Since images are two-dimensional signals, we have to extend the scheme to 2D space by applying the transform row- and column-wise, respectively (taking separability of the transform into account). As a consequence four sub bands arise from one level of the transform one low-pass sub band containing the coarse approximation of the source image called LL sub band, and three high pass sub bands that exploit image details across different directions HL for horizontal, LH for vertical and HH for diagonal details. In the next level of the transform, we use the LL band for further decomposition and replace it with respective four sub bands. This forms the decomposition image. IV. PROPOSED SYSTEM We have proposed an implementation of DWT-SPIHT coder and decoder for demonstration purposes. This coder is limited to images of square resolutions of power of 2 and grayscale (256 levels, that is 8 bit per pixel) information only. It produces a reconstructed output in spatial domain for comparison and also automatically computes PSNR (Peak Signal to Noise Ratio) as a measure of the difference between source and compressed (destination) images. SPIHT is very vulnerable to bit corruption, as a single bit error can introduce significant image distortion depending of its location. Much worse property is the need of precise bit synchronization, because a leak in bit transmission can lead to complete misinterpretation from the side of the decoder. Input parameters are: source image, wavelet name, transform depth and requested algorithm efficiency in bits per pixel (bpp). From wavelet name we recognize whether it is a part of MATLAB wavelet toolbox and use the appropriate wavelet toolbox functions [5] then, or employ a self-made CDF9/7 lifting scheme implementation (which is widely available). Both transforms produce the same decomposition image of specified depth. SPIHT coder is applied then, processing the exact number of bits computed from the bpp quantifier. Resulting bit stream is taken as a source for the decoder, that produces a reconstructed image. SPIHT is very vulnerable to bit corruption, as a single bit error can introduce significant image distortion depending of its location. Much worse property is the need of precise bit synchronization, because a leak in bit transmission can lead to complete misinterpretation from the side of the decoder. Fig.3. Frequency domain in the form of a decomposition image. Fig.4. SPIHT algorithm scheme.

4 G.RAMESH, V.S.R.K SHARMA the greedy search happens to grow the decomposition tree up to the fully grown tree, then Depending on the source image, the complexity generally falls between the two specified extremes. Thus, greedy approach is in general computationally less complex on the expense of potentially suboptimal performance. Fig.5. Method of Color Image Compression. Wavelet decomposition is dyadic in a case when only the LLk sub band is further transformed. It results in a new set of sub bands: HHkþ1, HLkþ1, LHkþ1, LLkþ1. Dyadic decomposition used in image compression will thus generate hierarchical pyramidal structure, as shown in Fig. 6. If the dyadic decomposition of N levels is performed (N times transforming the low low sub band) the result will be 3N þ 1 sub bands. The WP decomposition is a generalization of wavelet dyadic decomposition, where further wavelet transform on detail sub bands is possible, potentially producing up to 4N final sub bands. A single wavelet packet decomposition thus provides a multitude of choices from which the best representation with respect to a design objective (e.g.compression efficiency) can be found. In order to achieve compression gain while keeping the computational load reasonably low, two entities are needed: a criterion (cost function) for basis comparison and a fast search algorithm, which finds the best basis from the set of all possible bases. Therefore, in this work we use Shannon entropy of wavelet coefficients as cost function for WP bases comparison. The cost is expressed in terms of bits and it provides fast and fairly accurate estimation of the actual output bits that will be spent in coding of the coefficients. In order to find the best basis we apply adaptive search using single spatial tree algorithm. Fig.6.Pyramidal Structure of 3-level wavelet decomposition. In the proposed implementation, each of the N steps of the dyadic wavelet decomposition is followed by the greedy growing tree algorithm on the obtained high-pass sub bands. The growth of the tree is controlled with parameter dpn, that defines the coarsest scale of the su band that can be produced by the greedy growth. Each sub band that is on the scale finer than d is decomposed depending on the decision given by the cost function. The complexity for this type of decomposition is upper bounded with There are two possible approaches for finding the best tree fully grown tree and greedy grown tree. To compare complexity of each one the following notation is introduced here: L _ L is the size of a square image, N is the maximum allowed depth of the decomposition, a0 is the constant specifying per-pixel complexity of DWT for a wavelet filter of a specific length, and a1 is the constant specifying perpixel complexity of the computation of the specific cost measure. With this notation the complexity of the dyadic wavelet transform f D is Since the decomposition is fixed, no computation of cost measure is necessary and the complexity is dependent only on a0. In a case where the full growth is employed, the best basis is searched over all possible bases. With respect to the given cost measure, the optimal basis can always be found. If all the intermediate sub bands are preserved. In this case the complexity is independent of the finally selected tree. When (2) It can be seen that the complexity increases linearly with d. The lower bound of the complexity, for the case when only the immediately lower level of the dyadic decomposition tree is grown, is given by Since the algorithm execution time largely depends on the wavelet transform, this determines the overall WP SPIHT execution time to be at least two times longer than for the baseline SPIHT. Pseudocodes for WP decomposition/reconstruction and greedy tree growth algorithm are given in Tables 1 2. Dyadic wavelet decomposition interleaved with greedy growth decomposition of the high-pass sub bands is defined by function Decompose (Table 1), where the greedy growth itself is performed with the function WPanalysis (Table 2).

5 Color Image Compression using Set Partitioning in Hierarchical Trees Algorithm (3) The treeinfo vector contains a description of the chosen basis and is necessary at the decoder for proper reconstruction. This information represents a bit-stream overhead, but its influence on compression results is negligible. Note that when d ¼ this is equivalent to performing only dyadic decomposition and in this case the WP-SPIHT coder produces results identical to the baseline SPIHT coder. The image is reconstructed as shown in Table 2, using function. In the proposed implementation, each of the N steps of the dyadic wavelet decomposition is followed by the greedy growing tree algorithm on the obtained highpass sub bands. The growth of the tree is controlled with parameter dpn, that defines the coarsest scale of the sub band that can be produced by the greedy growth. Table.1. Pseudo code for wavelet packets decomposition Fig.7. Parent-children relations across sub bands. V. EXPERIMENTAL RESULTS WP-SPIHT algorithm has been tested on 256 X 256 all images compressions different levels with below wave lets names Haar, Biorthogonal, Symlets, Daubechies, ReverseBior, D a Meyer, Coif lets We present WP- SPIHT coding results both visually and in the terms of PSNR. Results only one level decomposition in the documentation and we show complete Results with MATLAB 7.2 VERSION. Table.2. Pseudo code for greedy tree growth algorithm Fig.8. WP-SPIHT coding results both visually and in the terms of PSNR. Present WP-SPIHT coding results with haar wave let image compression level one below figures. Fig.9. Original image.

6 G.RAMESH, V.S.R.K SHARMA Fig.10. Compressed image. Fig.13. Compressed image. Fig.11. Reconstructed image. We present WP-SPIHT coding results with Biorthogonal wave let image compression level one below figures. Fig.14. Reconstructed image. We present WP-SPIHT coding results with Symlets wave let image compression level one below figures Fig.12. Original image. Fig.15. Original image.

7 Color Image Compression using Set Partitioning in Hierarchical Trees Algorithm ofthe zerotree quantization. Using the entropy as a criterion for the best basis selection can result into many coarse scale high frequency subbands. If a coefficient somewhere at the bottom ofthe zerotree is found to be significant, the parent nodes need to be encoded even ifsome ofthem are insignificant. One way to improve those results would be in designing an advanced cost metric that would take into account the characteristics of SPIHT algorithm and provide optimal distortion value for a given bitrate. Fig.16. Compressed image. Fig.17. Reconstructed image. B. Future Scope Compressing color images efficiently are one of the main problems in multimedia applications. So we have tested the efficiency of color image compression using SPIHT algorithm. The SPIHT algorithm is applied for luminance (Y) and chrominance (Cb,Cr) part of RGB to YCbCr transformed image. Reconstructed image is verified using human vision and PSNR. Huffman and arithmetic coding can be added to increase the compression. We can test the channel behavior by sending compressed image between two computer and check the reconstructed image. VII. REFERENCES [1] An extensive study of compressed mammography image quality using SPIHT compression was performed at Stanford University. The paper can be downloaded in PostScript or portable document format (pdf) format. [2] SPIHT can also be efficiently used with DCTs, as shown by Z. Xiong, O. Guleryuz and M. T. Orchard, in their paper "A DCT-based embedded image coder." KITE ECE Page 71 [3]A new application for image compression proposed by Wm. Randolph Franklin. is described in Compressing Elevation Data (HTML version). [4] The importance of trees in the EZW algorithm was studied by V. Ralph Algazi and Robert R. Estes Jr.. You can read a HTML summary or download a postscript paper. [5] Antonini M, Barland M, Mathieu P, Daubechies I. Image coding using the wavelet transform. IEEE Transactions on Image Processing 1992;2: VI. CONCLUSION & FUTURE SCOPE This paper finally concluded and future scope will be enhanced below. A. Conclusion Efficient set of rules for establishing zerotree structures when used with WP decomposition is presented. The proposed solution enables the modification of popular SPIHT scheme, called WP-SPIHT a combination of WP as a decomposition method with SPIHT as an image compression scheme. The compression performance of WP-SPIHT has been compared to SPIHT both visually and in terms of PSN R. WP-SPIHT significantly outperforms SPIHT for textures. For natural images, which consist ofboth smooth and textured areas, the chosen cost function used in WP is not capable ofestimat ing correctly the true cost for a case when the subband is encoded with SPIHT. For those images the PSNR performance of WP-SPIHT is usually slightly worse. As opposed to wavelets, wavelet packet basis might not necessarily produce coefficients that help towards success

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