Comparative Analysis of Singular Value Decomposition (SVD) and Wavelet Difference Reduction (WDR) based Image Compression

Transcription

1 International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07) Comparatie Analysis of Singular Value Decomposition (SVD) and Waelet Difference Reduction (WDR) based Image Compression Ms. Sana Shafik Desai Research Scholar, Department of Electronics and elecommunication Engineering, Kolhapur Institute of echnology s College of Engineering, Gokul Shirgaon, Kolhapur, Maharashtra, India. Prof. Dr. M. S. Chaan Professor, Department of Electronics Engineering, Kolhapur Institute of echnology s College of Engineering, Gokul Shirgaon, Kolhapur, Maharashtra, India. Abstract his paper presents study of two lossy image compression techniques. he two techniques are Singular Value Decomposition (SVD) based image compression and Waelet Difference Reduction (WDR) based image compression. SVD based compression reduces the psychoisual redundancies present in the image through rank reduction method. WDR is a lossy image compression technique. It gains compression by taking the discrete waelet transform of the input image and then encodes the transform alues using difference reduction method. Various image compression parameters like PSNR, MSE and compression ratio are ealuated for the two techniques. he two techniques are compared on the basis of same compression parameters and isually similar compressed images. Keywords: Lossy Image Compression, Singular Value Decomposition, Waelet Difference Reduction. Introduction Compression is an important image processing technique that remoes the redundant information present in an image without much affecting the quality of that image. It maps a higher dimensional space into a lower dimensional space []. It is applied in different fields like signal processing, artificial intelligence, communication, etc. he aim of compression is to sae space so that huge data can be stored, transmitted and retrieed efficiently. Data to be compressed can be multimedia, documents, ideoconferencing information, medical images, etc. he uncompressed images require not only more storage capacity but also more transmission bandwidth. Compression will reduce the size of the file and allow more number of images to be stored in a gien memory space. Image compression deals with reducing the extra irreleant information present in an image. Redundancies present in an image can be:. Psycho isual redundancies: his information does not appear sensitie to human eye. It belongs to isually nonessential information of an image. Such an information can be discarded.. Inter pixel redundancy: his redundancy occurs due to presence of similar neighboring pixels. Such type of redundancy can be remoed. 3. Coding redundancy: his redundancy occurs due to use of longer code words to encode information of an image. Coding redundancies can be remoed by assigning shortest code words to most frequently occurring information. [] Remoing the redundancy will reduce the number of bits required to represent an image. his can be achieed by different compression techniques. here are two types of compression techniques- lossless compression and lossy compression [9]. In lossless image compression there is no loss of information as the original image is perfectly recoered from compressed one. his is needed in case of data like executables, documents, some medical images, etc. which need to be exactly reproduced when decompressed. In lossy compression, some amount of loss of information is tolerable and reconstruction of the original data is possible een after remoing some amount of redundant information [0]. Some images need not hae to be reproduced exactly after compression. An approximation of original image is enough for most of the purpose, as long as the error between the original and compressed image is not too high. Such lossy compression techniques are used to lessen the amount of data in order to store, handle, and transmit the represented content in effectie manner. In this paper two lossy image compression techniques are discussed. he two techniques are Singular Value Decomposition (SVD) based image compression and Waelet Difference Reduction (WDR) based image compression. Methods Singular Value Decomposition based image Compression Singular Value Decomposition (SVD) decomposes an image matrix into product of three matrices. Out of the three matrices, two are orthogonal matrices and third is diagonal matrix. he diagonal matrix contains the diagonal elements which are the singular alues of image matrix. [4][] 49

2 International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07) Let A be an image matrix, then singular alue decomposition of matrix A is: A yxz = U yxy S yxz (V zxz ) () where, A is a y z matrix, S is a y z diagonal matrix. he elements on the diagonal of S are singular alues of A. U is a y y matrix containing left singular ectors of A. V is a z z matrix containing right singular ectors of A. U and V are orthonormal matrices i.e. UU = I and VV = I. In matrix form the equation () becomes: σ σ σ A=[ u, u, u y ] σ z z In the aboe form, u i (for i=, y) represents columns of matrix U and are eigen ectors of AA with eigen alue σ i. u i is the left singular ector of image matrix A. he rows i (for i=,, z) of V are eigen ectors of A A, with eigen alue σ i. i is right singular ector of image matrix A. σ i (for i=,,.z) are the singular alues of matrix A. he singular alues are such that σ σ σ 3.. σ z 0. In order to bring out compression, the dimensions of diagonal matrix are reduced to S pxq where p y and q z. After application of SVD, only some singular alues from matrix S are kept while the lower singular alues are remoed. his can be done because of the fact that singular alues are arranged in descending order and that first singular alue contains much information than the following singular alues that contain decreasing amount of image information. So, the lower singular alues that contains less important information can be discarded.[4] redundancies present in an image without reducing isual quality of an image. [5] Algorithm for SVD based image compression:. Read the input image. Conert it into grayleel image. 3. Decompose the image using singular alue decomposition. 4. Discard the singular alues not required for compression. 5. Reconstruct the image. 6. Compute compression ratio, mse and psnr. Waelet Difference Reduction based Image Compression he Waelet Difference Reduction (WDR) is an encoding technique which is based on the difference reduction method. It gains compression by taking the discrete waelet transform of the input image and then encodes the transform alues using difference reduction method. [3] Input Image DW WDR encoding Figure : Block diagram for WDR based compression Initialisation Update threshold Significance pass Figure : WDR encoding WDR compressed bit stream Refinement pass he number of non-zero singular alues present in diagonal matrix S specifies the rank of the matrix A. If the singular alues after a certain rank are not zero, they are considered as redundant and can be remoed. Compressed bit stream WDR decoding Inerse DW Reconstructed image Equation () aboe can also be written as: A=u σ + u σ + ur σ r r +uz σ z z () where r is the rank of A. Reducing equation () till r alues does not gie much change in the image. he amount of compression achieed will be ery less and the image quality will also remain nearly same as quality of original image. For better compression, only the first k singular alues before r of equation () are taken so that equation () becomes A = u σ + u σ + uk σ k k (3); where k < r. he alue of k is chosen such that good amount of compression is achieed while maintaining the image quality. Ignoring lower singular alues remoes the psychoisual Figure 3: Block diagram for WDR decompression Discrete waelet transform diides image into four subbands- LL, LH, HL, HH. Waelet Difference Reduction encoding uses four steps for encoding: Initialisation, hresholding, Significant pass and refinement pass.. Initialisation: In this the scan order is decided. he scan order goes through sub-bands from higher leel to lower leels in zig-zag manner. A threshold 0 is selected.. Update hreshold: hreshold is updated to k = k-/, for k=, p and p is the number of pixels in an image. 3. Significance Pass: Here, alues of waelet transform are compared to a specific threshold alue. A alue is significant if it is greater than or equal to threshold alue. If an index is found to be significant then it is 49

3 International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07) remoed from the scan order. Next, difference of these index alues is taken and binary expansion of successie difference is done. Since the MSB in these expansions is always, we can ignore this bit and use the signs of the significant transform alues in its place in the symbol stream. he stream consists of four symbols that can be encoded using probabilistic model. 4. Refinement Pass: In this, standard bit plane quantization is carried out to gie refinement bit. Refined alue gies better approximation of transform alue. 5. Repeat steps () to (4) until you get desired bit budget. o reconstruct the image, WDR decoding and inerse DW is performed on compressed bit stream. [] he property of WDR is that it gies perceptually better image at high compression ratio while retaining the desirable features. Results PSNR = 0*log 0 [55*55/MSE] he experimental results for the two techniques are shown below. he codes of these algorithms are executed in MALAB software. For this MALAB ersion 03a was used. MALAB proides hundreds of built-in functions for technical computation, graphics, etc. For experimenting, a jpeg image is selected whose matrix size is 56x56. SVD based image compression results for different number of singular alues (k): Compression parameters he results of different compression techniques can be compared by using different compression parameters like compression ratio, mean square error (MSE) and peak signal to noise ratio (PSNR). hese parameters are used to measure the degree to which an image is compressed and estimate the quality of compressed image. [9]. Compression Ratio: It is defined as the ratio of original size of the image to compressed size of the image. It gies measure of the degree to which an image is compressed. (a) (b) (c) (d) (e) Compression ratio = (Size of original image/size of compressed image). Mean Square Error (MSE): It gies the measure of degradation of compressed image quality as compared to the original image. It is defined as square of the difference between original image pixel alues and the corresponding pixel alue of the compressed image, aeraged oer the entire image. (f) (g) (h) MSE = ( (g(x,y) g (x,y) ) ) / (m*n); where m*n gies total number of pixels present in original image; g(x,y) and g (x,y) are the pixel alue of original and compressed image respectiely. 3. Peak Signal to Noise Ratio (PSNR): It is the ratio of maximum signal power to the noise power. In Image compression, noise refers to the deiation of the compressed image from the original one. hus, PSNR gies the quality of the reconstructed images after compression. (i) (j) (k) he figures shown aboe are: (a) Original image (b) Gray scale image (c) Image for k=5 (d) Image for k=5 (e) Image for k=45 (f) Image for k=65 (g) Image for k= 85 (h) Image for k=05 (i) Image for k=5 (j) Image for k=45 (k) Image for k=65 493

4 psnr mse cr International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07).3.. WDR based image compression results: Input Image grayleel Image Number of Singular Values used Figure 4: Graph of compression ratio V/s number of singular alues used Figure 7: Original image WDR Compressed image Figure 8: Gray scale image Figure 9: WDR compressed image 0 0 Figure 5: Graph of MSE V/s number of singular alues used Number of Singular Values used able : Compression parameters ealuated for WDR compressed image Compression Parameters Values Compression Ratio.6875 MSE PSNR Result of comparison between two techniques with nearly equal compression parameters: SVD compressed image WDR Compressed image Number of Singular Values used Figure 6: Graph of PSNR (in db) V/s number of singular alues used able : Compression parameters ealuated for images constructed from different number of singular alues Number of singular alues used CR MSE PSNR Figure 0: SVD and WDR compressed images haing nearly equal compression parameters (compression ratio.7, MSE 40, PSNR 3.) Compression Parameters :CR :MSE 3:PSNR SVD WDR Figure : Bar graph representing nearly equal compression parameters for both the techniques with SVD at k=7. 494

5 International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07) Result of comparison between two techniques with nearly same isual quality of compressed image: SVD Compressed image WDR Compressed image performance of WDR will get enhanced. We can get isually better images with high amount of compression if WDR is used along with SVD. References Figure : SVD (at k=5) and WDR compressed images with nearly same isual quality but different compression parameters. For SVD, compression ratio=.40, MSE=7.0, PSNR= For WDR, compression ratio=.68, MSE=39.38, PSNR=3.7 Discussion SVD based image compression technique gies better isual quality at higher singular alues. With increase in number of singular alues used, compression ratio and MSE decreases. Hence, higher alues of k will gie lower compression. With increase in number of singular alues used, PSNR increases as shown in figure 6. WDR based image compression achiees perceptually good image with high compression. Conclusion On comparing SVD and WDR based compression for same compression parameters, it is seen that WDR compressed image appears perceptually better than SVD as shown in figure 0. For this SVD based compression with 7 singular alues was carried out. On comparing compressed images of the two techniques with same isual quality, it is found that compression ratio of WDR based compression is higher than SVD based compression. For this SVD based compression with 5 singular alues was carried out. Since reconstructed images are not exactly same as that of original image, both the techniques are lossy compression techniques. From the aboe discussion it can be concluded that WDR based compression gies good quality images with higher compression ratios. SVD based image compression gies better quality images at higher singular alues. For same compression parameters, the results obtained by WDR based compression are superior than that of SVD based compression. If we integrate both these techniques then [] K. Mounika. SVD based image compression. International journal of engineering research and general science, Vol 3, 05. [] Vaish and Kumar. WDR based compression technique using PCA. IEEE paper, 05. [3] Rufai, Anbarjafari and Demirel. Lossy image compression using singular alue decomposition and waelet difference reduction. A.M. Rufai et al. / Digital Signal Processing 4 (04) 7 3, Elseier, 04. [4] Samruddhi Kahu and Reena Rahate. Image compression using Singular Value Decomposition. International Journal of Adancements in Research & echnology, Volume, Issue 8, August-03. [5] Rowayda A. Sadek. SVD based image processing applications: State of the art, contributions and research challenges. (IJACSA) International Journal of Adanced Computer Science and Applications, Vol. 3, No. 7, 0. [6] S. J. Niedita. "Performance Analysis of SVD and SPIH Algorithm for Image Compression Application". International Journal of Adanced Research in Computer Science and Software Engineering, ol., no., 0. [7] S. Raja and A. Suruliandi. "Image Compression Using WDR and ASWDR echniques with Different Waelet Codecs". ACEEE Int. J. Inform. echnol. 0, pp. 3-6, 0. [8] S. Raja and A. Suruliandi. Performance Ealuation on EZW & WDR Image Compression echniques. International conference on Communication, Control and Computing echnologies, 00. [9] X. Zhang. "Lossy compression and iteratie reconstruction for encrypted image". Information Forensics and Security, IEEE ransactions on, ol. 6, no., pp , 0. [0] M. Boliek. "Beyond compression: a surey of functionality deried from still image coding". in Signals, Systems and Computers, Conference Record of the hirty-seenth Asilomar Conference on, pp , 003. [] R. C. Gonzalez and R. E. Woods. Digital Image Processing. Prentice Hall Upper Saddle Rier, NJ, 00. [] J. ian and R. O. Wells Jr. "Embedded image coding using waelet difference reduction in Waelet image and ideo compression. pp , 00. [3] M. D. Greenberg, Differential equations & Linear algebra, Prentice Hall, 00. [4] L. Knockaert, B. De Backer and D. De Zutter. "SVD compression, unitary transforms, and computational complexity," Signal Processing, IEEE ransactions on, ol. 47, no. 0, pp ,

6 International Journal of Engineering Research and echnology. ISSN Volume 0, Number (07) [5] J. ian and R. O. Wells Jr. "A lossy image codec based on index coding. IEEE Data Compression Conference,p. 456, 996. [6] J. ian and R. O. Wells Jr. "Image data processing in the compressed waelet domain". Signal Processing, 3rd International Conference on, pp , 996. [7] D. Kaiman. "A Singularly Valuable Decomposition". College Mathematics Journal, ol. 7, no., pp. -3, 996. [8] J.-F. Yang and C.-L. Lu. "Combined echniques of Singular Value Decomposition and Vector Quantization for Image Coding". IEEE ransactions on Image Processing, ol. 4, no. 8, pp. 4-46, 995. [9] 496