Image Compression with Singular Value Decomposition & Correlation: a Graphical Analysis

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1 ISSN -7X Volume, Issue June 7 Image Compression with Singular Value Decomposition & Correlation: a Graphical Analysis Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee Tripura University (A Central University), Suryamaninagar, Tripura First Author Phone No, Second Author Phone No, Third Author Phone No ABSTRACT Recognition system based on still images ( binary, grayscale and color images) in image processing domain requires high computational capability and large memory space. In this project our primary concern is to deal with both of the issues. Individually Singular Value Decomposition and Fast Fourier Transform tools are both have proven their importance in image analysis and face recognition. We have also tried to show the efficacy of Singular Value Decomposition while computing the Correlation between the original image and image in compressed form. We also studied the decomposed information received after Singular Value Decomposition transform; then calculated Fast Fourier Transform to compute cross-correlation to visualize the similarity of the images. After analysis it has been seen that the reconstructed image with less numbers of Singular values is as good as the original image (training image -set of face recognition system). The highest correlation pick is achieved with largest Singular Value. We have also analyzed the Full Width Half Maxima values along with both x and y axis in support of our observation. Finally, we have resolved that during pre-processing of image processing applications SVD can be used as a powerful tool for image compression and we have tried to visually present our observation using relative error calculations, correlation method followed by calculating Full Width Half Maxima. Keywords Singular Value Decomposition, Fast Fourier Transform, Full Width Half Maxima, Compression, Correlation.. INTRODUCTION Compression of images is an active field of science. Many works has been done in this domain [] []. Due to large spatial redundancy and intrusion of moderate erroneous data into the reconstructed images compression of images is possible. Compression of images has several applications in real life in the context of minimizing computational cost and maximum space utilization. Although very reliable forms of biometric personal identification exist, e.g., retinal or iris, fingerprint; these forms rely on the cooperation of the subject, whereas an identification system based on analysis of the face profile images if often effective in random conditions [] [7]. Now a day applications are being developed considering all nature and size of the hardware. Singular Value Decomposition (SVD) may be analyzed broadly from two view points. On the one hand, we can see it as a method for transforming a set of correlated variables into a set of uncorrelated ones which exposes various relationships with the original dataset, e.g. original dataset can be linearly represented by the decomposed sets of data [] [] [7]. On another hand, we can identify the point where the most variation occurs in the SVD transformed dataset which helps to find the best approximation of the original dataset using fewer dimensions []. That s why, SVD may be considered as an effective method for data reduction or compression. In the following section we tried to analyze the features and usability of decomposed matrices attained using SVD technique. Further we have tried to implement the cross-correlation technique between the original image and approximated images with much less singular values. To implement the correlation technique we have computed the Fast Fourier Transform (FFT) of the concerned image matrices. Implementing the mentioned methods we tried to compare and analysis the results we received working on different standard images (Gray-scale and binary) and Face images. Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

2 ISSN -7X Volume, Issue June 7. OBJECTIVE OF THE WORK The objective of this work is to apply SVD to midlevel image processing method, precisely to the area of image compression and recognition. We will study the application of SVD method in the image compression domain. This will minimize storage related complexities in any image processing application. As the size of the active data will be compressed; so, it will help to reduce computational cost too. SVD is factoring an image matrix (let, A) into two orthogonal matrices (U, V) and one diagonal matrix (S), in such way that A=USV T. We have conducted experiments with different ranks of S initializing from one and the outer product expansion of image matrix A for image compression. We have calculated the relative errors of reproduced image matrix with minimum number of Singular Values. Visualizing the output (i.e. approximated image) we applied cross-correlation method to establish the similarities with the original image and the approximated image which reflects that essential features are preserved in the image matrix reproduced with lower rank of singular valued matrix. We have used MATLAB for programming and experiments.. THEORITICAL BACKGROUND & EXPLAINATION A. Significance of SVD The SVD allows analyzing matrices and associated linear maps in detail, and solving a host of special optimization problems, from solving linear equations to linear least-squares [] []. It can also be used to reduce the dimensionality of high-dimensional data sets, by approximating data matrices with low-rank ones. Any nonzero real matrix with rank > can be factored as = with an matrix with orthogonal columns, = (,,,, ) and an matrix with orthogonal rows. This directly related to the spectral theorem which states that if B is a symmetric matrix ( = ) then we can write = where a diagonal matrix of eigenvalues and U is an orthonormal matrix of eigenvectors. The relationship can be found from below: = = = = These are both spectral decompositions, hence the are the positive square roots of the eigenvalues of. In the SVD, the matrices are rearranged so that. Reducing the SVD we can write an invertible matrix A as: = = (,,, ) = () i.e. the matrix A can be written as the sum of rank-one matrices. =, () where are the columns of U and V, respectively. We want to approximate the matrix A by using far fewer entries then in the original matrix by using the rank of a matrix, we remove the information that is not needed (the depended entries) where. = () since the singular values are always greater than zero. Adding on the dependent terms where the singular values are equal to zero does not affect the image i.e. the useful features of the original image is preserved. Removing the terms at the end of the equation zero out, leaving us with: = () One way to compress the image A is to approximate A by a matrix of smaller rank. If < then the closest approximation to A, (rank A=r)- by a matrix of rank K that is the truncation of the previous equation to the first K terms: = ) So, from the above equation we can approximate a matrix by adding only the first few terms of the series. It is noticed that the amount of memory required increases linearly as the dimension get larger, as opposed to exponentially in the case of representation of the original image. Thus, as the image gets larger, more memory is saved by using SVD. The Total storage of will be ( + + ). B. Low rank approximation in SVD If we consider a matrix R, with SVD given as in the theorem: Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

3 ISSN -7X Volume, Issue June 7 = () = (,,,,,,), where the singular values are ordered in decreasing order, >. A best approxmination is given by zeroing out the ( ) trailing singular values of A, that is =, = (,,,,,,) (9) C. Image Compression and measures with SVD: Total storage for will be ( + + ), where is the size of the original image. The integer can be chosen confidently less then, and the original image corresponding to the approximated image is seen very close to the original image. To measure the performance of the compression we have computed the compression factors. We have visualized the quality of the compressed image. Compression ratio ( ) = ( ) () To measure the quality of the compressed image w.r.t. the original image we have calculated the respective relative errors. The minimal error is given by the Euclidean norm of the singular values that have been zeroed out of the process: = + + () amount of similarity between two signals (-D or - D). In practice, correlation between (, ) and h(, ) can be written as [], (, ) h(, ) = (, )h( +, + ) () We have used -D Correlation to measure the similarity between the original image and the approximated images reconstructed with different numbers of Singular values. i.e. (, ) (, ) = (, ) ( +, + ) () We have computed the FFT of the images and then calculated the correlation by multiplying the FFT of the original image with the conjugate transpose of the approximated images acquired from different values of. F. Full Width Half Maxima (FWHM): Full width at half maximum (FW HM) is an expression of the extent of a function, given by the difference between the two extreme values of the independent variable at which the dependent variable is equal to half of its maximum value. FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. We have calculated the FWHM along with both X- axis and Y-axis of the correlation matrix calculated with increasing number of Singular Values. D. Relation to Fourier Analysis with reference to SVD: Data analysis with SVD has similarities to Fourier analysis. Fourier analysis also involves expansion of the original data in an orthogonal basis []. = () The connection with SVD can be illustrated by normalizing the vector and by naming it. = = () which generates the main equation =, similar to Eq. (). E. Two-dimensional Correlation: Correlation is deployed in any application to find the Fig.. Full Width Half Maxima. EXPERIMENTS AND RESULTS A. Images and Face Database used: We have used few benchmark-images those have been distributed freely for research purposes e.g. Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

4 ISSN -7X Volume, Issue June 7 Lena, Barbara, Cameraman, Baboon etc. [9] We have also used Face images from UGC-DDMC face database where face profiles are stored w.r.t. different poses. We have also developed binary images for testing purposes. B. Visualizing relative errors in approximated images: In Fig.(b), (b), (b), (b), (b) we have shown the relative errors i.e. w.r.t. the increasing number of diagonal elements i.e. singular values(eq.()), where (from Eq. (9)) Fig.. (a) Barbara Relative Error - Fig.. (b)relative error w.r.t. Relative Error = (,,,,, ) = = () Relative Error In the said figures we have plotted the log-value of the relative errors w.r.t. the different numbers of singular values (SV). The Graph reflects that the relative error doesn t change much with the increasing number of SVs. So, We can say that a well approximated image can be reconstructed using the largest SVs (initial two or three SVs) only. Fig.. (a) Lena (Original Image) Fig.. (b)relative error w.r.t. Fig.. (a) Cameraman Fig.. (a) Face Profile Image from UGC-DDMC FaceDB Fig.. (a) Baboon. Fig.. (b) Relative error w.r.t. Relative Error - - Fig.. (b)relative error w.r.t. Relative Error - Fig.. (b)relative error w.r.t. Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

5 ISSN -7X Volume, Issue June 7 Relative Error Fig. 7. (a) Binary S Fig. 7. (b)relative error w.r.t. C. Visualizing relative errors in approximated images: We have plotted (mesh plot) the correlation matrices. These matrices are the output of the cross-correlation between the original images and the reconstructed images with increasing number of SV. We have concentrated into first four SVs and tried to visualize the effect on the correlation matrix obtained. In Fig. (c,c,c,c), (c,c,c,c), (c,c,c,c), (c,c,c,c),(c,c,c,c),7(c,c,c,c) we can see the mesh plot of -D correlation between original image and images approximated with respectively one, two, three and We can see the just with single SV the correlation pick is attained. Slope difference of the mesh plot with two SVs and three SVs is less whereas there is almost no difference between the mesh plot with three and four SVs respectively and it is same with large number of SVs. So, from the correlation matrix we can correctly approximate the required number of SVs for image reconstruction. Considering the correlation pick as origin the FWHM is calculated along with both X-axis (Fig.(d), (d), (d), (d), (d), 7(d)) and Y-axis (Fig.(e), (e), (e), (e), (e), 7(e)). We plotted the respective results and we obtained two decay curves respectively for X and Y-axis. We have plotted (stem plot) the s of spatial frequencies f x (Fig..(g,g),.(g,g),.(g,g),.(g,g),.(g,g), 7.(g,g)) and f y (Fig..(h,h),.(h,h),.(h,h),.(h,h),.(h,h), 7.(h,h)) of the approximated images with respectively one, two, three and We noticed among them there are minimal changes or no change in some cases. Fig..(C) Mesh plot of -D Cross-correlation Fig..(C) Mesh plot with Fig..(C) Mesh plot with Fig..(C) Mesh plot with Fig..(C) Mesh plot of -D Cross-correlation Fig..(C) Mesh plot with Fig..(C) Mesh plot with Fig..(C) Mesh plot with 7 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

6 ISSN -7X Volume, Issue June 7. FWHM along X axis FWHM along X axis Fig..(d) Measurement of 7 9 Fig..(d) Measurement of Fig..(g) Magnitude of f x (spatial frequency of A Fig..(g) Magnitude of f x (spatial frequency of A.. FWHM along Y axis FWHM along Y axis Fig..(e) Measurement of. 7 9 Fig..(e) Measurement of Fig..(h) Magnitude of f y Fig..(h) Magnitude of f y correlation pick correlation pick pick pick Fig..(h) Magnitude of f y (spatial frequency of A. Fig..(h) Magnitude of f y (spatial frequency of A Fig..(f) Magnitude of Fig..(f) Magnitude of Fig..(g) Magnitude of f x Fig..(g) Magnitude of f x Fig..(C) Mesh plot of -D Cross-correlation Fig..(C) Mesh plot of - D Cross-correlation Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

7 ISSN -7X Volume, Issue June 7 FWHM along Y axis. FWHM along Y axis.... Fig..(C) Mesh plot with Fig..(C) Mesh plot with 7 9 Fig..(e) Measurement of 7 9 Fig..(e) Measurement of correlation pick correlation pick.9e-.9e-.99e-.99e- Fig..(C) Mesh plot with Fig..(C) Mesh plot with pick -.9e- pick -.9e- -.9e- -.9e- 7 9 Fig..(f) Magnitude of 7 9 Fig..(f) Magnitude of Fig..(C) Mesh plot with Fig..(C) Mesh plot with FWHM along X axis FWHM along X axis Fig..(g) Magnitude of f x Fig..(g) Magnitude of f x 7 9 Fig..(d) Measurement of 7 9 Fig..(d) Measurement of Fig..(g) Magnitude of f x (spatial frequency of A. Fig..(g) Magnitude of f x (spatial frequency of A. 9 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

8 International Journal of Electronics, Electrical and Computational System ISSN -7X Volume, Issue June Fig..(C) Mesh plot with Fig.7.(C) Mesh plot with Correlation with singular values Fig..(h) Magnitude of fy (spatial frequency of A Fig..(h) Magnitude of fy (spatial frequency of A.. -. Correlation with singular values Fig..(h) Magnitude of fy (spatial frequency of A Fig..(h) Magnitude of fy (spatial frequency of A. Fig..(C) Mesh plot with FWHM along X axis.7.9 Fig.7.(C) Mesh plot with FWHM along X axis Correlation with singular value Fig.7.(C) Mesh plot of D Cross-correlation Correlation with singular values 7 9 Fig..(d) Measurement of 7 9 Fig.7.(d) Measurement of FWHM along Y axis FWHM along Y axis.7. Fig..(C) Mesh plot of -D Cross-correlation Fig..(C) Mesh plot with Fig.7.(C) Mesh plot with 7 9 Fig..(e) Measurement of Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee 7 9 Fig.7.(e) Measurement of

9 ISSN -7X Volume, Issue June 7 correlation pick correlation pick pick Fig..(f) Magnitude of - Fig..(g) Magnitude of f x 7 9 Fig..(g) Magnitude of f x (spatial frequency of A Fig..(h) Magnitude of f y pick.9e-.99e- -.9e- -.9e- 7 9 Fig.7.(f) Magnitude of Fig.7.(g) Magnitude of f x Fig.7.(g) Magnitude of f x (spatial frequency of A Fig.7.(h) Magnitude of f y. CONCLUSION We have studied compression ability of SVD. Our analysis is based on rank-approximation and the correlation of the original image and the approximated image regenerated with different quantities of singular values which required very less space than the original image, also preserves the features and computational cost is less. REFERENCES [] L. Cao, Singular Value Decomposition Applied to Digital Image Processing, Division of Computing Studies, Arizona State University Polytechnic Campus, pp. -, [] M.E.Wall, A. Rechtsteiner, L. M. Rocha, "Singular value decomposition and principal component analysis, A Practical Approach to Microarray Data Analysis, Springer US, pp.9-9,. [] L. Zhao, W. Hu, L. Cui, Face Recognition Feature Comparison Based SVD and FFT, Journal of Signal and Information Processing,vol., pp. 9-, May. [] I.C.F.Ipsen, "Numerical Matrix Analysis: Linear systems and Least Squares," SIAM, Philadelphia, 9 [] G. Strang, "Introduction to Linear Algebra,", Wellesley-Cambridge Press, 99 [] G. Zeng, "Face Recognition with Singular Value Decomposition," CISSE Proceeding, [7] O. Bryt, M. Elad, "Compression of facial images using the K-SVD algorithm," Journal of Visual Communication & Image Presentation, Elsevier, pp. 7-, March. [] A. K. Jain, Fundamentals of Digital Image Processing, PHI Learning Pvt. Ltd., [9] Standard images used for experiment. ( processingplace.com/root_files_v/image_databases. htm.) Fig..(h) Magnitude of f y (spatial frequency of A. Fig.7.(h) Magnitude of f y (spatial frequency of A. Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee