Decomposition and denoise of an image using

(IJCSIS) International Journal of Computer Science and Information Security, Vol. 13, No. 5, 2015

Decomposition and denoise of an image using singular value decomposition Viji Paul

P.Selvaraju

Department of Information Technology Salalah College of Technology, Sultanate of Oman

Department of Computer Science & Engineering Vel Tech Multi Tech Dr.RR and Dr.SR Engineering College, Avadi, Chennai, India

D. Nagarajan Department of Information Technology Salalah College of Technology, Sultanate of Oman

A* = U* S* VT*. This process is termed dimensionality reduction, and A* is referred to as the Rank k Approximation of A or the "Reduced SVD" of A. The top k singular values are selected as a mean for developing the representation of A, which is now free from noisy dimensions [2].

Abstract - Image decomposition is now essential for transmission and storage in database. Singular Value Decomposition is a decomposition technique for calculating the singular values, pseudo-inverse and rank of a matrix. The conventional way of doing this was to convert a matrix to row echelon form. The rank of a matrix is then given by the number of nonzero rows or columns of the echelon form. Singular value decomposition is one of the methods to compress and denoise the images. The main focus of this paper is to decompose and denoise an image using singular value decomposition.

II. DATA BASE

Key words :SVD, decomposition, de noise .

I. INTRODUCTION Pixel of the image is 675 rows, 900 columns and 3 colours. Signal processing aims at extracting information from the raw signal. The difficulty in reaching this goal depends both on the characteristics of the noise-free signal and the noise. The signal-to-noise-ratio (SNR) is the ratio of the strength of the signal and the strength of the noise. The higher the ratio the easier it is to extract information and the more reliable are the results. SNR is the ratio, of the mean and the standard deviation of the measured signal.

A digital image is generally encoded as a matrix of grey level or color values. Each pair (i,u(i)),where u(i) is the value at i is called a pixel. The image accuracies are categorized as noise and blur. Noise is the perturbation of the image and blur is intrinsic to image acquisition system. A good quality image has a standard deviation of about 60. Image compression is one of the applications in SVD. Consider some matrix A with rank 1000; that is, the columns of this matrix span a 1000 dimensional space. Encoding this matrix on a computer is going to take quite a lot of memory. We might be interested in approximating this matrix with one of lower rank. An image is a section of random access memory that has been copied to another memory or storage location. Dimensionality reduction is a noise reduction process. Thus, SVD belongs to a class of dimensionality reduction techniques that deal with the uncovering of hidden data structures. If matrix A is in the form of A = USVT, where U is a matrix whose columns are the eigenvectors of the AAT matrix. These are termed the left eigen vectors. S is a matrix whose diagonal elements are the singular values of A. This is a diagonal matrix, so its nondiagonal elements are zero by definition. V is a matrix whose columns are the eigenvectors of the ATA matrix. These are termed the right eigenvectors. VT is the transpose of V [1].

SNR =

/s

Calculate the signal as the mean of pixel values. Calculate the Pnoise and the standard deviation or error value of the pixel values. Take the ratio or you may use SNR=10log10(Psignal / Pnoise) to express the result in decibel. 0.8620.The above image with standard deviation 66.04. III. MODEL DESCRIPTION The singular value decomposition closely associated to the companion theory of diagonal in a symmetric matrix. If A is a symmetric real n x n matrix there is an orthogonal matrix V and a diagonal D such that, A=VDVT. Here the columns of V are latent vectors for A and diagonal entries of D are eigen values of A for Singular Value Decomposition begin with m x n real matrix. There

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are orthogonal matrices U and V and a diagonal matrix S, such that A = USVT Here U is m x m and V is n x n, so that S is rectangular with the same dimensions as A .The matrix S can be formatted to be non-negative and in order of decreasing order. The columns of U and V are called left and right Singular vectors for A[5],[8].

approx_S = S; approx_S(1:ns, 1:ns) = diag(approx_sigmas); approx_ncduck = U * approx_S * V'; subplot(4, 2, j + 1), imshow(approx_ncduck), title(sprintf('Rank %d ncduck', ranks(j))); end Approximation rank of the image as follows it shown below:

IV. ANALYSIS The analysis is done in MATLAB software. The image can be converted to black and white, and then the image is treated as a matrix. a=imread('ncduck.jpg'); imshow(a) [m,n,k]=size(a) ncduck = rgb2gray(imread('ncduck.jpg')); ncduck = im2double(imresize(ncduck, 0.5)); [U, S, V] = svd(ncduck); From the above algorithm the image is converted to the singular value decomposition. sigmas = diag(S); figure; plot(log10(sigmas)); title('Singular Values (log10 Scale)');

From the above images it shows that, higher the singular value, better the quality of the image. Singular value decomposition compressed a 675x900 pixel image into a 675 x 675 for U, 30 x 30 singular value and a 30 x 900 matrix. Singular values can be used to highlight which dimensions are affected the most when a vector is multiplied by a matrix.

The above algorithm shows the singular value of the image, which is a base 10 log scale.

V. DE NOISE. The singular value decomposition , T T T A = USV T , A = (USV ) The discrete noisy v = { v ( i ) / i ∈ I } ω ( p , q ) image , depends on the

similarity between the pixels p and q and satisfies the condition of the probability

Approximately first thirty ranks get the largest singular values. figure; plot(cumsum(sigmas) / sum(sigmas)); title('Cumulative Percent of Total Sigmas');

0 ≤ ω ( p, q ) ≤ 1 and

∑ ω ( p, q ) = 1 q

[4]Computing the similarity between the images pixels will depend on the similarity of the intensity grey level of vector p referred as black and q as white. Then,

The above algorithm lines shows the cumulative percentage of the singular value .

p = pTU K S K−1 and q = qTU K S K−1 . The similarity of p −1

−1

and q sim ( p , q ) = sim ( p U K S K , q U K S K ) .The large weight of similarity windows are similar and smaller because of the intensity grey values in the similarity windows are varying. White noise alters the distance between windows in a uniform way. Impixel region creates a Pixel Region associated with the image displayed in the current figure, called the target image. The Pixel Region tool opens a separate figure window containing an extreme close-up view of a small region of pixels in the target image [7]. T

figure; subplot(4, 2, 1), imshow(ncduck), title('Full-Rank ncduck');ranks = [50, 30, 20, 10, 3,2]; for j = 1:length(ranks) approx_sigmas = sigmas; approx_sigmas(ranks(j):end) = 0; ns = length(sigmas);

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T



VI. COMPRESSION MODEL

REFERENCES [1] Chan.T , “An improved algorithm for computing the singular value Decomposition”, ACM Transactions on Mathematical Software, Vol 8, No. 1 (1982). [2] Berry, M., S. Dumais, and G. O'Brien .,”Using Linear Algebra for Intelligent information” Retrivel. SIAM Review 37(4): 573-595. (1995). [3] T. F. Chan and H. M. Zhou, “Total variation improved wavelet thresholding in image compression”, in Proceedings of the IEEE International Conference on Image Processing, Vol. 2,Vancouver, BC, Canada, 2000, pp. 391–394. [4] T. Weissman, E. Ordentlich, G. Seroussi, S. Verdu, and M. Weinberger,Universal Discrete denoising: Known channel, IEEE Trans. Inform. Theory, 51 (2005), pp. 5–28. [5] E. Tadmor, S. Nezzar, and L. Vese, A multiscale image representation using hierarchical (BV,L2) decompositions, Multiscale Model. Simul., 2 (2004), pp. 554–579. [6] H. Choi and R. G. Baraniuk, (2004) “Multiple wavelet basis image denoising using Besov ball projections,” IEEE Signal Processing Letters, Vol. 11, No. 9, 2004, pp. 717–720. [7] Z. Tongzhou; W. Yanli R. Ying; L. Yalan “Approach of Image Denoising Based on Discrete Multi-wavelet Transform International Workshop on Intelligent Systems and Applications, 2009, pp: 1-4. [8] D.Nagarajan ,P.Sunitha, V.Nagarajan and V.Seethalekshmi (IJCSIS) International Journal of Computer Science and Information Security, Vol. 7, No. 1, 2010.

Use the compress button to bring up the Wavelet Packet 2-D Compression window. Select the remove near zero option from the Select thresholding method. Threshold of the image is 4.995. [3],[6].

Notice that the default threshold (1.5 higher degree) provides about 43.61% compression while retaining virtually all the energy of the original image. Depending on criteria, it may be worthwhile experimenting with more aggressive thresholds to achieve a of compression. This can be considered a precompression step in a broader compression system. Peak Signal to Noise Ratio ( PSNR) and Mean Square Error (MSE) are used to compare the squared error between the original image and the reconstructed image. There is an inverse relationship between PSNR and MSE. Higher PSNR value indicates the higher quality of the image.

PSNR of the compressed image is 28.02, which means, 28% of the noise removed from the image. VII. CONCLUSION The results show that, one of the main decomposition approach based on singular value decomposition for adaptive noise is de-noising. Experimental results are proposed for performance of PSNR on visual effect in color images, even in presence of high ratio of noise. High PSNR value reveals good image quality and less error introduced in to the image. In case of loss less compression PSNR will be high. Further the work can be extended to 3D framework image denoesing and image restoration.

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