Abstract
The purpose of this paper is to study an important application of Singular Value Decomposition (SVD) to image processing. The idea is that by using the smaller number of vectors, one can reconstruct an image that is closer to the original. The clarity of the image depends on how many singular values are used to reconstruct it. In this paper, SVD was applied to the image and also using the Matlab software we developed the code. We also demonstrated how the SVD is used to minimize the size needed to store an image. Keywords: Singular value decomposition, image compression, image processing.
Highlights
The singular value decomposition (SVD), one of the most useful tools of linear algebra, is a factorization and approximation technique which effectively reduces any matrix into a smaller invertible and square matrix
As we see the 10th Iteration the image contain the 100 entries, form the 30th Iteration we get the image near to original image, and form the 70th Iteration i.e. A 70x70 matrix, with 4900 entries is significantly reduced the original image of size 497x498 matrix, with 247506 entries
We focus on Error in the output image; we take the mean square error (Bakwad et al, 2009), between the original image and noisy image that is compressed image
Summary
The singular value decomposition (SVD), one of the most useful tools of linear algebra, is a factorization and approximation technique which effectively reduces any matrix into a smaller invertible and square matrix. One special features of SVD is that it can be performed on any real m× n matrix. It factors A into three matrices U, S, V, such that A = USV T , 1998) where, U and V are orthogonal matrices and S is a diagonal matrix. Singular value decomposition Given a m× n matrix A, there exists decomposition (Ogden & Huff, 1997), such that A = USV T , where U and. V are orthogonal matrices and S is a diagonal matrix with nonnegative diagonal entries in decreasing order. The diagonal entries of S are the positive square roots of the eigen values of AAT and are called the singular values of.
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