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International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013 244

ISSN 2278-7763

Image Compression using Singular Value

Decomposition

Miss Samruddhi Kahu Ms. Reena Rahate

Associate Engineer Professor, Department of Electronics and Communication

Marvell Technologies Shri Ramdeobaba College of Engg. and Management

Pune, India. Nagpur, India.

samruddhikahu@gmail.com reenapalashn@rediffmail.com

Abstract: The Singular Value Decomposition expresses image

data in terms of number of eigen vectors depending upon the

dimension of an image. The psycho visual redundancies in an

image are used for compression. Thus an image can be

compressed without affecting the image quality. This paper

presents one such image compression technique called as SVD.

Basic mathematics of SVD is dealt with in detail and results of

applying SVD on an image are also discussed. The MSE and

compression ratio are used as thresholding, parameters for

reconstruction. SVD is applied on variety of images for

experimentation. The work is concentrated to reduce the number

of eigen values required to reconstruct an image.

Keywords – SVD, Image Compression techniques, Image

Processing

I. INTRODUCTION

With the advancement in technology, multimedia content

of digital information is increasing day by day. Which mainly

comprises of images either pictures or video frames. Hence

storage and transmission of these images requires a large

memory space and bandwidth. The solution to this problem is

to reduce the storage space required for these images which

can be done by compressing the image while maintaining

acceptable image quality. Many methods are available for

compression of still images. But the most widely used image

compression technique today is JPEG (Joint Photographic

Experts Group) which uses DCT (Discrete Cosine Transform)

for compression of images. In this paper we are discussing a

image compression technique called SVD (Singular Valued

Decomposition).

Even though DCT gives high energy compaction as

compared to SVD which gives optimal energy compaction,

SVD performs better than DCT in case of images having high

standard deviation (i.e. higher pixel quality).

SVD is a linear matrix transformation used for

compressing images. Using SVD an image matrix is

represented as the product of three matrices U, S, and V where

S is a diagonal matrix whose diagonal entries are singular

values of matrix A. The image A can also be represented by

using less number of singular values, thus, presenting

necessary features of an image while compressing it. The

compressed image requires less storage space as compared to

the original image. To choose the value of k i.e. number of

Eigen values for compression and reconstruction of the image

is an important decision for acceptable reconstruction. It

varies with application and in this work compression ratio is

used to select the number of Eigen values out of the

maximum. It is observed that if the value of k chosen is equal

to the rank of the image, the reconstructed image is closer to

the original image. As the value of k decreases from the rank

image quality degrades. Second observation is that as the

compression ratio is high image quality is poorer and if

compression ratio is low, image with superior quality can be

reconstructed but with less compression. Therefore

compression ratio and image quality is required to select

appropriately.

Section I presents literature survey and brief outline

of the paper. Section II presents the SVD computations on

images. Section III presents the experimentation and the

proposed procedure to estimate the percentage of eigenvectors

required to reconstruct the original image using the SVD and

PSNR. Section IV presents the results and the discussions.

Finally Section V presents the conclusion.

II. B

ASIC MATHEMATICS

Singular Value Decomposition(SVD) is a linear

image matrix transformation in which an image matrix G is

decomposed into 3 component matrices L, D & R such that

G = LDR

(1)

where G is a m × n matrix

D is a m × n diagonal matrix in which the entries

along the diagonal of D are singular values of G. Singular

value of a matrix is calculated by taking square root of its

eigen value.

L is a m × m matrix containing left singular vectors

of G and

R is a n × n matrix containing right singular vectors

of G.

L and R are orthonormal matrices which means

= I and RR

= I

In matrix form equation (1) can be written as

0 0……… 0 r

0 σ

0 0 ………….0 r

G = [l

, l

,………,l

] 0 0 σ

0 0 0 …… .

. . .

. σ

0 0 ……………. σ

IJOART

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