International Journal of Advancements in Research & Technology, Volume 2, Issue 8, August-2013 244
ISSN 2278-7763
Copyright © 2013 SciResPub. IJOART
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
T
(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
LL
T
= I and RR
T
= I
In matrix form equation (1) can be written as
σ
1
0 0……… 0 r
1
0 σ
2
0 0 ………….0 r
2
G = [l
1
, l
2
,………,l
m
] 0 0 σ
3
0 0 0 …… .
. . .
. σ
r
.
0 0 ……………. σ
N
r
n
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