Image Compression with Singular Value Decomposition &...

International Conference on Advancement in Engineering, Applied Science and Management (ICAEASM-2017)

at Centre for Development of Advanced Computing (C-DAC), Juhu, Mumbai, Maharashtra (India)

on 18th June 2017 ISBN: 978-81-934083-4-6

376 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee

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, gray-

scale 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.

1. INTRODUCTION

Compression of images is an active field of science.

Many works has been done in this domain [1] [2].

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 [6] [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 [1] [3]

[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 [4]. 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.



on 18th June 2017 ISBN: 978-81-934083-4-6


2. OBJECTIVE OF THE WORK

The objective of this work is to apply SVD to mid-

level 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=USVT. 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.

3. 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 [4] [5]. 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 𝑚 × 𝑛 matrix 𝐴 with rank 𝑟 > 0

can be factored as 𝐴 = 𝑈 ∑𝑉𝑇 with 𝑈 an 𝑚 × 𝑟

matrix with orthogonal columns, ∑ =𝑑𝑖𝑎𝑔(𝜎1, 𝜎2, 𝜎3,⋯ , 𝜎𝑟) and 𝑉𝑇 an 𝑟 × 𝑛 matrix

with orthogonal rows. This directly related to the

spectral theorem which states that if B is a

symmetric matrix (𝐵𝑇 = 𝐵) then we can write 𝐵 =𝑈 ⋀𝑈𝑇 where ⋀ 𝑖𝑠 a diagonal matrix of eigenvalues

and U is an orthonormal matrix of eigenvectors.

The relationship can be found from below:

𝐴𝑇𝐴 = 𝑉∑𝑇𝑈𝑇𝑈∑𝑉𝑇 = 𝑉∑2𝑉𝑇

𝐴𝐴𝑇 = 𝑈∑𝑉𝑇𝑉∑𝑈𝑇 = 𝑈∑2𝑈𝑇

These are both spectral decompositions, hence the 𝜎𝑖

are the positive square roots of the eigenvalues of

𝐴𝑇𝐴. In the SVD, the matrices are rearranged so that

𝜎1 ≥ 𝜎2 ≥ ⋯ ≥ 𝜎𝑛. Reducing the SVD we can write

an 𝑛 × 𝑛 invertible matrix A as:

𝐴 = 𝑈∑𝑉𝑇 = (𝑢1, 𝑢2, ⋯ , 𝑢𝑛) [𝜎1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝜎𝑛

] [𝑉1

𝑇

⋮𝑉𝑛

𝑇]

= 𝑢1𝜎1𝑣1𝑇 + 𝑢2𝜎2𝑣2

𝑇 + ⋯ + 𝑢𝑛𝜎𝑛𝑣𝑛𝑇 (3)

i.e. the 𝑚 × 𝑛 matrix A can be written as the sum of

rank-one matrices.

𝐴 = ∑ 𝜎𝑖𝑢𝑖𝑣𝑖𝑇𝑟

𝑖=1 , (4)

where 𝑢𝑖 𝑎𝑛𝑑 𝑣𝑖 are the 𝑖𝑡ℎ columns of U and V,

respectively.

We want to approximate the 𝑚 × 𝑛 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 ≤ 𝑚 𝑜𝑟 𝑟 ≤ 𝑛 .

𝐴 = 𝑢1𝜎1𝑣1𝑇 + 𝑢2𝜎2𝑣2

𝑇 + ⋯+ 𝑢𝑟𝜎𝑟𝑣𝑟𝑇 + ⋯ (5)

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:

𝐴 = 𝑢1𝜎1𝑣1𝑇 + 𝑢2𝜎2𝑣2

𝑇 + ⋯+ 𝑢𝑟𝜎𝑟𝑣𝑟𝑇 (6)

One way to compress the image A is to approximate

A by a matrix of smaller rank. If 𝑘 < 𝑟 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:

𝐴 ≈ 𝐴𝑘 = 𝑢1𝜎1𝑣1𝑇 + 𝑢2𝜎2𝑣2

𝑇 + ⋯+ 𝑢𝑘𝜎𝑘𝑣𝑘𝑇 7)

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 𝐴𝑘 will be 𝑘(𝑚 + 𝑛 + 1).

B. Low rank approximation in SVD

If we consider a matrix ∈ ℝ𝑚×𝑛 , with SVD given as

in the theorem:

𝐴 = 𝑈�̃�𝑉𝑇 (8)



on 18th June 2017 ISBN: 978-81-934083-4-6


𝑆 = 𝑑𝑖𝑎𝑔(𝜎1, 𝜎2,⋯ , 𝜎𝑟, 0,⋯ ,0)

, where the singular values are ordered in decreasing

order,

𝜎1 ≥ 𝜎2 ≥ ⋯ ≥ 𝜎𝑟 > 0 .

A best 𝑘 − 𝑟𝑎𝑛𝑘 approxmination �̃�𝑘 is given by

zeroing out the (𝑟 − 𝑘) trailing singular values of A,

that is

�̃�𝑘 = 𝑈�̃�𝑘𝑉𝑇,�̃�𝑘 = 𝑑𝑖𝑎𝑔(𝜎1, 𝜎2,⋯⋯ , 𝜎𝑘 , 0, …… ,0)

(9)

C. Image Compression and measures with SVD:

Total storage for 𝐴𝑘 will be 𝑘(𝑚 + 𝑛 + 1), where

𝑚 × 𝑛 is the size of the original image.

The integer 𝑘 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 (𝐶𝑅) =𝑚×𝑛

𝑘×(𝑚+𝑛+1) (10)

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:

‖𝐴 − �̃�𝑘‖𝐹

= √𝜎𝑘+12 + ⋯+ 𝜎𝑟

2 (11)

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 [3].

𝑎𝑖𝑗 = ∑ 𝑐𝑖𝑘𝑒 �̂�2𝜋𝑗𝑘 𝑚⁄𝑘 (12)

The connection with SVD can be illustrated by

normalizing the vector 𝑒 �̂�2𝜋𝑗𝑘 𝑚⁄ and by naming it

𝑣𝑘′ .

𝑎𝑖𝑗 = ∑ 𝑐𝑖𝑘𝑣𝑘′

𝑘 = ∑ 𝑢𝑖𝑘′ 𝑠𝑘

′ 𝑣𝑗𝑘′

𝑘 (13)

which generates the main equation 𝐴′ = 𝑈′𝑆 ′𝑉 ′𝑇,

similar to Eq. (1).

E. Two-dimensional Correlation:

Correlation is deployed in any application to find the

amount of similarity between two signals (1-D or 2-

D).

In practice, correlation between 𝑓(𝑥, 𝑦) and ℎ(𝑥, 𝑦)

can be written as [8],

𝑓(𝑥, 𝑦) ⊛ ℎ(𝑥, 𝑦) = ∑ ∑ 𝑓∗(𝑚, 𝑛)ℎ(𝑥 + 𝑚, 𝑦 +𝑁−1𝑛=0

𝑀−1𝑚=0

𝑛) (14)

We have used 2-D Correlation to measure the

similarity between the original image and the

approximated images reconstructed with different

numbers of Singular values. i.e.

𝐴𝑘(𝑥, 𝑦) ⊛ 𝐴(𝑥, 𝑦) = ∑ ∑ 𝐴𝑘∗ (𝑚, 𝑛)𝐴(𝑥 +𝑁−1

𝑛=0𝑀−1𝑚=0

𝑚, 𝑦 + 𝑛) (15)

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 𝑘.

F. Full Width Half Maxima (FWHM):

Full width at half maximum (FWHM) 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.

4. 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.

Image of Lena, Barbara, Cameraman, Baboon etc.

[9]

Fig. 1. Full Width Half Maxima



on 18th June 2017 ISBN: 978-81-934083-4-6


0 5 10 15 20 25 30 35 40 45 50

10-3

10-2

10-1

100

Relative Error

Number of Singular Values

Rela

tive E

rror

valu

e

0 5 10 15 20 25

10-1

100

Relative Error


Rel

ativ

e E

rror

val

ue

0 5 10 15 20 25

10-1

100

Relative Error


Rela

tive E

rror

valu

e

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.2(b), 3(b), 4(b), 5(b), 6(b) we have shown the

relative errors i.e. ‖𝐴−�̃�𝑘‖𝐹

‖𝐴‖𝐹 w.r.t. the increasing

number of diagonal elements i.e. singular

values(Eq.(12)), where (from Eq. (9))

𝐴 = (𝑢1, ⋯ , 𝑢𝑘, 0,⋯ , 𝑢𝑛)

[ 𝜎1 ⋯ ⋯ ⋯ 00⋮⋮

⋱⋮⋮

⋯ ⋯ 0𝜎𝑘 ⋯ ⋮⋮ ⋱ ⋮

0 ⋯ ⋯ ⋯ 0]

[ 𝑣1

𝑇

⋮𝑣𝑘

𝑇

⋮𝑣𝑛

𝑇]

≈ 𝐴𝑘 = 𝑈𝑘∑𝑘𝑉𝑘𝑇

𝐴𝑘 = 𝑢1𝜎1𝑣1𝑇 + 𝑢2𝜎2𝑣2

𝑇 + ⋯ + 𝑢𝑘𝜎𝑘𝑣𝑘𝑇 (16)

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. 2. (a)

Image of

Lena

(Original

Image) Fig. 2. (b)Relative error w.r.t.

different numbers of singular values

Fig. 3. (a)

Image of

Barbara

Fig. 3. (b)Relative error w.r.t.


Fig. 4. (a)

Image of

Cameraman

Fig. 4. (b) Relative error w.r.t.


Fig. 5. (a)

Face Profile

Image from

UGC-DDMC

FaceDB Fig. 5. (b)Relative error w.r.t.


Fig. 6. (a)

Image of

Baboon.



0 5 10 15 20 25

10-1

100

Relative Error


Rela

tive E

rror

valu

e

0 5 10 15 20 25

10-1

100

Relative Error


Rela

tive E

rror

valu

e



on 18th June 2017 ISBN: 978-81-934083-4-6


Fig. 7. (a)

Image of

Binary ‘S’



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.

2(c1,c2,c3,c4), 3(c1,c2,c3,c4), 4(c1,c2,c3,c4),

5(c1,c2,c3,c4),6(c1,c2,c3,c4),7(c1,c2,c3,c4) we can

see the mesh plot of 2-D correlation between original

image and images approximated with respectively

one, two, three and four SVs. 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.2(d), 3(d), 4(d), 5(d), 6(d), 7(d)) and Y-axis

(Fig.2(e), 3(e), 4(e), 5(e), 6(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 magnitudes of

spatial frequencies fx (Fig.2.(g1,g2), 3.(g1,g2),

4.(g1,g2), 5.(g1,g2), 6.(g1,g2), 7.(g1,g2)) and fy

(Fig.2.(h1,h2), 3.(h1,h2), 4.(h1,h2), 5.(h1,h2),

6.(h1,h2), 7.(h1,h2)) of the approximated images

with respectively one, two, three and four SVs. We

noticed among them there are minimal changes or no

change in some cases.

Fig.2.(C1) Mesh plot of

2-D Cross-correlation

between Original Image

and Images constructed

with one SV.

Fig. 3.(C1) Mesh plot of




with one SV.

Fig.2.(C2) Mesh plot with

two SVs.

Fig. 3.(C2) Mesh plot with

two SVs.


three SVs.


three SVs.


four SVs.


four SVs.

0 10 20 30 40 50 60 70 80 9010

-15

10-10

10-5

100

Relative Error


Rela

tive E

rror

valu

e



on 18th June 2017 ISBN: 978-81-934083-4-6


1 2 3 4 5 6 7 8 9 10

101.7

101.8

FWHM along X axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 10

102.1

102.2

102.3

102.4

102.5

FWHM along Y axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 1010

1

102

103

FWHM along X axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 1010

-1

100

101

correlation pick


pic

k m

agnitude

1 2 3 4 5 6 7 8 9 1010

-1

100

101

correlation pick


pic

k m

agnitude

Fig.2.(d) Measurement of

FWHM of Correlation

matrix along X-axis.


FWHM of Correlation


Fig.2.(e) Measurement of

FWHM of Correlation

matrix along Y-axis.


FWHM of Correlation


Fig.2.(f) Magnitude of

correlation picks with

different singular values.




Fig.2.(g1) Magnitude of fx

(spatial frequency of A1.



Fig.2.(g2) Magnitude of

fx (spatial frequency of

A4.



Fig.2.(h1) Magnitude of fy












with one SV.

Fig.5.(C1) Mesh plot of 2-

D Cross-correlation



with one SV.

1 2 3 4 5 6 7 8 9 10

102.1

102.2

102.3

102.4

FWHM along Y axis


valu

es o

f F

WH

M

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

16fx spatial frequency

fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

16fy spatial frequency

fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude



on 18th June 2017 ISBN: 978-81-934083-4-6


1 2 3 4 5 6 7 8 9 1010

1

102

103

FWHM along X axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 1010

1

102

103

FWHM along Y axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 10

10-4.00196e-14

10-2.00098e-14

100

101.99616e-14

104.00196e-14

correlation pick


pic

k m

agnitude

1 2 3 4 5 6 7 8 9 10

102.2

102.3

FWHM along X axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 10

102.1

102.2

102.3

102.4

102.5

FWHM along Y axis


valu

es o

f F

WH

M


two SVs.


two SVs.


three SVs.


three SVs.


four SVs.


four SVs.


FWHM of Correlation



FWHM of Correlation



FWHM of Correlation



FWHM of Correlation
















1 2 3 4 5 6 7 8 9 10

10-4.00196e-14

10-2.00098e-14

100

101.99616e-14

104.00196e-14

correlation pick


pic

k m

agnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude



on 18th June 2017 ISBN: 978-81-934083-4-6


1 2 3 4 5 6 7 8 9 1010

1

102

103

FWHM along Y axis


valu

es o

f F

WH

M













with one SV.

Fig.7.(C1) Mesh plot of 2-

D Cross-correlation



with one SV.


two SVs.


two SVs.


three SVs.


three SVs.


four SVs.


four SVs.


FWHM of Correlation



FWHM of Correlation



FWHM of Correlation



FWHM of Correlation


-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

1

Correlation with 1 singular value

020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

1

Correlation with 2 singular values

020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

1


020

4060

80100

0

50

1000

0.2

0.4

0.6

0.8

1


1 2 3 4 5 6 7 8 9 10

101.8

101.9

FWHM along X axis


valu

es o

f F

WH

M

1 2 3 4 5 6 7 8 9 1010

1.5

101.6

101.7

FWHM along X axis


valu

es o

f F

WH

M

2 3 4 5 6 7 8 9 10

101.39

101.41

101.43

101.45

101.47

FWHM along Y axis


valu

es o

f F

WH

M



on 18th June 2017 ISBN: 978-81-934083-4-6


1 2 3 4 5 6 7 8 9 10

10-4.00196e-14

10-2.00098e-14

100

101.99616e-14

104.00196e-14

correlation pick


pic

k m

agnitude























5. 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 [1] L. Cao, “Singular Value Decomposition Applied to

Digital Image Processing,” Division of Computing

Studies, Arizona State University Polytechnic

Campus, pp. 1-15, 2006

[2] 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.91-109, 2003.

[3] L. Zhao, W. Hu, L. Cui, “Face Recognition Feature

Comparison Based SVD and FFT,” Journal of Signal

and Information Processing,vol.3, pp. 259-262, May

2012.

[4] I.C.F.Ipsen, "Numerical Matrix Analysis: Linear

systems and Least Squares," SIAM, Philadelphia,

2009

[5] G. Strang, "Introduction to Linear Algebra,",

Wellesley-Cambridge Press, 1993

[6] G. Zeng, "Face Recognition with Singular Value

Decomposition," CISSE Proceeding, 2006

[7] O. Bryt, M. Elad, "Compression of facial images

using the K-SVD algorithm," Journal of Visual

Communication & Image Presentation, Elsevier, pp.

270-282, March 2008.

[8] A. K. Jain, “Fundamentals of Digital Image

Processing,” PHI Learning Pvt. Ltd., 2013

[9] Standard images used for experiment. (www.image

processingplace.com/root_files_V3/image_databases.

htm.)

1 2 3 4 5 6 7 8 9 1010

-1

100

101

correlation pick


pic

k m

agnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10


fx Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14


fy Cycle/Pixel

magnitude

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

5

10


fy Cycle/Pixel

magnitude

Image Compression with Singular Value Decomposition &...

Documents

Transcript of Image Compression with Singular Value Decomposition &...