Post on 13-Sep-2020
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.
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
377 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
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)
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
378 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
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
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
379 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
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
Number of Singular Values
Rel
ativ
e E
rror
val
ue
0 5 10 15 20 25
10-1
100
Relative Error
Number of Singular Values
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.
different numbers of singular values
Fig. 4. (a)
Image of
Cameraman
Fig. 4. (b) Relative error w.r.t.
different numbers of singular values
Fig. 5. (a)
Face Profile
Image from
UGC-DDMC
FaceDB Fig. 5. (b)Relative error w.r.t.
different numbers of singular values
Fig. 6. (a)
Image of
Baboon.
Fig. 6. (b)Relative error w.r.t.
different numbers of singular values
0 5 10 15 20 25
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
Number of Singular Values
Rela
tive E
rror
valu
e
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
380 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
Fig. 7. (a)
Image of
Binary ‘S’
Fig. 7. (b)Relative error w.r.t.
different numbers of singular values
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
2-D Cross-correlation
between Original Image
and Images constructed
with one SV.
Fig.2.(C2) Mesh plot with
two SVs.
Fig. 3.(C2) Mesh plot with
two SVs.
Fig.2.(C3) Mesh plot with
three SVs.
Fig. 3.(C3) Mesh plot with
three SVs.
Fig.2.(C4) Mesh plot with
four SVs.
Fig. 3.(C4) Mesh plot with
four SVs.
0 10 20 30 40 50 60 70 80 9010
-15
10-10
10-5
100
Relative Error
Number of Singular Values
Rela
tive E
rror
valu
e
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
381 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
1 2 3 4 5 6 7 8 9 10
101.7
101.8
FWHM along X axis
Number of Singular Values
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
Number of Singular Values
valu
es o
f F
WH
M
1 2 3 4 5 6 7 8 9 1010
1
102
103
FWHM along X axis
Number of Singular Values
valu
es o
f F
WH
M
1 2 3 4 5 6 7 8 9 1010
-1
100
101
correlation pick
Number of Singular Values
pic
k m
agnitude
1 2 3 4 5 6 7 8 9 1010
-1
100
101
correlation pick
Number of Singular Values
pic
k m
agnitude
Fig.2.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.3.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.2.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
Fig.3.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
Fig.2.(f) Magnitude of
correlation picks with
different singular values.
Fig.3.(f) Magnitude of
correlation picks with
different singular values.
Fig.2.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.3.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.2.(g2) Magnitude of
fx (spatial frequency of
A4.
Fig.3.(g2) Magnitude of fx
(spatial frequency of A4.
Fig.2.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.3.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.2.(h2) Magnitude of fy
(spatial frequency of A4.
Fig.3.(h2) Magnitude of fy
(spatial frequency of A4.
Fig.4.(C1) Mesh plot of
2-D Cross-correlation
between Original Image
and Images constructed
with one SV.
Fig.5.(C1) Mesh plot of 2-
D Cross-correlation
between Original Image
and Images constructed
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
Number of Singular Values
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
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4
6
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14
16fx spatial frequency
fx Cycle/Pixel
magnitude
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16fx spatial frequency
fx Cycle/Pixel
magnitude
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fx Cycle/Pixel
magnitude
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fx Cycle/Pixel
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fy Cycle/Pixel
magnitude
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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
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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
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16fy spatial frequency
fy Cycle/Pixel
magnitude
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
382 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
1 2 3 4 5 6 7 8 9 1010
1
102
103
FWHM along X axis
Number of Singular Values
valu
es o
f F
WH
M
1 2 3 4 5 6 7 8 9 1010
1
102
103
FWHM along Y axis
Number of Singular Values
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
Number of Singular Values
pic
k m
agnitude
1 2 3 4 5 6 7 8 9 10
102.2
102.3
FWHM along X axis
Number of Singular Values
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
Number of Singular Values
valu
es o
f F
WH
M
Fig.4.(C2) Mesh plot with
two SVs.
Fig.5.(C2) Mesh plot with
two SVs.
Fig.4.(C3) Mesh plot with
three SVs.
Fig.5.(C3) Mesh plot with
three SVs.
Fig.4.(C4) Mesh plot with
four SVs.
Fig.5.(C4) Mesh plot with
four SVs.
Fig.4.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.5.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.4.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
Fig.5.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
Fig.4.(f) Magnitude of
correlation picks with
different singular values.
Fig.5.(f) Magnitude of
correlation picks with
different singular values.
Fig.4.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.5.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.4.(g2) Magnitude of fx
(spatial frequency of A4.
Fig.5.(g2) Magnitude of fx
(spatial frequency of A4.
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
Number of Singular Values
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
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
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
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
16fx spatial frequency
fx Cycle/Pixel
magnitude
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
383 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
1 2 3 4 5 6 7 8 9 1010
1
102
103
FWHM along Y axis
Number of Singular Values
valu
es o
f F
WH
M
Fig.4.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.5.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.4.(h2) Magnitude of fy
(spatial frequency of A4.
Fig.5.(h2) Magnitude of fy
(spatial frequency of A4.
Fig.6.(C1) Mesh plot of
2-D Cross-correlation
between Original Image
and Images constructed
with one SV.
Fig.7.(C1) Mesh plot of 2-
D Cross-correlation
between Original Image
and Images constructed
with one SV.
Fig.6.(C2) Mesh plot with
two SVs.
Fig.7.(C2) Mesh plot with
two SVs.
Fig.6.(C3) Mesh plot with
three SVs.
Fig.7.(C3) Mesh plot with
three SVs.
Fig.6.(C4) Mesh plot with
four SVs.
Fig.7.(C4) Mesh plot with
four SVs.
Fig.6.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.7.(d) Measurement of
FWHM of Correlation
matrix along X-axis.
Fig.6.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
Fig.7.(e) Measurement of
FWHM of Correlation
matrix along Y-axis.
-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
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
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
16fy spatial frequency
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
Correlation with 3 singular values
020
4060
80100
0
50
1000
0.2
0.4
0.6
0.8
1
Correlation with 4 singular values
1 2 3 4 5 6 7 8 9 10
101.8
101.9
FWHM along X axis
Number of Singular Values
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
Number of Singular Values
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
Number of Singular Values
valu
es o
f F
WH
M
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
384 Tamojay Deb, Anjan K Ghosh, Anjan Mukherjee
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
Number of Singular Values
pic
k m
agnitude
Fig.6.(f) Magnitude of
correlation picks with
different singular values.
Fig.7.(f) Magnitude of
correlation picks with
different singular values.
Fig.6.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.7.(g1) Magnitude of fx
(spatial frequency of A1.
Fig.6.(g2) Magnitude of fx
(spatial frequency of A4.
Fig.7.(g2) Magnitude of fx
(spatial frequency of A4.
Fig.6.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.7.(h1) Magnitude of fy
(spatial frequency of A1.
Fig.6.(h2) Magnitude of fy
(spatial frequency of A4.
Fig.7.(h2) Magnitude of fy
(spatial frequency of A4.
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
Number of Singular Values
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
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
5
10
15fx 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
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
5
10
15fx 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
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
5
10
15fy 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
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
5
10
15fy spatial frequency
fy Cycle/Pixel
magnitude