ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
Designing Quantization Table for Hadamard Transform based on Human
Visual System for Image Compression
K.Veeraswamy* , S. Srinivaskumar#, B.N.Chatterji§
*Research scholar, ECE Dept., JNTUCE, Kakinada, A.P, India.
#Professor ,ECE Dept., JNTUCE, Kakinada, A.P, India.
§
Former Professor, E&ECE Dept., IIT, Kharagpur,W.B, India.
[kilarivs@yahoo.com, samay_ssk2@yahoo.com,bnchatterji@gmail.com]
the transform domain [1]. Wavelet Transform (WT) [2,3]
and Contourlet Transform (CT) [4] are used as a method
of information coding. The ISO/CCITT Joint
Photographic Experts Group (JPEG-2000) has selected
the WT for its baseline coding technique [5]. The
Discrete Cosine Transform has attracted widespread
interest as a method of information coding. JPEG has
selected the DCT for its baseline coding technique [6]. In
this work, Hadamard transform is used for image
compression. The elements of the basis vectors of the
Hadamard Transform take only the binary values ± 1
and are, therefore, well suited for digital hardware
implementations of image processing algorithms.
Hadamard Transform offers a significant advantage in
terms of a shorter processing time as the processing
involves simpler integer manipulation (compared to
floating point processing with DCT) and the ease of
hardware implementation than many common transform
techniques. So it is computationally less expensive than
many other orthogonal transforms. Integer transforms are
very much essential in video compression. Hadamard
Transform is used for video compression [7]. The
quantization table is central to the compression. Several
approaches have been tried in order to design
quantization tables for particular distortion or rate
specifications. The most common of these is to use a
default table and scale it up or down by a scalar
multiplier to vary the quality and compression. Methods
for determining the quantization table are usually based
on rate-distortion theory. These methods do achieve
better performance than the JPEG default quantization
table [8, 9]. However, the quantization tables based on
rate distortion theory methods are image-dependent and
the complexity of the encoder is rather high. In this work,
quantization table is designed based on the human visual
system (HVS) [10]. HVS model is easy to adapt to the
specified resolution for viewing. The mind does not
perceive everything the eye sees. This knowledge is used
to design new quantization table for Hadamard
Transform.
Abstract
This paper discusses lossy image compression using
Hadamard Transform (HT). Quantization table plays
significant role in image compression (lossy) that
improves the compression ratio without sacrificing visual
quality. In this work, Human Visual system (HVS) is
considered to derive the quantization table, which is
applicable for Hadamard Transform. By incorporating
the human visual system with the uniform quantizer, a
perceptual quantization table is derived. This
quantization table is easy to adapt to the specified
resolution for viewing. Results show that this
quantization table is good in terms of improving peak
signal to noise ratio (PSNR), Normalized Cross
correlation (NCC) and to reduce blocking artifacts. This
work is extended to test the robustness of watermarking
against various attacks.
Keywords: Image compression, Hadamard Transform,
human visual system, quantization table, watermarking.
1. Introduction
There are many applications requiring image
compression, such as multimedia, internet, satellite
imaging, remote sensing, and preservation of art work,
etc. Decades of research in this area has produced a
number of image compression algorithms. Most of the
effort expended over the past decades on image
compression has been directed towards the application
and analysis of orthogonal transforms. The orthogonal
transform exhibits a number of properties that make it
useful. First, it generally confirms to a parseval constraint,
in that the energy present in the image is same as that
displayed in the image’s transform. Second, the
transform coefficients bear no resemblance to the image
and many of the coefficients are small and, if discarded,
will provide image compression with nominal
degradation of the quality of reconstructed image. Third,
certain sub scenes of interest, such as some targets or
particular textures, have easily recognized signatures in
31
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
This paper is organized as follows. Human visual system
is discussed in section 2. The 2D-Hadamard Transform is
discussed in section 3. The proposed method is presented
in section 4. Experimental results are given in section 5.
Concluding remarks are given in section 6.
f (u ) =
The human visual system has been investigated by
several researchers [11, 12]. Moreover, the simplicity and
visual sensitivity and selectivity to model and improve
perceived image quality are the requirements for the
design of HVS. The HVS is based on the psychophysical
process that relates psychological phenomena (contrast
and brightness etc.) to physical phenomena (light
sensitivity, spatial frequency and wavelength etc.). The
HVS is complicated, it is a nonlinear and spatial varying
system. Putting its multiple characteristics into single
equation, especially one that is linear, is not an easy task.
Mannos and Sakrison’s work [13] may be first
breakthrough to incorporate the HVS in image coding.
HVS as a nonlinear point transformation followed by the
modulation transfer function (MTF) is given by
(
d
)
π
f (u, v) =
f (u) + f (v)
2
1
180arcsin
1 + dis2
where, u , v = 1,2...N
2
(4)
Finally, to account for variations in visual MTF as a
function of viewing angle, θ, these frequencies are
normalized by an angular-dependent function,
s (θ (u , v )) , such that
∧
f (u, v ) =
f (u, v )
s (θ (u , v ))
(5)
where, s (θ (u , v )) is given by Daly as
(1)
where, f is the radial frequency in cycles/degree of the
visual angel subtended and a, b, c and d are constants.
HVS model proposed by Daly [14] is applied for
generating the quantization table. This HVS model is a
modified version of Mannos and Sakrison’s work with
a=2.2, b=0.192, c=0.114 and d=1.1 respectively. The
MTF of HVS has been successfully applied to the
optimal image half toning and image compression.
s (θ (u, v )) =
with
1−ω
1+ ω
cos(4θ (u, v )) +
2
2
(6)
ω being a symmetry parameter, and
⎛ f (u ) ⎞
⎟⎟
⎝ f (v ) ⎠
θ (u , v ) = arctan⎜⎜
The MTF as reported by Daly [14] is
⎧ ⎛
⎞
∧
⎪ 2 . 2 ⎜ 0 . 192 + 0 . 114 f (u , v )⎟
⎟⎟
⎪ ⎜⎜
⎠
⎪ ⎝
1 .1
∧
⎪
⎞
⎛ ⎛
H (u , v ) = ⎨ . exp ⎜ − ⎜ 0 . 114 f (u , v )⎞⎟ ⎟
⎜ ⎝
⎪
⎠ ⎟⎠
⎝
⎪
∧
if f (u , v ) > f ;
⎪
⎪
1 .0
otherwise
⎩
(3)
Converting these to radial frequencies, and scaling the
result to cycles/visual degree for a viewing distance (dis)
in millimeters gives
2. Human Visual System
H ( f ) = a (b + cf ) exp − (c( f ) )
u −1
v −1
, f (v ) =
ΔN
ΔN
Equation (6) indicates that as
decreases.
ω
(7)
decreases, s (θ (u, v ))
3. 2D-Hadamard Transform
The 2D-HT has been used in image processing and image
compression [15]. Let x represents the original image
and [T] the transformed image. The 2D-Hadamard
transform is given by [16]
[]
(2)
[T ] = H n [x]H n
N
(8)
where H n represents an NxN Hadamard matrix, N=2n,
∧
f (u , v ) is radial spatial frequency in cycles/degree and
f is the frequency of 8 cycles/degree at which the
n=1,2,3,…, with element values either +1 or -1. The
Hadamard transform H n is real, symmetric, and
exponential peak. To implement this, it is necessary to
convert
discrete
horizontal
and
vertical
frequencies, { f (u ), f (v )} into radial visual frequencies.
orthogonal [17] that is
For a symmetric printing grid, the horizontal and vertical
discrete frequencies are periodic and given in terms of
the dot pitch Δ and the number of frequencies N by
The inverse 2D-Hadamard transform is given as
H n = H n* = H nt = H n−1
[x ] = H n [T ]H n
N
32
(9)
(10)
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
Table 1: The human visual frequency weighting matrix
for Hadamard transform
The Hadamard matrix of the order n is generated in terms
of Hadamard matrix of order n-1 using Kronecker
product ‘ ⊗ ’ [18] given by
H1 =
1 ⎛1 1 ⎞
⎜⎜
⎟⎟ , H n = H n −1 ⊗ H1
2 ⎝1 − 1⎠
1.0000 0.6571 1.0000 0.9599 1.0000 0.7684 1.0000 0.8746
0.6571 0.1391 0.4495 0.3393 0.6306 0.1828 0.5558 0.2480
(11)
1.0000 0.4495 0.7617 0.6669 1.0000 0.5196 0.8898 0.5912
0.9599 0.3393 0.6669 0.5419 0.9283 0.3930 0.8192 0.4564
HT matrix has its AC components in a random order. The
processing is performed based on the 8x 8 sub-blocks of
the whole image, the third order HT matrix H3 is used. By
applying (11) H3 becomes:
⎡1
⎢1
⎢
⎢1
⎢
1 ⎢1
H3 =
8 ⎢1
⎢
⎢1
⎢1
⎢
⎣⎢1
1
−1
1
1
1
−1
1
1
1
1
−1
1
1 −1 −1 1 1
−1 −1 1 1 −1
1
1 1 −1 −1
−1 1 −1 −1 1
1 −1 −1 −1 −1
−1
−1
−1
−1
1
−1 −1
1
1
−1
1
1⎤
− 1⎥⎥
− 1⎥
⎥
1⎥
− 1⎥
⎥
1⎥
1⎥
⎥
− 1⎦⎥
1.0000 0.6306 1.0000 0.9283 1.0000 0.7371 1.0000 0.8404
0.7684 0.1828 0.5196 0.3930 0.7371 0.2278 0.6471 0.2948
1.0000 0.5558 0.8898 0.8192 1.0000 0.6471 0.9571 0.7371
0.8746 0.2480 0.5912 0.4564 0.8404 0.2948 0.7371 0.3598
The quantization table is
Q(u , v ) =
(12)
q
H (u , v )
(14)
where, q is the step size of the uniform quantizer.
Therefore, the HVS-based quantization table can be
derived by
QHVS (u, v ) = Round [Q(u , v )]
4. Proposed method
Given Hadamard matrix, the number of sign changes in
each row of the Hadamard Transform matrix is given as
0,7,3,4,1,6,2 and 5 in the rows 1 to 8 respectively. The
number of sign changes in each column of the Hadamard
Transform matrix is given as 0,7,3,4,1,6,2 and 5 in the
columns 1 to 8 respectively. The number of sign changes
is referred to as sequency. The concept of sequency is
analogous to frequency for the Fourier transform.
Therefore,
R = [0 7 3 4 1 6 2 5], C= [0 7 3 4 1 6 2 5]
The horizontal and vertical discrete frequencies in the
Hadamard domain are given in Equation (13)
R(u )
for u =1, 2,….,N
Δ ∗ 2N
C (v )
f (v ) =
for v =1, 2,….,N
Δ ∗ 2N
Table 2: Proposed quantization table,
16 24 16
f (u ) =
(13)
(15)
QHVS (u, v )
17 16 21
16 18
24 115 36 47 25 88
29 65
16 36 21
24 16 31
18 27
17 47 24
30 17 41
20 35
16 25 16
17 16 22
16 19
21 88 31
41 22 70
25 54
18 65 27
35 19 54
22 44
Table 2 shows the proposed luminance quantization table
derived with q=16. Varying levels of image compression
and quality are obtainable through selection of specific
quantization matrices by varying ‘q’. This enables the
user to decide on quality levels ranging form 1 to 100,
where 1 gives the worst quality and highest compression,
while 100 gives the best quality and lowest compression.
Above quantization matrix works well for quality level
50.
The dot pitch ( Δ ) of the high resolution computer
display is about 0.25mm. High resolution computer
display is about 128mm height and 128mm width to
display a 512x512 pixel image. The appropriate viewing
distance is four times of height. Hence, distance is
considered as 512mm. Constant ω is a symmetric
parameter, derived from experiments and set to 0.7[19].
Thus, the human visual frequency weighting matrix
H (u , v ) of (2) is calculated for HT using equations (4)
and (5) as given in Table 1. The human visual frequency
weighting matrix H (u , v ) indicates the perceptual
importance of the transform coefficients. After
multiplying the 64 hadamard coefficients with human
visual frequency weighting matrix H (u , v ) , the
weighted hadamard coefficients contribute the same
perceptual importance to human observers.
The compression method is given as:
1. The input image is first divided into non overlapping
8 x 8 blocks.
2. Each block is transformed into 64 Hadamard
coefficients via 2D-HT.
3. These 64 HT coefficients are then uniformly quantized
by HVS based quantization table and rounded.
The image is reconstructed through decompression, a
process that uses the inverse HT. Nonzero coefficients
are used to reconstruct the original image. Flow charts for
33
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
the proposed method are shown in Figure 1and 2
respectively.
co efficients as follows
AC p = ACa − T to embed bit ‘0’
AC p = ACa + T to embed bit’1’,
Image,
where, ‘p’ and ‘a’ are different locations.
x
5. Experimental results
Experiments are performed on two gray images as given
in Figure 3 [20] to verify the proposed compression
technique.
Calculate the Hadamard
Transform of image (block wise)
Quantize each block using proposed HVS
based quantization table
(a)
Figure 3: (a) Lena
Round of the quantized coefficients
(b)
(b) Peppers
These two images are represented by 8 bits/pixel and
each image size is 512 x 512. The entropy (E) [21] is
calculated as
Image in HT domain in
compressed form
E = −∑ p(e) log 2 p (e)
(16)
e∈s
Figure 1. Flowchart for image compression
p(e) is the
where, s is the set of coefficients and
probability of coefficient (e) . An often used global
objective quality measure is the mean square error (MSE)
defined as
Image in HT domain in
compressed form
MSE =
Multiply each block using proposed
quantization table to obtain coefficients
[
n −1 m −1
1
'
xij − xij
∑∑
(n − 1)(m − 1) i =0 j =0
]
2
where, n is the number of total pixels and
(17)
xij and xij
'
are the pixel values in the original and reconstructed
image. The peak to peak signal to noise ratio (PSNR in
dB) is calculated as
Calculate inverse HT of coefficients
(block wise)
PSNR = 10 log10
Reconstructed image
2552
MSE
(18)
where, the usable gray level values range from 0 to 255.
The other metric used to test the quality of the
reconstructed image is Normalized Cross Correlation.
Normalized Cross Correlation (NCC) is defined as given
in equation (19).
Figure 2. Flowchart for image reconstruction
The human visual frequency weighting matrix for
Hadamard transform indicates that, middle and high
frequency bands in HT are distributed in a random order.
This property increases the reliability of watermark. The
steps of the algorithm for watermarking are as follows.
NCC=
1.Identify two AC coefficients in each
transformed (HT) block of image.
2. Embed the watermark bit in one of the AC
34
−
⎛ − − ⎞⎛ ' ' ⎞
−
⎜ xij x ⎟⎜ xij x ⎟
∑∑
⎠⎝
i
j ⎝
⎠
2
⎡
⎛ ' −' ⎞
⎛ x − x− ⎞ ⎤⎡
⎢∑∑⎜ ij ⎟ ⎥⎢∑∑⎜ xij − x ⎟
⎠ ⎦⎢⎣ i j ⎝
⎠
⎣i j ⎝
2
⎤
⎥
⎥⎦
(19)
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
achieve high performance without increasing any
complexity. Performance comparison of proposed HVSbased Hadamard quantization table with other
quantization tables is shown in Figures 4 and 5
respectively.
−
where, x indicates the mean of the original image
−
'
and x indicates the mean of reconstructed image.
Comparative performance is studied in terms of PSNR
considering the following methods.
1.
2.
3.
Standard quantization matrix as used in JPEG
algorithm is applied to quantize the hadamard
coefficients denoted as Q1.
HVS-based quantization matrix (as applied for
DCT) is used to quantize the Hadamard
coefficients denoted as Q2.
Proposed HVS- based quantization matrix to
quantize the Hadamard coefficients.
The experiments are done on images of Lena and Peppers
and are presented in Table 3 and 4.
Table 3: PSNR results for Lena image using different
quantization tables
Entropy
(Bits/pixel)
PSNR
Q1
Q2
0.3
27.1810
27.9678
Proposed
method
28.0495
0.5
29.2018
30.4706
30.9096
0.7
30.7308
32.3234
32.8423
0.9
32.1647
33.7089
34.3915
1.0
32.7591
34.4589
35.2633
Figure 4: Performance comparison of Lena image
Table 4: PSNR results for Peppers image using different
quantization tables
Entropy
(Bits/pixel)
PSNR
Q1
Q2
0.3
26.5786
27.5926
Proposed
method
27.7241
0.5
28.9478
30.2501
30.5620
0.7
30.5664
32.0616
32.3596
0.9
31.8695
33.3461
33.8205
1.0
32.6403 34.2340 34.3731
The bit rate and the decoded quality are determined
simultaneously by the quantization table, and therefore,
the proposed quantization table has a strong influence on
the whole compression performance. Experimental
results indicate that the proposed method can achieve
better performance in terms of PSNR at the same level of
compression. Proposed HVS based quantization table
Figure 5: Performance comparison of Peppers image
The process of quantization results in both blurring and
blocking artefacts. The effect of blocking artifacts occurs
due to the discontinuity at block boundaries. The
blockiness is estimated as the average differences across
block boundaries. This feature is used to constitute a
quality assessment model to calculate the score [22]. The
score typically has a value between 0 and 10 (10 and 0
35
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
represent more and less quality respectively).
Performance comparison in terms of Normalized Cross
Correlation (NCC) and score is given in Table 5 for Lena
and Peppers at 0.9 bits per pixel.
PSNR (less than 1 dB) than the proposed method. This is
because DCT is high gain transform.
To test the reliabilility of watermarking, binary logo of
size 64x64 and Lena image of size 512 x 512 are
considered. Watermark bits are embedded in the location
AC(2,2) by modifying the contents of AC(2,6).
Threshold value T=8 is considered for experimentation.
Logo and watermarked images are shown in Figure 7.
Table 5: NCC and Score results for Lena and Peppers
using different quantization tables
Lena
Q1
Q2
Proposed
NCC
0.9913
0.9940
0.9949
Score
3.4139
3.8274
4.1027
Peppers
NCC
Q1
0.9936
Q2
0.9954
Proposed
0.9959
Score
3.5986
4.1087
4.4115
(a)
The results of retrieved watermarks with DCT and HT
techniques are given in Table 7.
Table 6: PSNR results for different images using
different quantization tables
Entropy
Q1
Q2
Proposed
Mandrill
Barbara
Zelda
Airplane
Washsat
2.17
1.47
0.89
1.11
0.92
29.0
31.8
35.3
34.2
34.0
30.4
33.4
36.5
36.0
35.0
31.2
34.2
37.1
36.7
35.6
Table 7: Retrieved watermarks and NCC results
The reconstructed images Lena and Peppers using the
proposed HVS-based quantization table are shown in
Figure 6.
(a)
(c)
Figure 7: (a) Logo(64x64) (b)Watermarked Lena image
using DCT (c)Watermarked Lena image using HT
The results of the proposed method with other images are
given in Table 6.
Image
(b)
Attack
64x64
logo
embedded
Salt and Pepper
noise
(Density 0.01)
DCT
PSNR=34.2dB
HT
PSNR=47.1dB
NCC=0.8320
NCC=0.8687
Bit
plane
removal
(4th bit plane =0)
NCC=0.4888
NCC=0.5190
Cropping
(25%)
NCC=0.9909
NCC=0.9932
Histogram
equalization
NCC=0.6896
NCC=0.9277
The watermarked images are attacked by image cropping,
histogram equalization, bit plane removal and noise
attacks. Extracted watermarks are given in Table 7.
Experiments demonstrate that the HT based
watermarking scheme is robust to various attacks than
DCT. PSNR of watermarked image is high in HT based
watermarking scheme.
(b)
Figure 6: (a) Reconstructed Lena image(0.5bpp and
PSNR is 30.90) (b) Reconstructed Peppers image(0.5bpp
and PSNR is 30.56)
The proposed method performs well in all tests. The error
introduced by quantization of a particular Hadamard
coefficient will not be visible if its quantization error is
less than the just noticeable difference. By using JPEG
method, PSNR is 34.9759 dB and 36.1517 dB for
reconstructed Peppers and Lena images respectively (1.0
bpp). Experimental results indicate that JPEG gives better
6. Conclusions
In this paper, a simple approach to the generation of
optimal quantization table based on HVS model is
presented. This quantization table is considered to
quantize the HT coefficients and to obtain the superior
image compression over standard quantization tables
available in the literature. Superiority of this method is
36
ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
observed in terms of PSNR and NCC. It is observed that
the Hadamard transform has more useful middle and high
frequency bands for image watermarking than several
high gain transforms, such as DCT, DFT (Discrete
Fourier Transform) and DST (Discrete Sine Transform).
HT matrix has its AC components in a random order and
is exploited for digital image watermarking. The scheme
of digital image watermarking presented here with HT is
robust compared to DCT based watermarking scheme.
international journal of computer science and
network security, Volume No.6,168-174, Sept 2006.
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Acknowledgements
First author thank Prof B. Chandra Mohan and Sri CH.
Srinivasa rao research scholars at JNTU College of
Engineering, Kakinada for the valuable discussions
related to this work. The authors would like to thank the
reviewers for the review of the paper and valuable
suggestions.
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ICGST-GVIP Journal, Volume 7, Issue 3, November 2007
Biographies
B.N.Chatterji is a former
Professor
in
E&ECE
Department IIT, Kharagpur. He
received B.Tech. and Ph.D.
(Hons.)
from
E&ECE
Department IIT, Kharagpur in
1965 and 1970, respectively. He
has served the Institute under
various
administrative
capabilities
as
Head
of
Department, Dean (Academic), etc. He has chaired many
international and national symposium and conferences
organized in India and abroad, apart from organizing 15
short term courses for Industries and Engineering college
teachers. He has guided 35 Ph.D. scholars. Presently, he
is active in research by guiding three research scholars.
He has published more than 150 papers in reputed
international and national journals apart from authoring
three scientific books. His research interests are low-level
vision, computer vision, image analysis, pattern
recognition and motion analysis.
K.Veeraswamy is currently
working as Associate Professor
in ECE Department, Bapatla
Engineering College, Bapatla,
India. He is working towards
his Ph.D. at JNTU College of
Engineering, Kakinada, India.
He received his M.Tech. from
the same institute. He has nine
years experience of teaching
undergraduate students and post graduate students. His
research interests are in the areas of image compression,
image watermarking and networking protocols.
S.Srinivas Kumar is currently
Professor and HOD in ECE
Department, JNTU College of
Engineering, Kakinada, India.
He received his M.Tech. from
Jawaharlal Nehru Technological University, Hyderabad,
India. He received his Ph.D.
from E&ECE Department IIT,
Kharagpur. He has nineteen
years experience of teaching undergraduate and postgraduate students and guided number of post-graduate
theses. He has published 15 research papers in National
and International journals. Presently he is guiding five
Ph.D students in the area of Image processing. His
research interests are in the areas of digital image
processing, computer vision, and application of artificial
neural networks and fuzzy logic to engineering problems.
38