Inception of Hybrid Wavelet Transform using Two Orthogonal Transforms and It’s use for Image Compression

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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 6, 2011

Inception of Hybrid Wavelet Transform using Two
Orthogonal Transforms and It’s use for Image
Compression
Dr. H.B.Kekre, Dr.Tanuja K. Sarode Sudeep D. Thepade
Senior Professor, Assistant Professor Associate Professor
Computer Engineering Department, Computer Engineering Computer Engineering Department,
SVKM’s NMIMS (Deemed-to-be Department, SVKM’s NMIMS (Deemed-to-be
University) Thadomal Shahani Engineering University)
Vile Parle(W), Mumbai, India. College, Bandra(W), Mumbai, India. Vile Parle(W), Mumbai, India.
hbkekre@yahoo.com, tanuja_0123@yahoo.com sudeepthepade@gmail.com

Abstract—The paper presents the novel hybrid wavelet transform century [19,20]. Generally, wavelets are purposefully crafted to
generation technique using two orthogonal transforms. The have specific properties that make them useful for image
orthogonal transforms are used for analysis of global properties processing. Wavelets can be combined, using a "shift, multiply
of the data into frequency domain. For studying the local and sum" technique called convolution, with portions of an
properties of the signal, the concept of wavelet transform is unknown signal(data) to extract information from the unknown
introduced, where the mother wavelet function gives the global signal. Wavelet transforms are now being adopted for a vast
properties of the signal and wavelet basis functions which are number of applications, often replacing the conventional
compressed versions of mother wavelet are used to study the local Fourier transform [23,24,25,26]. They have advantages over
properties of the signal. In wavelets of some orthogonal
traditional fourier methods in analyzing physical situations
transforms the global characteristics of the data are hauled out
better and some orthogonal transforms might give the local
where the signal contains discontinuities and sharp
characteristics in better way. The idea of hybrid wavelet spikes[27,28,29]. In fourier analysis the local properties of the
transform comes in to picture in view of combining the traits of signal are not detected easily. STFT(Short Time Fourier
two different orthogonal transform wavelets to exploit the Transform)[29] was introduced to overcome this difficulty.
strengths of both the transform wavelets. However it gives local properties at the cost of global
The paper proves the worth of hybrid wavelet properties. Wavelets overcome this shortcoming of Fourier
transforms for the image compression which can further be analysis [28,29] as well as STFT. Many areas of physics have
extended to other image processing applications like seen this paradigm shift, including molecular dynamics,
steganography, biometric identification, content based image astrophysics, optics, quantum mechanics etc. This change has
retrieval etc. Here the hybrid wavelet transforms are generated also occurred in image processing, blood-pressure, heart-rate
using four orthogonal transforms alias Discrete Cosine transform and ECG analyses, DNA analysis, protein analysis,
(DCT), Discrete Hartley transform (DHT), Discrete Walsh climatology, general signal processing, speech, face
transform (DWT) and Discrete Kekre transform (DKT). Te recognition, computer graphics and multifractal analysis.
comparison of the hybrid wavelet transforms is also done with Wavelet transforms are also starting to be used for
the original orthogonal transforms and their wavelet transforms. communication applications. One use of wavelet
The experimentation results have shown that the transform approximation is in data compression. Like other transforms,
wavelets have given better quality of image compression than the wavelet transforms can be used to transform data then, encode
respective original orthogonal transforms but for hybrid
the transformed data, resulting in effective compression [24].
transform wavelets the performance is best. Here the hybrid of
DCT and DKT gives the best results among the combinations of
Wavelet compression can be either lossless or lossy. The
the four mentioned image transforms used for generating hybrid wavelet compression methods are adequate for representing
wavelet transforms. high-frequency components in two-dimensional images.
Earlier wavelets of only Haar transform have been studied.
Keywords-Orthogonal transform; Wavelet transform; Hybrid In recent work [4,7,11,13] the wavelets of few orthogonal
Wavelet transform; Compression. transforms alias Walsh [16,17,18], DCT [14,15], Kekre [21,22]
and Hartley[1,2,3] are proposed. The wavelet transforms in
I. INTRODUCTION many applications are proven to be better than respective
Wavelets are mathematical tools that can be used to extract orthogonal transforms [8,9,10,12]. The paper presents the
information from many different kinds of data, including innovative hybrid wavelet transform generation method, which
images [4,5,6,7]. Sets of wavelets are generally needed to generates hybrid wavelet transform of any two orthogonal
analyze data fully. A set of "complementary" wavelets will transforms. This concept of hybrid wavelet transform can
reconstruct data without gaps or overlap so that the acquire the positive traits from both the orthogonal transforms
deconstruction process is mathematically reversible and is with used to generate it. The hybrid wavelet generation concept
minimal loss. The wavelets are results of the thought process of opens up new avenues of selection of orthogonal transforms for
many people starting with with Haar's work in the early 20th hybrid and their use in particular image processing application

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to gain some upper edge over individual orthogonal transforms
or respective wavelet transforms. The paper presents the use of [K ] ∗ [K ]t = [D ] (3)
hybrid wavelet transforms generated using Discrete Walsh where, D is the diagonal matrix. The hybrid wavelet
Transform (DWT), Discrete Kekre Transform (DKT), Discrete transform of size NxN generated from any two orthogonal
Hartley Transform (DHT) and Discrete Cosine Transform transforms satisfies this property and hence it is orthogonal.
(DCT) for image compression. The experimental results prove
that the hybrid wavelet transforms are better than the respective
B. Non Involutional
orthogonal transforms as well as their wavelet transforms.
An involutionary function is a function that is it’s own
inverse. So involutional transform is a transform which is
II. GENERATION OF HYBRID WAVELET TRANSFORM inverse transform of itself. The Hybrid wavelet transform is
non involutional transform
⎡ a11 a12 L a1 p ⎤ ⎡b11 b12 L b1q ⎤
⎢a L a2 p ⎥ ⎢b
a 22 b22 L b2 q ⎥ C. Transform on Vector
A=⎢ ⎥ B=⎢ ⎥
21
(1)
21
⎢ M M M M ⎥ ⎢ M M M M ⎥ The hybrid wavelet transform (say ‘K’) of one-dimensional
⎢ ⎥ ⎢ ⎥
⎢a p1
⎣ a p2 L a pp ⎥
⎦ ⎢bq1 bq 2 L bqq ⎥
⎣ ⎦ vector q is given by.

Q = K ∗q [] (4)
And inverse is given by

Q
q = K [ ]t ∗ ij
(5)
μ ∗μ
Ti Tj
Where Qij is the value at ith row, jth column of matrix Q and
(2) the term μ in normalization factor can be computed as given
below through equations 6, 7 and 8.
t
μ =T T (6)
T AB AB
Such that

μ =μ μ
T1 A1 B1 ,
μ =μ μ
The hybrid wavelet transform matrix of size NxN (say T2 A1 B2 ,
‘TAB’) can be generated from two orthogonal transform μ =μ μ
matrices ( say A and B respectively with sizes pxp and qxq, T3 A1 B3
where N=p*q=pq) as given by equations 1 and 2.Here first ‘q’ M
number of rows of the hybrid wavelet transform matrix are
calculated as the product of each element of first row of the μ =μ μ (7)
Tq A1 Bq
orthogonal transform A with each of the columns of the
orthogonal transform B. For next ‘q’ number of rows of hybrid μ =μ =L=μ =μ
Tq +1 Tq + 2 T2q A2
wavelet transform matrix the second row of the orthogonal
transform matrix A is shift rotated after being appended with μ =μ =L=μ =μ
zeros as shown in equation 2. Similarly the other rows of T2q +1 T2q + 2 T3q A3
hybrid wavelet transform matrix are generated (as set of q rows M
each time for each of the ‘p-1’ rows of orthogonal transform
matrix A starting from second row upto last row). μ =μ =L=μ =μ
T(p −1)q +1 T2q + 2 T3q A3

III. PROPERTIES OF HYBRID WAVELET TRANSFORM
Where with reference to equation 1, μ A and μ B can be
The crossbreed of two orthogonal transforms results into
given as equation 8.
hybrid wavelet transform, which itself satisfies the following
properties. ⎡μ A1 0 L 0 ⎤
t ⎢ 0 μ
A2
L 0 ⎥
A. Orthogonal AA = μ
A
= ⎢ M M M M
⎥ (8)
The transform matrix K is said to be orthogonal if the ⎢ 0 ⎥
following condition is satisfied. ⎣ 0 L μ
Ap ⎦

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⎡μ B1 0 L 0 ⎤ Where Iij is the pixel intensity value at ith row, jth column of
image I and the calculation of the term μ in normalization
t ⎢ 0 μ
B2
L 0 ⎥
BB = μ
B
= ⎢ ⎥ factor is as given above through equations 6, 7 and 8.
M M M M
⎢ 0 ⎥
⎣ 0 L μ
Bq ⎦ IV. RESULTS AND DISCUSSION
D. Transform on Two-Dimensional Image The test bed used in experimentation for proving the worth
The hybrid wavelet transform of two-dimensional image I is of hybrid wavelet transform consists of 11 color images of size
given by. 256x256x3and is shown in figure 1. On each image all the
three alias orthogonal transform, wavelet transform and hybrid
[]
Q = K ∗I∗ K [ ]t (9) wavelet transform are applied. In transform domain the high
frequency data is removed and the images are transformed
inversely back to spatial domain. To judge the performance of
And inverse is given by the orthogonal transform, wavelet transform and hybrid
wavelet transform in compression; the original images are
I
compared with these modified images (having the data loss as
q = K [ ]t ∗ ij
[]
∗ K
(10) compression) using mean squared error (MSE). In all size data
⎛μ ∗ μ ⎞
⎜ ⎟
⎝ Ti Tj ⎠ compression percentages are considered as 95%, 90%, 85%,
80%, 75% and 70%. The average of such MSEs for all images
for respective transform and considered percentage of data
compression is taken for performance analysis.

Figure 1: The test bed of eleven original color images belonging to different categories and namely (from left to right and top to bottom) Aishwarya, Balls, Bird,
Boat, Flower, Dagdusheth-Ganesh, TajMahal, Strawberry, Scenery, Tiger and Viharlake-Powai.

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Figure 2: Performance comparison of Image compression using Discrete Cosine transform (DCT), cosine wavelet transform (DCT wavelets) and the hybrid
wavelet transforms of DCT taken with Hartley (DCT_DHT) and Kekre transforms (DCT_DKT) with respect to 95% to 70% of data compression.

Figure 2 shows the average of mean squared error (MSE) differences of the original and respective compressed image pairs
plotted against percentages of data compression from 955 to 70% for image compression done using Discrete Cosine transform
(DCT), cosine wavelet transform (DCT wavelets) and the hybrid wavelet transforms of DCT taken with Hartley (DCT_DHT) and
Kekre transforms (DCT_DKT). Here the performance of hybrid wavelet transforms (DCT_DKT and DCT_DHT) is the best as
indicated by minimum MSE values over the respective DCT and DCT wavelet transform.

Figure 3: Performance comparison of Image compression using Discrete Walsh transform (DWT), Walsh wavelet transform (Walsh wavelets) and the hybrid
wavelet transforms of Walsh transform taken with Hartley (DWT_DHT) and Cosine transforms (DWT_DCT) with respect to 95% to 70% of data compression

The average of mean squared error (MSE) differences of the original and respective compressed image pairs for image compression
done using Discrete Walsh transform (DWT), Walsh wavelet transform (Walsh wavelets) and the hybrid wavelet transforms of
Walsh transform taken with Hartley (DWT_DHT) and Cosine transforms (DWT_DCT) with respect to 95% to 70% of data
compression are plotted in figure 3. Here the performance of hybrid wavelet transforms (DWT_DHT and DWT_DCT) are better
than the Walsh transform and are almost similar to the Walsh wavelet transform. The DWT_DCT hybrid wavelet transform
marginally performs better in case of 95% and 80% data compression.

Figure 4: Performance comparison of Image compression using Discrete Hartley transform (DHT), Hartley wavelet transform (Hartley wavelets) and the hybrid
wavelet transforms of Hartley transform taken with Walsh (DHT_DWT) and Cosine transforms (DHT_DCT) with respect to 95% to 70% of data compression.

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Figure 5: Performance comparison of Image compression using Discrete Kekre transform (DKT), Kekre wavelet transform (Kekre wavelets) and the hybrid
wavelet transforms of Kekre transform taken with Cosine transforms (DKT_DCT) with respect to 95% to 70% of data compression

Figure 4 gives the average of mean squared error (MSE) differences of the original and respective compressed image pairs for
image compression done using Discrete Hartley transform (DHT), Hartley wavelet transform (Hartley wavelets) and the hybrid
wavelet transforms of Hartley transform taken with Walsh (DHT_DWT) and Cosine transforms (DHT_DCT) with respect to 95%
to 70% of data compression. Here except 95% data compression in al other percentages of data compression, the performance of
hybrid wavelet transforms (DHT_DWT and DHT_DCT) are better than the Hartley transform and are almost similar to the
Hartley wavelet transform with DWT_DCT proved to be marginally better.

In case of image compression using hybrid wavelet transform (DKT_DCT) generated using discrete Kekre transform (DKT) and
discrete Cosine transform (DCT), the performance is almost similar to the Kekre wavelet transform but better than the Kekre
transform as shown in figure 5.

Figure 6: Overall performance analysis of Image compression using the orthogonal transforms, their respective wavelet transforms and newly introduced hybrid
wavelet transforms for Cosine, Kekre, Walsh and Hartley transforms with respect to 95% to 70% of data compression

Figure 6 gives overall performance comparison of image compression using all the proposed hybrid wavelet transforms with
respective orthogonal transform and wavelet transform based compression methods for various percentages of data compression
from 70% to as high as 95%. Overall the best performance is given by DCT_DKT (hybrid wavelet transform of Cosine transform
with Kekre transform) followed by DCT_DWT and DCT_DHT (hybrid wavelet transform of Cosine transform taken respectively
with Walsh transform and Hartley transform). In all the respective orthogonal transforms the hybrid wavelet transforms have shown
better quality of image compression.

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Figure 7: The compression of flower image using the hybrid wavelet transform (DCT_DHT Wavelet) generated using Discrete Cosine transform and Discrete
Hartley transform with respect to 95% to 70% of data compression

Figure 8: The compression of flower image using the hybrid wavelet transform (DCT_DKT Wavelet) generated using Discrete Cosine transform and Discrete
Kekre transform with respect to 95% to 70% of data compression

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Figure 9: The compression of flower image using the hybrid wavelet transform (DCT_DWT Wavelet) generated using Discrete Cosine transform and Discrete
Walsh transform with respect to 95% to 70% of data compression.

Figures 7, 8 and 9 have shown the compression of flower in a Greyscale Image”, International Journal of Computer
image for various hybrid wavelet transforms with respect to Applications (IJCA), Volume 1, Number 11, December 2010, pp 32-
38.
the 955 to 70 % of data compression. The subjective quality
[5] Dr. H.B.kekre, Sudeep D. Thepade, Adib Parkar “Storage of Colour
of compression in all cases is quite acceptable as negligible Information in a Greyscale Image using Haar Wavelets and Various
distortion is observed in original and compressed images Colour Spaces”, International Journal of Computer Applications
even at the 95% data compression. Even the objective (IJCA), Volume 6, Number 7, pp.18-24, September 2010.
criteria (i.e. mean squared error) values of differences [6] Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal,
between the original and compressed images are minimal. “IRIS Recognition using Texture Features Extracted from Walshlet
Pyramid”, ACM-International Conference and Workshop on
Emerging Trends in Technology (ICWET 2011),Thakur College of
V. CONCLUSION Engg. And Tech., Mumbai, 26-27 Feb 2011. Also will be uploaded
on online ACM Portal.
The innovative concept of the hybrid wavelet transforms [7] Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Face
generation using any two orthogonal transforms is proposed Recognition using Texture Features Extracted form Walshlet
in the paper. Here the hybrid wavelet transforms are Pyramid”, ACEEE International Journal on Recent Trends in
Engineering and Technology (IJRTET), Volume 5, Issue 1,
generated using Discrete Walsh Transform (DWT), Discrete www.searchdl.org/journal/IJRTET2010
Kekre Transform (DKT), Discrete Hartley Transform [8] Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal,
(DHT) and Discrete Cosine Transform (DCT) for image “Performance Comparison of IRIS Recognition Techniques using
compression. The experimental results prove that the hybrid Wavelet Pyramids of Walsh, Haar and Kekre Wavelet Transforms”,
International Journal of Computer Applications (IJCA), Number 2,
wavelet transforms are better than the respective orthogonal Article 4, March 2011,
transforms as well as their wavelet transforms. The various http://www.ijcaonline.org/proceedings/icwet/number2/2070-aca386
orthogonal transforms can be considered for crossbreeding [9] Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Face
to generate the hybrid wavelet transform based on the Recognition using Texture Features Extracted from Haarlet
expected behavior of the hybrid wavelet transform for Pyramid”, International Journal of Computer Applications (IJCA),
Volume 12, Number 5, December 2010, pp 41-45. Available at
particular application. After proving the worth of hybrid www.ijcaonline.org/archives/volume12/number5/1672-2256
wavelet transforms for the image compression future work [10] Dr.H.B.Kekre, Sudeep D. Thepade, Juhi Jain, Naman Agrawal,
could include the extension of the concept to other image “IRIS Recognition using Texture Features Extracted from Haarlet
processing applications like steganography, biometric Pyramid”, International Journal of Computer Applications (IJCA),
Volume 11, Number 12, December 2010, pp 1-5, Available at
identification , content based image retrieval etc. www.ijcaonline.org/archives/volume11/number12/1638-2202.
[11] Dr.H.B.Kekre, Sudeep D. Thepade, Akshay Maloo, “Performance
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Article 2, August 2010. head in the Department of Computer Engg. at Thadomal Shahani
http://www.ijcaonline.org/specialissues/casct/number2/1006-41 Engineering. College, Mumbai. Now he is Senior Professor at MPSTME,
[13] Dr.H.B.Kekre, Sudeep D. Thepade, “Image Retrieval using Color- SVKM’s NMIMS. He has guided 17 Ph.Ds, more than 100 M.E./M.Tech
Texture Features Extracted from Walshlet Pyramid”, ICGST and several B.E./ B.Tech projects. His areas of interest are Digital Signal
International Journal on Graphics, Vision and Image Processing processing, Image Processing and Computer Networking. He has more than
(GVIP), Volume 10, Issue I, Feb.2010, pp.9-18, Available online 270 papers in National / International Conferences and Journals to his
www.icgst.com/gvip/Volume10/Issue1/P1150938876.html credit. He was Senior Member of IEEE. Presently He is Fellow of IETE
[14] N. Ahmed, T. Natarajan and K. R. Rao, “Discrete Cosine and Life Member of ISTE Recently 11 students working under his guidance
Transform”, IEEE Transaction Computers, C-23, pp. 90-93, January have received best paper awards. Two of his students have been awarded
1974. Ph. D. from NMIMS University. Currently he is guiding ten Ph.D. students.
[15] W. Chen, C. H. Smith and S. C. Fralick, “A Fast Computational
Algorithm For The Discrete Cosine Transform”, IEEE Transaction Dr. Tanuja K. Sarode has Received Bsc.(Mathematics) from Mumbai
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Patel School of Technology, Management and
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Vile-Parle (W), Mumbai, INDIA. She has more
[18] Zhibin Pan, Kotani K., Ohmi T., “Enhanced fast encoding method than 12 years of experience in teaching.
for vector quantization by finding an optimally-ordered Walsh Currently working as Assistant Professor in
transform kernel”, ICIP 2005, IEEE International Conference, Dept. of Computer Engineering at Thadomal
Volume 1, pp I - 573-6, Sept. 2005. Shahani Engineering College, Mumbai. She is life member of IETE,
[19] Charles K. Chui, “An Introduction to Wavelets”, Academic Press, member of International Association of Engineers (IAENG) and
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of Multidisciplinary Research and Advances in Engineering
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journals.com 2003. M.E. in Computer Engineering from
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Prathamesh V., Suraj S., “Kekre Transform over Row Mean, currently submitted thesis for Ph.D. at SVKM’s
Column Mean and Both using Image Tiling for Image Retrieval”, NMIMS, Mumbai. He has more than 08 years
International Journal of Computer and Electrical Engineering of experience in teaching and industry. He was
(IJCEE), Volume 2, Number 6, October 2010, pp 964-971, is Lecturer in Dept. of Information Technology at
available at www.ijcee.org/papers/260-E272.pdf Thadomal Shahani Engineering College,
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Computer Engineering at Mukesh Patel School
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and Wavelet Transform” Prentice-hall International, Inc., New International Advisory Committee for many International Conferences. He
Jersey, 1998. is reviewer for various International Journals. His areas of interest are
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Science and Engineering, vol. 2, num. 2, Summer 1995, USA. papers in National/International Conferences/Journals to his credit with a
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Program for Non-Harmonic Sounds Based on a Sinusoidal Best Paper Award at ThinkQuest-2009 National Level paper presentation
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AUTHORS PROFILE
Dr. H. B. Kekre has received B.E. (Hons.) in Telecomm. Engineering.
from Jabalpur University in 1958, M.Tech
(Industrial Electronics) from IIT Bombay in
1960, M.S.Engg. (Electrical Engg.) from
University of Ottawa in 1965 and Ph.D.
(System Identification) from IIT Bombay
in 1970 He has worked as Faculty of
Electrical Engg. and then HOD Computer
Science and Engg. at IIT Bombay. For 13
years he was working as a professor and

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