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Laporan Akhir Projek PenyelidikanJangka Pendek
Image Data Compression using DCT(Discrete Cosine Transform) and
Interpolation and Allied Topics in DigitalImage Processing Applied to Satellite
Imaging
byDr. Pabbisetti Sathyanarayana
Dr. Aftanasar Md Sahar
·nrtERfMA? 3 APR 2007
UNIT E·..... ·UNIVERSITI SAINS MALAYSIA "J '. < , . , I
. Lr"IVi~{ :;;!I S.inllP.lh·PEJABAT PENGURUSAN & KREATIVITI PENYELlDIKAN·-·---·····------....~· .
RESEARCH CREATIVITY AND MANAGEMENT OFFICE [RCMOl
1) Nama Ketua Penyelidik :Name of Research Leader: Dr. PABBISETTI SATHYANARAYANA
PTJSchool/Centre
Aerospace Engineering/Engineering campus
Ketua PenyelidikResearch Leader
Dr. PABBISETTI SATHYANARAYANA
LAPORAN AKHIR PROJEK PENYELlDIKAN JANGKA PENDEKFINAL REPORT OF SHORT TERM RESEARCH PROJEC '\
Nama Penyelidik Bersama(Jika berkaitan) :Name/s of Co-Researcher/s
(if applicable)
Penyelidik Bersama PTJCo-Researcher School/Centre
Dr. AFTANASAR Md sAlIAR Aerospace Engineering/Engineering campus
2) Tajuk Projek: '" .. , , '" .Tit/e of Project:
Image data compression using DCT(Discrete cosine
transform) and interpolation and allied topics in Digital Image
Processing applied to satellite Imaging.
3) Abstrak untuk penyelidlkan anda(Pertu disediakan di antara 100 - 200 perkataan dl dalam Bahasa Malaysia dan Bahasa IRggerls. Ini kemudiannya akandimuatkan ke dalam Laporan Tahunan Sahagian Penyelidikan & Inovasi sebagal satu cara untuk menyampaikan dapatanprojek tuan/puan kepada pihak Universiti & luar).
Abstract ofResearch
(Must be prepared in 100 - 200 words in Bahasa Malaysia as well as in English. This abstract will later be included in theAnnual Report of the Research and Innovation Section as a means of presenting the project findings of the researcherIs tothe university and the outside community)
Digital image processing plays an important role in modern scientific endeavors. It has .specific uses in satellite imaging, remote sensing, telemetry and medical imaging. Image .processing requires huge memory space to store the data, and to process the data in realtime high speed computers are required. For transmission, the channel capacityrequirement is much more stringent. To circumvent such problem in storage and'transmission one of the methods is to compress the data as much as possible beforetransmission and after reception decompression is to be applied with out loss of much .information. The compression process can be carried out by using DCT (Discrete cosine 'transform), Discrete Hartley transform and also Discrete Fourier transform. Algorithms aredeveloped fQr all the three methods. Matlab software is used for the compression anddecompression process. Numbers of images are tested for the process. Detailed analysis is 'carried out. Mean square error estimation of the image obtained by compression anddecompression process is carried out with respect to the original image. Interpolation is theother technique to recover the original image from the sampled image. Interpolation in twodimensions is the method to recover replica of the original image from the sampled image.There are different methods to implement interpolation.2-D FFT is used for this and forthis method also error estimation is done and compared with other methods.
2
.j
Sila sediakan Laporan teknikallengkap yang menerangkan keseluruhan projek ini.[Sila gunakan kertas berasingan]Kindly prepare a comprehensive technical report explaining the project(Prepare report separately as attachment)Refer to Attachment -1
Senaraikan Kata Kunci yang boleh menggambarkan penyelidikan anda :List a glosssary that explains or reflects your research:
Iiannya akan<an dapatan
~/uded in thelearoher/s to
4)
Sahasa Malaysia Bahasa Inggeris
rs. It hasg. Imageta in realcapacity
'age ande beforeof muchte cosinethIns aresion andlalysis islion andon is then in two:l image.and for
Discrete cosine transformDisorete Hartley TransformImage compressionInterpolationMean square errorImage processing
5) Output Dan Faedah ProjekOutput and Benefits of Project
(a) * Penerbitan (termasuk laporan/kertas seminar)Publications (inclUding reports/seminar papers)(Sila nyatakan janis, tajuk, pengarang, tahun terbitan dan di mana telah diterbitldibentangkan).(Kindly state each type, title, author/editor, publication year and joumalls containing pUblication)
1. International Conference: "Image data compression techniques using discrete Hartleytype transform and FFTs: A comparison", Pabbisetti Sathyanarayana, Hamid R.Saeedipour, Aftanazar Md. Sahar and Radzuan Razali,International Conference onRobotics, Vision, Information and Signal Processing,(ROVISP-2005) 21-22 July 2005Pinang, Malaysia
2. International Conference: "Data compression techniques using CAS-CAS transformapplied to remotely piloted vehicle(RPV) digital· images before transmission to groundstation", Pabbisetti Sathyanarayana, Hamid R. Saeedipour, and K.S. Rama Rao 9th
International Conference on Mechatronics Technology (ICMT-2005)2005, 5-8 December2005, Kuala Lumpur, Malaysia.
3. International Conference:"Digital image compression and decompression using threedifferent transforms and comparison oftheir performance", Pabbisetti Sathyanarayana, andHamid R. Saeedipour, International Conference on Man-Machine Systems(ICoMMS2006), 15-16 September 2006. Langkawi, Malaysia
(b) Faedah-Faedah Lain Seperti Perkembangan Produk, Prospek Komersialisasi DanPendaftaran Paten atau impak kepada dasar dan masyakarat.Other benefits such as product development, product commercialisation/patent registration orimpact on source and society
Different methods of image compression and decompression are tested and merits anddemerits ofthem are also discussed."the algorithm is developed. The implementation inreal time using hard ware and software is to be developed as further work in this project.
* Sile berikan salinan• Kindly provide copies
3
(c) Latihan Gunatenaga ManusiaTraining in Human Resources
KOME
i) Pelajar Siswazah : '" .Postgraduate students:(perincikan nama, ijazah dan status)(Provide names, d6grees and status)............. w ~ ..
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KOMEN JAWATANKUASA PENYE,-IDIKAN PUSATPENGAJIANComments of the Research Committees of Schools/Centres
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APPENDIX-I
TECHNICAL REPORT
Project Title: Image data compression using OCT (Discrete Cosine transform) and interpolation andallied topics in digital image processing applied to satellite imaging
Project Leader: Dr. Pabbisetti Sathyanaryana
The project work started in the month of April 2005. The literature survey of the project work iscompleted after three months of the start of the project. The project work is completed. Digital imageprocessing plays an important role in many modem scientific applications.!t has specific applications insatellite imaging, remote sensing and medical imaging. Digital images require huge memory space andlarger band width for transmission, since for a reasonable pixel size of512x 512 ofdigital image requiresabout few MBs memory space. To reduce the storage space and burden of transmission of data,compression ofdata is one ofthe important applications.There are different techniques used for data compression. They arei. Discrete cosine transformii. Fast Fourier transformiii. Fast Hartley Transformiv. Interpolation using FFTv. Karhumen - Loeve transform
The present work started with analysis ofcompression and decompression using DCT and Hartleytransforms. The Discrete Cosine Transform (DCT) and Fast Hartley type transforms are applied for datacompression in two dimensions for digital images. The Hartley transform (HT) was developed as asubstitute to Fourier transform (FT) in applications where the data is in real domain. [1] to [16]. The HThas also defmed in two dimensions [11] as a separable Hartley type transform, named the CAS-CAStransform (CCl). CAS stands for Cos plus Sine. Algorithms are developed for data compression anddecompression using CAS- CAS transform in two dimensions. Comparison of compressed anddecompressed image with the original image with respect to mean square error is carried out. Discretecosine transform method of compression and decompression process is also carried out on the sameexamples. The Fourier Transform is also used for Gompression and decompression process by developingan algorithm for the compression'These three transform methods are compared in terms·ofmean squareerror with respect to original image.
I.CAS-CAS Transform method:
Hartley transform (Hl) relations in 2-D are [11]
M-IN-l mu nvH (u, v) = LLx(m,n)cas27£(-+-) '" (1)
m..on=O M N
1 M-IN-l mu nvX(m,n) LIH(u,v)cas27£(-+-) ... (2)
MxN u..ov-O M N
mu nvCas 27£(- +-) is not a separable Kernel like
M N
Exp [
compucase ofCAS-CT(u, v)
T (u, v)
X(m,n
Where~·
Cas (a
H(u, v
In this
1.1 The1. ScanInsteadconsid
2. Co3. Mod'of4.
ConC (u, v)
C (u, vmatrix
4. Thison the c
5. Attto origi
D(u, v:
:ion and
work isI image:ions in,ce and~quires
f data,
Exp [ j27Z"(mu + nv)] for 2-D DFT, hence the row- column decomposition method [13] [14] ofM N
computing a 2-D transform from I-D fast Fourier transform algorithm can not be applied directly in thiscase ofHT. To take advantage ofthe fast I-D HT algorithms, a separable Hartley like transform namelyCAS-CAS transform is developed.
T(u, v), is defined as [11]
M-IN-I 2mnu 27lflvT(u, v) == LLx(m,n)cas--cas-- ...(3)
m=On=O M N
1 M-IN-l 2mnu 27lflvX (m, n) = ILT(u,v)cas---cas-- ....(4)
MxN u=Ov=O M N
Where
Cas (a ) = cos ( a ) + sin (a ) and
1H (u, v) ="2 [T(u, v) + T (-u, v)+T(u, -v,) - T(-u, -v)] ....(5)
In this way 2-D Discrete Hartley transform can be computed.
artleyr dataas a
eRT·CASI and
andcrete;amepinguare
1.1 The algorithmic steps for compression and decompression using CAS-CAS transform (CCT):
1. Scan the analog Image to get digital image with a size ofpixels (256 x 256) as reference data matrix A.Instead of particular size of the digital image data, general size of the original digital image matrix A isconsidered as (M, N).
2. Com.pute CCT ofthis matrix A using equation (3) as Matrix B which is of same size A.
3. Modify the CCT coefficients matrix B to reduce its size to (M/2, N/2), to achieve a compression factorof4.
Construct a new sequence C (ll, v) from B (ll, v) as follows
C (u, v) == B (u, v) for u == 0, 1 (M/4) -1
v == 0, 1 (N/4)-I= B (ll, v+N/2) for u == 0, 1 (M/4)-I
v = (N/4) (N/2)-I
for u == 0, 1. .. (M/4)-I
v == 0, 1. .. (N/4)-I
== B (u+M/2, v) for u =M/4... (M/2)-I
v = 0, 1. ..N/4== B (u+M/2,v+N/2) for u ==M/4... (M/2)-I
v ==N/4 '" (N/2)-I
C (u, v) is matrix of size (M/2, N/2) a compression ratio of 4 with respect to original size of the imagematrix A of size (M, N) is achieved.
4. This reduced CCT coefficient matrix is to be transtpitted instead of the original matrix, so that burdenon the channel can be reduced by a factor of4. .
5. At the receiver, after receiving the modified CCT matrix C (u, v), is appended with zeros to make itto original size as follows.
D(u, v) == C (u, v)
= 0 for u = 0, 1. .. (M/4)-1
v =N/4 (3N/4)-1= C (u, v-N/2) for u = 0, 1 (M/4)-1
v = 3N/4 (N-l)
= 0 for u = M/4 (3M/4)-1
v=O, 1,2 (N-l)= C (u-M/2, v) for u =3M/4 (M-l)
v =0, 1. .. (N/4)-1
=0 for u =3M/4... (M-l)
v =N/4 .,. (3N/4)-1=C (u-M/2, v-N/2) for u = 3M/4 (M-l)
v = 3N/4 (N-l)
6. Multiply each D (u, v) by the square ofthe compression factor.
7. Perform the inverse CCT using equation (4) on the sequence D (u, v) of size (M, N) to obtain thereplica oforiginal image.
x(u, v) ,:b~
Wherem'
X(m, n)
Where '
EquatiodFourierimage.;:
•ID.l. TJ
ll. Data compression and decompression using discrete cosine transform;
The discrete cosine transform relations in 2 -D are
The proFast Foalso by rofCCT.
ID. Fast Fourier transform method
Where m = 0, 1. ... M-l And n = 0, 1. .. N-l
•••• (7)
a v =~ for v=l, ... M-l
a = [2: for u=l, ... M-lu VA!foru= 0
forv=O
M-IN-l 7T(2m + l)u 7T(2n + l)vX (u, v) = auav L L x(m, n) cos cos ... (6)
m=On=O 2M 2N
And 2-D inverse discrete cosine Transform is
M-IN-l 7T(2m + l)u 7T(2n + l)vX (m, n) = auav LLX(u,v) cos cos~----:..-
u=Ov=O 2M 2N
Where u =0, 1 M-l
And v = 0, 1 N-l
Equations (6) are forward transform and equation (7) is r~verse transform. By using the above twoequations discrete cosine transform coefficients and inverse discrete cosine transform coefficients can becomputed for the digital image.
in the
twon be
The Fourier transform relations in 2 -D are
M-IN-l ·2 (mu nv)
X (U, v) = L L x(m,n)e -} 11: ~N (6)m=O n=O
Where m = 0, 1 ... M-l, and n:;: 0, 1. .. N-l
M-IN-l j211:(~+.'!!:)
X(m,n)= LLX(u,v)e M N (7)u=O v=O
Where u = 0, 1 ... M-l, and v = 0, 1 ... N-l
Equation (6) is forward transform and equation (7) is reverse transform. By using the above two equationsFourier transform coefficients and inverse Fourier transform coefficients can be computed for the digitalimage.
ID.l. The algorithmic steps for compression and decompression using FFT
The procedure is similar to that of Hartley type transform for data compression and decompression usingFast Fourier transform. The algorithmic steps given above for Hartley transform, are to be used for thisalso by replacing CCT with FFT. FFT magnitude coefficients are also concentrated at the comers as thatofCCT.
Procedure:1. CCT methodThe above algorithmic steps are applied on the original scanned digital image of size (256,256) pixelsshown as figure 1. For this image data CCT coefficients are computed. Then data is reduced bydiscarding some of the coefficients by using the algorithmic step (3). Most of CCT coefficients arehaving larger magnitudes in all the four comers of the matrix instead of central region of the matrix.Based on this principle the discarding of negligibly small v~lue data points is done. To bring back to theoriginal size of the matrix, zeros are appended where ever the data points are discarded. Now inverseCCT is applied on this data with a proper multiplier. The resultant replica of the original image is shownas figure 2. These two images figure 1 and figure 2 are look alike, and there is no noticeable loss ofinformation. In this a reduction of CCT coefficients to be transmitted is 1I4th of that of original imagedata points.
A second example, with a compression ration of 16, is shown as figure 3. The resultant image figure 3 isnot up to the mark, and further compression will completely distorts the image. The advantage ofCCT isthat it is a real transform and hence the storage and transmission is not complex but FFT algorithm ofrowcolumn decomposition can be applied.
2. nCT method
For the same data (figure 1), equation (6) is applied to get the DCT coefficients. DCT coefficients areconcentrated near the origin. To achieve the compression ratio of 4 the DCT coefficient matrix istruncated by retaining (M/4, N/4) coefficients from the origin.
Now zeros are appended to get back the original size of the matrix and with proper multiplier inversetransform equation (7), is applied to get the replica of the original image as shown figure 4. Figure 1 and
figure 4 are similar with out appreciable differences. Hence there is a compression ratio of 4. is achieved .with out distortion. As the compression ratio is increased further say 16, by decreasing the data points tobe transmitted, the recovered image is distorted as shown in figure 5.
3. FFT method
For the same data (figure 1), equation (6) is applied to get the Fourier coefficients. As in the CCT methodsome of the coefficients are discarded by following same algorithmic steps as in CCT method. Thediscarded coefficients are substituted with zeros, and with proper multiplier inverse transform equation(7), are applied to get the replica of the original image as shown figure 6. Figures 1 and figure 6 aresimilar with out appreciable differences. Hence there is a· compression ratio of 4 is achieved with outdistortion. The only difference in this method in comparison with CCT is that the data to be transmitted iscomplex instead of real and hence effective reduction in memory space is 50 %. As the compression ratiois increased further, decreasing the data points to be transmitted, the recovered image is distorted asshown in figure 7 with a compression ratio of 16. Figure 8 shows the distribution of magnitudes ofFourier coefficients for the figure 1 data points. It shows that the magnitudes are less in the middle region,which are discarded and in the reverse transform they are assumed to be zeros.
Results and conclusions
CCT and DCT methods are tried for different compression factors. The mean square error estimation withrespect to the original image data (figure 1) are presented in table 1.
Serial Compression DCTMethod CCTMethod FFTMethodNumber factor Mean Square error Mean Square Error Mean square error
1 4 0.0019 0.0018 0.00182 16 0.0050 0.0048 0.0050
Table 1. Mean square error with CCT and FFT methods for different data compression ratios.
The mean square error in the three methods is almost the same for different compression factors. It showsthat CCT method can be used for image compression process as that of DCT method. The advantage ofCCT method over DCT method is that the CCT implementation in two dimensions is simpler and wellestablished techniques available for FFT can be used for this. Hence digital image data compression anddecompression process can be effectively implemented by using CCT. RPV continuously takingphotographs for remote sensing application, and sending '~them to ground station, the band widthrequirement is a stringent problem with out compression. By using the compression process the channelbandwidth reduces drastically with out loss ofmuch information and storage space also reduces.
Fig. 5
fact
~hieved
oints to
nethodd. Theluation~ 6 areith outitted isn ratioted asdes of'egion,
1 with
lOWS
~e ofwelland
kingridthnnel
Fig. Original Image
Fig. 2 CCT Method - Compression Factor 4
Fig. 5 OCT method compression
factor of 16
Fig. 3 CCT method compression
factor of 16
Fig.4 OCT method Compressionfactor of 4
Fig.6 FFT method Compression factor of 4
Fig. 7 FFT method compression factor of 16
Fig. 8 FFT coefficients magnitudesof Fig. 1
References
[1] H. W. Musmann, P. Pirsch, and H.G. Grallert," Advances in picture coding," Proc. IEEE Vol. 73, pp,523-548, Apr. 1985.
[2] W. H. Chen, C. H. Smith, and S. Fralick," A fast computational algorithm for the discrete cosinetransform," IEEE Trans. Commun. Vol. com- 25, pp. 1004-1009, Sep.1977[3] "-, Image data compression: A review," Proc. IEEE Vol. 69, pp. 366-406, Mar. 1981
[4] P.Yip and K R Rao." A fast computational algorithm for the discrete cosine transform," IEEE Trans.Commun., vol. com- 38, pp. 304-307, Feb.1980
[5J K.R Rao and R Yip, Discrete cosine transform algorithm, Advantages and applications, Newyork,academic 1990
[6} K Rose, A. Heiman and I. Dintein, "DCT/DST Alternate transform image coding", IEEE Trans. Com.Vol. com-38, pp. 94-101, Jan. 1990
[7J Ziad Alkachouch and Maurice G. Bellengaer," Fast DCT based spatial domain interpolation ofblocks in images", IEEE Trans. On image processing, vol. 9 Apr. 2000[8] T.P. O'Rourkr and Rl. Stevenson, "Improved image
decompression for reduced transform coding artifacts, IEEE Trans. Circuits Systems. Video Technol.,Vo1.5, pp. 490-499, Dec. 1995
[9] T. K Truong,l.J. Wang, I.S. Reed, W.S. Hsieh, " Image data compression using cubic splineinterpolation", IEEE Trans. Image processing Vol. 9, Issue 11, Nov. 2000, pp. 1985-1995[10] Bracewell, R. N.," The Fast Hartley Transform," Proc. IEEE, Vol.72, pp.1 010-1018, Aug, 1986[11] Bracewell,R. N." Buneman, O.,Hao., and Villasenor, J.," Fast two Dimensional Hartley Transform," Proc.IEEE, vol. 74, pp. 1282 - 1283, September 1986..
[12] Reddy G R., Sathyanarayana P. &; Swamy M. N. S. "CAS- CAS Transform for 2-D Signals: A fewapplications," Journal Circuits Systems Signal Process, Vol. 10, No.2 1991[13]. KP. Prasad and P. Sathyanarayana, "Fast interpolationalgorithm using FFT", Electron. Letters, lEE vol. 22, pp.185-187, Feb. 1986[14] Sathyanarayana P., Reddy, P. S. and Swamy, M. N. S., "Interpolation of 2-D, signal" IEEE Trans. Circuits
and Systems, Vol. 37, pp. 623-625, May 1990.
[15] Ioannis pitas and Anestis Karasaridis, " Multi-channel Transforms for signal/image processing," IEEETrans. on Image processing, vol. 5, No. 10, October 1996.'"[16J Sung Cheol Park, Moon Gi Kang, Segall, C. A. ,
Submissions should include: Katsaggelos, A. K.," Spatially adaptive high resolution imag~ reconstruction ofDCTbased compressed images," IEEE Trans. on Image Processing, VoU3, No.4, pp. 573-585, April 2004
~ - - - - - -- --- - - -- ,
:
Proceedings of the International Conference onRobotics, Vision, Information and Signal Processing ROVISP2005
Image Data Compression Techniques Using Discrete Hartley Type Transform AndFfts: A Comparison.
Pabbisetti Sathyanarayana I , Hamid R. Saeedipour 2 , Aftanasar Md. Sahar 3, Radzuan Razali 4ICorresponding Author; Lecturer; PhD, MSc, BEng.
2Lecturer; PhD, MSc, MBS, BEng, IT.
3Lecturer; PhD, MSc, BEng.
4 Associate Professor; PhD, MSc, BScSchool ofAerospace Engineering
Universiti Sains Malaysia, Engineering Campus14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, Malaysia
Tel: +604-5937788 ext.5884, Fax: +604-5941026, E-mail:[email protected]
des
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Abstract
The Image processing plays an important role in modemscientific advancements. It has wide applications in thefields of remote sensing, satellite imaging, medical imagingand telemetry. Digital image processing requires largememory space for storage , high speed computers forprocessing and wider band widths for transmission. Toprocess these images in real time it will be much moreinvolved. Data compression is an important tool in digitalimage processing to reduce the burden on the storage andtransmission systems. The basic idea ofdata compression isto reduce the number of the image pixel elements directly,say by sampling or by using transforms and truncate thetransformed image coefficients, so that the total number ofpicture elements or coefficients are reduced . The imageinformation now requires lesser storage and also lesserband width for transmission. When ever the image is to berecovered or received after transmission the imageinformation is to be decompressed i.e. brought back to theoriginal size. There are different methods for compressionand decompression process. In this paper discrete Hartleytype transforms and Fast Fourier transforms areconsidered for both data compression and decompression.Algorithms are developed and tested using twodime1Jsional Hartley type transform (HT) and Fast Fouriertransform (FFT )for data compression and decompressionof the digital image data. The main advantage ofHartleytransform is, it is a real transform, hence the storage andthe processing of the coefficients require less space andfaster operation in comparison with Fast Fouriertransforms. Number ofdigital images are compressed anddecompressed and the mean square error estimation iscarried out as a comparison with the original images forboth the methods.
Keywords:
Digital Image Processing, Discrete Hartley Transform,Data Compression, Fast Fourier Transform, RemoteSensing,
489
Introduction
Active research work is going on, in digital signalprocessing in recent years. This has good number ofapplications in satellite imaging, medical imaging andtelemetry. Data compression plays an important role insatellite image processing. The satellite image in digitalmode requires huge memory space and higher bandwidthsfor transmission. Data compression before transmissionreduces the channel band width requirement. [ 1 ] to [ 8 ]There are different transforms for data compression. Thereare Karhumen - Loeve transform, Discrete cosinetransform and interpolation process. [ 1 ] to [ 16 ]
In this paper discrete Hartley type transform, and fastFourier transform are studied for data compression in twodimensions, which is directly applicable to digital imageprocessing. The Hartley transform ( HT ) was developed asa substitute to Fourier transform ( FT ) in applicationswhere the data is in real domain. [ 10] , [ 11 ]. The HT hasalso defined in two dimensions [ 12] as a separable Hartleytype transform, named the CAS-CAS transform (CCT ).Algorithms are developed for data compression anddecompression using Hartley type transform in twodimensions. Comparison of compressed anddecompressed image with the original image with respect tomean square error is carried out.
FFT is applied for interpolation after sampling in one andtwo dimensions. [ 13 ] , [ 14] Interpolation is very usefulfor reconstruction of the sampled image. sampled image is apattern of compressed image. In this paper instead ofinterpolation process, direct FFT is evaluated, andcompression algorithm is applied on the FFT data. In there-conversion process zeros are substituted to the FFTcoefficients where ever they are taken off in thetransmission process and inverse FFT is applied on themodified data which presents a replica of original image.This is the process of compression and decompression ofthe image using FFT. Error analysis is also carried out.
Where
(5)
l.CCT me
The abovescanned difigure 1. FoThen data isby using theare having Imatrix instprinciple thTo bringbawhereeveris applied 0
replica oftwo imagesno noticeablpoints to be
A second eshown as fithe mark,the image.and hence
And
Where
X(m, n)=
And n=O,t
Where m=O.
3. Modify the ccr coefficients matrix B to reduce its size to(M/2 , NI2 ), to achieve a compression factor of 4.
Construct a new sequence C (u, v) fromB(u, v) as follows
C (u, v) = B (u, v) for u = 0,1, , (Ml4)-1
v == 0,1, , (N/4)-1
= B(u, v+N/2) for u = 0,1, ,(M/4)-1
v =(N/4), , (N/2)-1
= B(u+M/2, v) for u =M/4, , (MI2)-1
v:>;:: 0,1, ,N/4
= B(u+Ml2,v+N/2) for u =M/4, , (Ml2)-1
v ==N/4, ,(N/2)-1
qu, v) is matrix of size (Ml2, N/2) • a compression ratio of4 with respect to original size of the image matrix A of size(M, N) is achieved.
4. This reduced ccr coefficient matrix is to be transmittedinstead of the original matrix, so that burden on the channelcan be reduced by a factor of 4.
5. At the receiver, after receiving the modified CCTmatrix qu, v), is appended with zeros to make it to originalsize as follows.
D(u, v) = qu, v) for u = 0,1, , (M/4)-1
v = 0,1, , (N/4)-1
= 0 for u == 0,1, , (MI4)-1
v == N/4, ... •(3N/4)-1
= C (u, v-N/2) for u = 0,1, ,(MI4)-1
v =3N/4, , (N-I)
=0 for u == M/4, ,(3M14)-1
v == 0,1,2 (N-1)
= C (u-Ml2, v) for u =3M14, , (M-I)
v =0,1, ,(N/4)-1
=0 for u =3M/4, , (M-I)
v =N/4, '" ,(3N/4)-1
== C (u-Ml2, v-N/2) for u = 3M14, , (M-1)
v =3N/4, (N-1)
6. Multiply each D(u. v) by the square of the compressionfactor.
7. Perform the inverse CCT using equation (4) on thesequence D (u, v) of size (M, N) to obtain the replica oforiginal ~ge.
Data compression and decompression usingfast Fourier transform :
The Fourier transform relations in 2 -D are
Proceedings ofthe International Conference on'Robotics, Vision, Infonnation and Signal Processing ROVISP2()()S
•.(1)
M-IN-I mu nvH(u, v)= EEx(m,n)cas2Jr(-+-)
m;()n;() M N
Hartley transform relations in 2-D are [ 12 ]
Cas(a)=cos(a)+sin(a) and
1H (u, v)=- [T(u, v) + T (-u, v)+T(u, -v,) - T(-u, -v) ]
2
mu nvExp [ j2 1&(-+-)] for 2-D Dfl, hence the row-
M Ncolumn decomposition method [ 14] of computing a 2-Dtransform from 1-D fast Fourier transform algorithm can notbe applied directly in this case of lIT. To take advantage ofthe fast I-D lIT algorithms, a separable Hartley liketransform namely CAS-CAS transform
T (u, v), is defined as [ 12 ]
M-IN-I 2mnu 2JZnvT(u, v) =EEx(m, n)cas--cas-- (3)
m;()n;() M N
1 M-IN-I 2mnu 2JZnvX(m, n) =---E..ET(u,v)cas---cas--
MxN u;()=Q M N
(4)
mu nvCas 21&(-+-) is not a separable Kernel like
M N
1 M-IN-I mu nvX(m, n)==-- EEH(u, v)cas27E'(-+-) (2)
MxN u~Ov-O M N
Data compression and decompression usingHartley type transform
In this way 2-D Discrete Hartley transform can be computed.
The algorithmic steps for compression and
decompression using DHT:
1. Scan the analog Image to get digital image with a size ofpixels (256 x 256 ) as reference data matrix A. Instead ofparticular size of the digital image data, general size of theoriginal image matrix A is considered as (M, N).
2. Compute CCT of this matrix A using equation (3) asMatrix B which is of same size A.
490
ceon~
'200S
[zeto
ows
M-1N-1 -j2Jr(~+!!!:.)
X(u,v)= EEx(m,n)e M N
m=O n=O
(6)
Proceedings of the International Conference onRobotics, Vision, Information and Signal Processing ROVISP200S
2. FFT method
lof:ize
tednel
Where m= 0,1,.... M-l
And n=O,I, ... N-I
M-1N-1 j2Jr(mu~)
X(m,n)= EEX(u,v)e M N (7)u=O v=O
Where u= 0,1, M-l
And v= 0,1, N-l
Equations (6) is forward transform and equation (7) isreverse transform. By using the above two equations Fouriertransform coefficients and inverse Fourier transformcoefficients can be computed for the digital image.
The algorithmic steps for compression anddecompression using FFT:
The procedure is similar to that of Hartley transform for datacompression and decompression using Fast Fouriertransform. In the algorithmic steps given above for Hartleytransform, ccr is to be replaced with FFT, for FFT baseddata compression and decompression process.
Procedure:
1. CCT method
The above algorithmic steps are applied on the originalscanned digital image of size (256,256) pixels shown asfigure I. For this image data ccr coefficients are computed.Then data is reduced by discarding some of the coefficientsby using the algorithmic step (3). Most ofccr coefficientsare having larger magnitudes in all the four corners of thematrix instead of central region of the matrix. Based on thisprinciple the discarding of small value data points are done.To bring back to the original size matrix, zeros are appendedwhere ever the data points are discarded. Now inverse CCTis applied on this data with a proper multiplier. The resultantreplica of the original image is shown as figure 2. Thesetwo images figure I and figure 2 are looks alike, and there isno noticeable loss of information. In this a reduction of datapoints to be transmitted is 1/4th of that of original image.
A second example, with a compression ration of 16, isshown as figure 3. The resultant image figure 3, is not up tothe ~ark, and fuqher compression will completely distortsthe Image. the advantage of ccr is that it is a real transformand hence the storage and transmission is not complex.
For the same data (figure I), equation (6) is applied to get theFourier coefficients and as in the ccr method some of thecoefficients are discarded and substituted with zeros, andwith proper multiplier inverse transform equation (6), isapplied to get the replica of the original image as shownfigure 4. Figures 1 and figure 4 are similar with outappreciable differences. Hence there is a compression ratioof 4 is achieved with out distortion. The only difference inthis method in comparison with ccr, is that the data to betransmitted is complex instead of real and hence effectivereduction is 50 %. As the compression ratio is increasedfurther, decreasing the data points to be transmitted , therecovered image is distorted as shown in figure 5 with acompression ratio of 16. Figure 6 shows the distribution ofmagnitudes of Fourier coefficients for the figure I datapoints. It shows that the magnitudes are less in the middleregion, which are discarded and at the reverse transform theyare assumed as zeros.
Results and conclusions
CCT and FFf methods are tried for different compressionfactors . The mean square error estimation with respect tothe original image data ( figure 1 ) are presented in table I.
Serial Compression CCT FFTNumber factor Method Method
Mean MeanSquare error Square Error
I 4 0.0019 0.00182 16 0.0050 0.0048
Table 1. Mean square error with CCT and FFT methods fordifferent data compression ratios.
The mean square error in both the methods is almost thesame for different compression factors. It shows that CCTmethod can be used for image compression process as that ofFFr method. The advantage of ccr method over FFrmethod is that the CCT is a real transform and hence thenumber of multiplications and additions are less than that ofFFr and storage also reduces to half that of FFT. The bandwidth required for transmission also reduces with CCTinstead of FFT. Hence digital image data compression anddecompression process can be effectively implementedusing ccr.
Acknowledgements
We express our sincere thanks to RCMO authorities ofUniversiti Sains malaysia for providing financial assistancethrough short term grant for this work.
491
Fig. 2 CCT method Compression factor of 4
Proceedings of the Intemational Conference O1iRobotics, Vision, Information and Signal Processing ROVISP2005
Fig.4 FFT method Compression factor of 4
492
Re
[1]pict[2]comTranScpo[3]
pp.3[4]dispp.3[5]
Adv[6]trans94-1
FFT coefficients magnitudes of Fig. 1" ., ~:-:.-'
References
[1] H. W. Musmann, P. Pirsch, and H.G. Grallert," Advances inpicture coding," Proc. IEEE Vol. 73, pp. 523-548, Apr. 1985.[2] W. H. Chen, C. H. Smith, and S. Fralick," A fastcomputational algorithm for the discrete cosine transform," IEEETrans. Commun. Vol. com- 25, pp. 1004-1009,Sep.1977[3] "_, Image data compression: A review," Proc. IEEE Vol. 69,
pp. 366-406, Mar. 1981[4] P.Yip and K. R. Rao." A fast computational algorithm for thediscrete cosine transform," IEEE Trans. Commun., vol. com- 38,pp. 304-307, Feb.1980[5] K.R. Rao and R. Yip, Discrete cosine transform algorithm,
Advantages and applications, Newyork, academic 1990[6] K. Rose, A. Heiman and I. Ointein, "OCTIDST Alternatetransform image coding", IEEE Trans. Com. Vol. com-38, pp.94-101, Jan. 1990
Proceedings ofthe International Conference onRobotics, Vision, Information and Signal Processing ROVISP2005
[7] ZiadAlkachouch and Maurice G. Bellengaer," Fast OCT basedspatial domain interpolation of blocks in images", IEEE Trans. Onimage processing, vol. 9 Apr. 2000[8] T.P. O'Rourkr and R.L. Stevenson, "Improved imagedecompression for reduced transform coding artifacts, IEEE Trans.Circuits Systems. Video Technol., Vol.5, pp. 490-499, Dec. 1995[ 9] T. K. Truong,L.J. Wang, 1.S. Reed, W.S. Hsieh, " Image data
compression using cubic spline interpolation", IEEE Trans. Imageprocessing Vol. 9, Issue 11, Nov. 2000, pp. 1985- 1995[lO] Bracewell, R. N., " The Fast Hartley Transform," Proc.IEEE, Vol.72, pp.10lO-1018, Aug,1986[11] Bmcewell,R. N." Buneman, O.,Hao., and Villasenor,J.," Fast two Dimensional Hartley Transform," Proc. IEEE,vol. 74, pp. 1282 - 1283, September 1986..[12] Reddy G. R., Sathyanarayana P. & Swamy M. N. S."CAS- CAS Transform for 2-D Signals: A fewapplications," Journal Circuits Systems Signal Process,Vol. lO, No.2 1991[ 13]. K.P. Pmsad andP.Sathyanarayana," Fast interpolationalgorithm using FFr', Electron. Letters, lEE vol. 22, pp.185-187,Feb. 1986[14] Sathyanarayana P., Reddy, P. S. and Swamy, M. N. S.,
"Interpolation of 2-D, signal" IEEE Trans. Circuits andSystems, Vol. 37, pp. 623-625, May 1990.[15] Ioannis pitas and Anestis Karasaridis, " Multi-channelTransforms for signal/image processing," IEEE Trans. onImage processing, vol. 5, No. 10, October 1996.[16] Sung Cheol Park, Moon Gi Kang, Segall, C. A. ,Submissions should include: Katsaggelos, A. K.,"Spatially adaptive high resolution image reconstruction ofDCT based compressed images ," IEEE Trans. on ImageProcessing, Vol. 13, No.4, pp. 573-585, April 2004
493
Proceedings ofIntemational Conference on Man-Machine Systems 2006September 15-16 2006, Langkawi, Malaysia
Digital Image Compression and Decompression Using Three Different Transformsand Comparison of Their Performance
And reverse
Pabbisetti Sathyanarayana I , and Hamid R. Saeedipour,ICorresponding Author
Faculty ofSchool ofAerospace EngineeringUniversiti Sains Malaysia, Engineering Campus
14300 Nibong Tebal, Seberang Perai Selatan, Pulau Pinang, MalaysiaTel: +604-599-5946, Fax: +604-5941026,
E-mail: [email protected]@eng.usm.my
X(m,n)=
In this wcomputed.only CCT is
m
2. ComputMatrixB wis that theamplitudesnegligibly s
Where
X(m,n)=
T(u,v)=
column dectransform frused. In casbe appliedalgorithms,CAS-CAS
Cas(a)=
1H (u, v)=-
2
M-l>N-1 mu nvH(u, v)= LLx(m,n)cas27r(-+-)
m;On;O M N
form will take few MB (mega bits) hence the transmissionand storage in the same form will occupy more bandwidthand larger storage space. Data compression beforetransmission reduces the channel band width requirementand memory space [1] to [8]. There are differenttransforms available for data compression. These areKarhumen - Loeve transform, discrete cosine transform,and sampling and interpolation process using fast Fouriertransform. [1] to [16].
In this paper Discrete Hartley type transform andFast Fourier transform are used for data compression in twodimensions, which is directly applicable to digital imageprocessing of RPV photographed images. The performanceis compared with the existing discrete cosine transform.The Hartley transform (Hl) was developed as a substituteto Fourier transform (F1) in applications where the data isin real domain. [10], [11]. The HT was also defined in twodimensions in which kernel is not separable [12]. Anothermodified separable Hartley type transform, named theCAS-CAS transform (CCl) was also defined. CAS standsfor cos plus sine. And CCT for two dimensional cos plussine transform. Algorithms are developed for datacompression and decompression using the three transformsnamely CCT, FFT, and DCT in two dimensions.Comparison of compressed and decompressed image withthe original image with respect to mean square error iscarried out.
Discrete Hartley type transform in two dimensions asseparable transform is the CAS-CAS transform and iscalled as CCT.
Hartley transform relations in 2-D in which thekernel is not separable are [12].
2. CCT Method
(1)
Active research work is going on, in digital signalprocessing in recent years. This has good number ofapplications in remote sensing, medical imaging andtelemetry. Digital image processing comprises of many subareas, some of them are image enhancement, imagerestoration, image encoding, and data compression. Datacompression play's an important role in image processingespecially in remote sensing using remotely piloted vehicle(RPV). The image taken by the RPV in digital moderequires huge memory space and higher bandwidths fortransmission to ground station. Each raw image in digital
Abstract
1 Introduction
Data compression is an important tool in digital imageprocessing to reduce the burden on the storage andtransmission systems. The basic idea ofdata compression isto reduce the number of the image pixel elements directly,say by sampling, or by using transforms and truncate thetransformed image coefficients, so that the total number ofpicture elements or its coefficients are reduced The imageinformation now requires lesser storage and also lesserband width for transmission. When ever the image is to berecovered or received after transmission the imageinformation is to be decompressed i.e. brought back to theoriginal size and form. This compression process isessential for images taken by satellites, or unmanned aerialvehicles (UAVs) for remote sensing and weatherapplication. By·applying compression algorithm the imagedata may take one fourth or even less size with out loss ofmuch information. There are different methods forcompression and decompression process. In this paperthree methods are used for both data compression anddecompression process, they are i. Discrete Hartley typetransform, ii. Fast Fourier transform (FFI), iii. Discretecosine transforms (DCI). Algorithms are developed andtested using the three methods on different images. Acomparison with respect to mean square error with theoriginal image also presented The main advantage ofdiscrete Hartley type transform is, it is a real transform.Discrete cosine transform also has similar performance asthat of discrete Hartley type transform. An algorithm forcompression using FFT method is also presented
And reverse transform is
1 M-IN-I mu nvX(m,n) = LLH(u, v)cas21r(-+-)
MxN u=Ov-O M N
(2)
mu nvCas 21r(- +-) is not a separable Kernel.
M N
But for 2-D DFT the kernel is
Proceedings ofIntemationai Conference on Man-Machine Systems 2006September 15-162006, Langkawi, Malaysia
3. Based on the above principle, modify the CCTcoefficients matrix B to reduce its size to (Ml2, N/2), toachieve a compression factor of 4 by neglecting smallamplitude coefficients.
Construct a new matrix C (ll, v) from B (u, v) which isreduced in size as follows
C (ll, v) = B (ll, v) for u= 0, I .,. (Ml4)-1
v =0, I ... (N/4)-1
= B (ll, v+N/2) for u = 0, 1... (Ml4)-1
v= (N/4) ... (N/2)-1
= B (u+Ml2, v) for u =Ml4... (Ml2)-1
v= 0,1... N/4
=B (u+Ml2, v+N/2) for u=Ml4 (MI2)-1
v=N/4 (N/2)-1
C (u, v) is matrix of size (Ml2, N/2), a compression ratio of4 with respect to original size of the image matrix A of size(M, N) is achieved.
4. This reduced CCT coefficient matrix is to be transmittedinstead of the original matrix, so that memory space issaved for storage and burden on the transmission channel isreduced by a factor of 4.
5. At the receiver, after receiving the modified CCTmatrix C (ll, v), is appended with zeros at respective regions,to make it to reference image size as follows.
mu nvExp [j21r(- +-)] which is separable. Hence the row-
M Ncolumn decomposition method [14] of computing 2-Dtransform from I-D fast Fourier transform algorithm can beused. In case of 2-D Hartley transform row column can notbe applied directly. To take advantage of the fast I-D HTalgorithms, a separable Hartley like transform namelyCAS-CAS transform that is CCT is shown.
T (ll, v), is defmed as [12]
M-IN-I 2mnu 2nnvT(ll,v)= LLx(m,n)cas--cas--
m=On=O M N
(3)
1 M-IN-I 2mnu 27lnVX (m, n) = --LLT(u,v)cas--eas-
MxN U=Ov=O M N
(4)
Where
Cas ( a ) = cos (a ) + sin (a ) and
1H (ll, v) =- [T (ll, v) + T (-u, v) +T (u, -v,) - T (-u, -v)]
2
(5)
In this way 2-D Discrete Hartley transform can becomputed. For the compression and decompression processonly CCT is used instead ofHartley transform.
2.1 The Algorithmic Steps for Compression and DeCompression Using CCT
1. Scan the analog Image to get digital image with a size ofpixels (256 x 256) as reference data matrix A. Instead ofparticular size of the digital image data, general size of theoriginal image matrix A is considered as (M, N).
2. Compute CCT of this matrix A using equation (3) asMatrix B which is of same size as A. Important observationis that the CCT coefficient matrix is having largeramplitudes at the four comers and the value diminishes,negligibly small towards central region
D (ll, v) = C (ll, v)
=0
= C (u, v-N/2)
=0
= C (u-Ml2, v)
=0
for u= 0,1... (MI4)-1
v = 0, 1... (N/4)-1
foru=O, 1... (MI4)-1
v=N/4 ... (3N/4)-1
for u = 0, 1... (MI4)-1
v= 3N/4 ... (N-I)
for u=Ml4 (3M14)-1
v= 0,1,2 (N-I)
for u =3M/4 ... (M-I)
v= 0,1... (N/4)-1
for u =3M14 (M-I)
v=N/4 (3N/4)-1
Proceedings ofIntemational Conference on Man-Machine Systems 2006September 15-16 2006, Langkawi, Malaysia
=C (u-Ml2, v-NI2) for u=3M14 (M-I)
v=3N/4 (N-I)
6. Multiply each D (u, v) by the square of the compressionfactor.
7. Perform the inverse CCT using equation (4) on thesequence D (11, v) of size (M, N) to obtain the replica oforiginal image A.
3. Fast Fourier Transform Method
~I~l ( ) Jr(2m + l)u Jr(2n + l)va u a v .LJ .LJ x m, n COS COS -'----'----
m=On=O 2M 2N
(8)
Where m = 0, 1.. .. M-I; and n = 0,1... N-I
1a =-- Foru=O
u JM
dimi'
4.decommatrix
5. Comodifiof the
The discrete cosine transform relations in 2-D are
5. Pr
5.2.For thethe Focoefficistepssubstitutransfooriginalsimilarcompreonly dithat thehence eis inertransmifigure 5distrib
Forv=O
Where u= 0, I ... M-I, and v= 0, I ... N-I
Equation (8) is forward transform and equation (9) isreverse transform. By using the above two equationsdiscrete cosine transform coefficients and inverse discretecosine transform coefficients can be computed for thedigital image.
(9)
~I~X( ) Jr(2m+l)u Jr(2n+l)vauav .LJ.LJ u, v COS cos----
u=Ov=O 2M 2N
X(m, n)=
a v =& Forv=l ... M-I
a = f2 Foru=I ... M-1u fii
And 2-D inverse discrete cosine Transform is
4.1. The Algorithmic Steps for Compression andDecompression Using DCT
1. Scan the analog Image to get digital image with a size ofpixels (256 x 256) as reference data matrix A. Instead ofparticular'"size of the digital image data, general size of theoriginal image matrix A is considered as (M, N).
2. Compute DCT of this matrix A using equation (8) asMatrix B which is of same size as A. The DCT coefficientmatrix is having larger magnitudes near the origin and is
(7)
(6)
M-IN-I '2 (mu nv)
X(m,n)= LLX(u,V)/ 7f M+/i
u=O 1'=0
X (11, v)=
3.1. The Algorithmic Steps for Compression andDecompression Using FFT
The procedure is similar to that of Hartley type transformfor data compression and decompression using Fast Fouriertransform. The algorithmic steps given above for Hartleytransform, are to be used for this also by replacing CCTwith FFT. FFT magnitude coefficients are also concentratedat the comers as that of CCT.
4. Discrete Cosine Transform Method
The Fourier transform relations in 2 -D are
Where u = 0, 1... M-I, and v = 0, 1. .. N-I
Equation (6) is forward transform and equation (7) isreverse transform. By using the above two equationsFourier transform coefficients and inverse Fouriertransform coefficients can be computed for the digitalimage.
Where m= 0, I ... M-I, and n= 0,1... N-I
Proceedings ofInternalional Conference on Man-Machine Systems 2006September 15-16 2006, Langkawi, Malaysia
6. Results and Conclusions
Table 1. Mean square error with the three methods fordiffi d ..
Three methods are tried for different compression factors.The mean square error estimation of resultant images withrespect to the original image data (figure 1) are presented intable 1.
The mean square error for all the methods is almost thesame for different compression factors. It shows that CCTmethod can be used for image compression process as thatof FFT method. The advantage of CCT method over FFTmethod is that the CCT is a real transform and hence thenumber ofmultiplications and additions are less than that ofFFT and storage also reduces to half that of FFT. The bandwidth required for transmission also reduces with CCTinstead of FFT. Hence digital image data compression anddecompression process can be effectively implementedusing CCT. DCT method also gives similar performance asthat of CCT. Either nCT or CCT can be implemented forany practical system.
figure 1 data points. It shows that the magnitudes are less inthe middle region, which are discarded and in the reversetransform they are assumed to be zeros.
5. 3. DCT Method
For the same data (figure 1), equation (8) is applied to getthe nCT coefficients. nCT coefficients are concentratednear the origin. To achieve the compression ratio of 4 thenCT coefficient matrix is truncated by retaining (Ml2, N/2)from the origin.
Now zeros are appended to get back the originalsize of the matrix and with proper multiplier inversetransform equation (9), is applied to get back the replica ofthe original image. Hence there is a compression ratio of 4is achieved with out distortion. As the compression ratio isincreased further say 16, by decreasing the data points to betransmitted, the recovered image is distorted as shown infigure 7.
erent ata compreSSiOn ratiOS.
'0 4>
= '0 ..,'0 ~
Q ...~ c:> .. -= el,Q .. c:> .. - ='" .... = ,Q = 4> 0"'" .., C''t~ ~r:IJ~ ~oo
-; =- .. Eo< = .. ~ = .. E-< = ..·C = e.s .. c:> Eo< .. c:> el c:>
c:> <.> U .., .. "'" .., .. U 4> ..
"5z u~ u~~ "'"~~ ~~~
1 4 0.0019 0.0018 .00192 16 0.0050 0.0048 .0050
5. Procedure
5.1. CCT Method
The above algorithmic steps referred in chapter 2.1 areapplied on the original scanned digital image of size(256,256) pixels shown as figure 1. For this image dataCCT coefficients are computed. Then data size is reducedby discarding some of the coefficients by using thealgorithmic step (3). Most of CCT coefficients are havinglarger magnitudes in all the four comers of the matrixinstead of central region of the matrix. Based on thisprinciple the discarding of small value data points isachieved. To bring back to the original size matrix, zerosare appended where ever the data points are discarded.Now inverse CCT is applied on this data with a propermultiplier. The resultant replica of the original image isshown as figure 2. These two images figure 1 and figure 2are looks alike, and there is no noticeable loss ofinformation. In this a reduction of data points to betransmitted is 1/4th ofthat of original image.
A second example, with a compression ratio of 16for the same image is applied and the resultant image isshown as figure 3. Figure 3 is not up to the mark, andfurther compression will completely distorts the image. Theadvantage of CCT is that it is a real transform and hence thestorage and transmission is not complex and FFT algorithmof row column decomposition can be applied.
5. 2. FFT Method
For the same data (figure 1), equation (6) is applied to getthe Fourier coefficients. As in the CCT method some of thecoefficients are discarded by following same algorithmicsteps as in CCT method. The discarded coefficients aresubstituted with zeros, and with proper multiplier inversetransform equation (7), are applied to get the replica of theoriginal image as shown figure 4. Figures 1 and figure 4 aresimilar with out appreciable differences. Hence there is acompression ratio of 4 is achieved with out distortion. Theonly difference in this method in comparison with CCT isthat the data to be transmitted is complex instead of real andhence effective reduction is 50 %. As the compression ratiois increased further, decreasing the data points to betransmitted, the recovered image is distorted as shown infigure 5 with a compression ratio of 16. Figure 6 shows thedistribution of magnitudes of Fourier coefficients for the
diminishing as the (m, n) increases.
3. Based on this principle truncate the Matrix B keeping thenCT coefficients for m=O to (Ml2)-1, and n = 0 to (N/2)-1so that the size of the matrix formulated C is (MI2, N/2) acompression ratio of4 is achieved.
4. The matrix c is transmitted and received. Nowdecompression process is applied by appending the Cmatrix zeros to get back the size ofthe original matrix A.
5. Compute inverse discrete cosine transform on thismodified matrix using equation (9) to get back the replicaof the original digital image.
Acknowledgements
We express our sincere thanks to RCMO authorities ofUniversiti Sains Malaysia for providing fmancial assistancethrough short tenn grant for this work.
References
[1] H. W. Musmann, P. Pirsch. and H.G. Grallert,"Advances in picture coding," Proc. IEEE Vol. 73, pp. 523.548, Apr. 1985.
[2] W. H. Chen, C. H. Smith. and S. Fralick," A fastcomputational algorithm for the discrete cosine transfonn,"IEEE Trans. Commun. Vol. com- 25, pp. 1004-1009,Sep. 1977
[3] "-" Image data compression: A review," Proc. IEEEVol. 69, pp. 366-406, Mar. 1981
[4] P.Yip and K R Rao." A fast computational algorithmfor the discrete cosine transfonn," IEEE Trans. Commun.,vol. com- 38, pp. 304·307, Feb.1980
[5] KR Rao and R. Yip, Discrete cosine transfonnalgorithm, Advantages and applications, Newyork,academic 1990
[6] K Rose, A. Heiman and I. Dintein, "DCT/DSTAlternate transfonn image coding", IEEE Trans. Com. Vol.com-38, pp. 94-101, Jan. 1990
[7] Ziad Alkachouch and Maurice G. Bellengaer," FastDCT based spatial domain interpolation of blocks inimages", IEEE Trans. On image processing, vol. 9 Apr.2000
[8] T.P. O'Rourkr and R.L. Stevenson, "Improved imagedecompression for reduced transfonn coding artifacts, IEEETrans. Circuits and Systems. Video Technol., Vol.5, pp.490-499, Dec. 1995
[ 9] T. K Truong,L.J. Wang, I.S. Reed, W.S. Hsieh. "Image data compression using cubic spline interpolation",IEEE Trans. Image processing Vol. 9, Issue 11, Nov.2000,pp. 1985- 1995
[10] Bracewell, R. N.," The Fast Hartley Transfonn," Proc.IEEE, Vol.72, pp.101Q-1018, Aug, 1986[11] Bracewell,R. N." Buneman, O.,Hao., and Villasenor,J.," Fast two Dimensional Hartley Transfonn," Proc. IEEE,vol. 74, pp. 1282 - 1283, September 1986..[12] Reddy GR., Sathyanarayana P. & Swamy M. N. S."CAS- CAS Transfonn for 2-D Signals: A fewapplications," Journal Circuits Systems Signal Process,Vol. 10, No.2 1991[13]. KP. Prasad and P. Sathyanarayana, "Fastinterpolation algorithm using FFT", Electron. Letters, IEEvol. 22, pp.185-187, Feb. 1986
[14] Sathyanarayana P., Reddy, P. S. and Swamy, M. N. S.,"Interpolation of 2-D, signal" IEEE Trans. Circuits andSystems, Vol. 37, pp. 623-625, May 1990.
Proceedings ofInternational Conference on Man-Machine Systems 2006September 15-16 2006, Langkawi, Malaysia
[15] Ioannis pitas and Anestis Karasaridis, " Multicharmel Transfonns for signal/image processing," IEEETrans. on Image processing, vol. 5, No. 10, October 1996.[16] Sung Cheol Park, Moon Gi Kang, Segall, C. A.,Submissions should include: Katsaggelos, A.K.,"Spatially adaptive high resolution image reconstruction ofDCT based compressed images ," IEEE Trans. on ImageProcessing, VoU3, No.4, pp. 573-585, April 2004
Fgure 1. Original Image
Fig. 2 CCT methodCompression factor of 4
Fig. 3 CCT method compressionFactor of 16
Fig.4 FFT method Compression factor of 4
Fig. 5 FFT method compression factor of 16
Proceedings ofIntemational Conference on Mao-Machine Systems 2006September 15-16 2006, Langkawi, Malaysia
Fig. 6 FFT coefficientsmagnitudes of Fig. 1
Fig. 7 OCT method compressionfactor of 16
I
I,I
Data compression technique using CAS-CAS transform applied to remotely pilotedvehicle (RPV) digital images before transmission to ground station x
Pabbisetti Sathyanarayana l, Hamid R. Saeedipour 2 and K S. Rama Rao 3
I Co"esponding Author1,2 School ofAerospace Engineering
3School ofElectrial and Electronic EngineeringUniversiti Sains Malaysia, Engineering Campus
14300 Nihong TebaJ, Seberang Perai Selatan, Pulau Pinang, MalaysiaTel: +604-5937788 ext 5946, Fax: +604-5941026, E-moil: sotyamp1@J'ediffmaiLcom
Abstract
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Active research work is going on, in remotely pilotedvehicles in recent years. This has good number ofapplications in remote sensing, and telemetry and militaryapplications. Data compression plays an important role inremotely piloted vehicle image processing. The RPV inJagein digital mode requires huge memory space and higherbandwidths for transmission to ground station. Datacompression before transmission reduces the channel bandwidth requirement [1] - [8]. There are different transformsavailable for data compression. These are Karhumen Loeve transform discrete cosine transform and interpolationprocess. [1] - [16]
ill this paper CAS- CAS transform and OCT are studiedfor data compression in two dimensions, which is directlyapplicable to digital image processing of RPVphotographed images. The Hartley transform (HT) wasdeveloped as a substitute to Fourier transform (FT) inapplications where the data is in real domain. [10], [11].The HT was also defined in two dimensions [12] as aseparable Hartley type transform, named the CAS-CAStransform (CCT). CAS stands for cos +Sin. Algorithms aredeveloped for data compression and decompression usingCAS- CAS transform in two dimensions. Comparison ofcompressed and decompressed image with the originalimage with respect to mean square error is carried out.
Discrete cosine transform method of compression anddecompression process is also carried out on the sameexamples. These two methods are compared in terms ofmean square error.
1. Introduction
2. CAS-CAS Transform method:
Remotely piloted vehicle plays an important role inmodem scientific advancements. It has wide applications inthe field of remote sensing, and military applications. Oneof the important tasks of remotely piloted vehicle (RPV) isto take images continuously and send them to groundstation for further processing. If the raw images are to besent directly to ground station wider band widths arerequired for data transmission. To reduce the band widthrequirement inJages are to be compressed beforetransmission. Data compression is an important tool indigital image processing to reduce the burden on the storageand transmission systems. The basic idea of datacompression is to reduce the number of the image pixelelements directly, say by sampling or by using transformsand truncate the transformed image coefficients, so that thetotal number ofpicture elements or coefficients are reduced .The image information now requires lesser storage and alsolesser band width for transmission. When ever the image isto be recovered or received after transmission the imageinformation is to be decompressed i.e. brought back to theoriginal size. There are different methods for compressionand decompression process. In this paper CAS-CAStransform is used for both data compression anddecompression process. Algorithms are developed andtested using two dimensional CAS-CAS transform. Acomparison is also presented with this transform to that ofdiscrete cosine transform (DCT). The main advantage ofCAS-CAS transform is, it is a real transform, and is similarto that of fast Fourier transform (FFT). The fast algorithmsof FFT can be applied directly to this. Row columndecomposition method can be used. Number of digitalimages is compressed and decompressed and the meansquare error estimation is carried out.
Keywords: Hartley transform relations in 2-D are [12]
C
C
Digital Image Processing, CAS-CAS transform, DataCompression, Remotely piloted vehicle, Remote Sensingand Discrete cosine transform.
M-1N-1 mu nvH(n, v)= LLx(rri,n)cas2n-(-+-)
m=On=O M N... (1) j
iIi
1 M-1N-I mu nvX (m, n) =-- LLH(u,v)cas27T(-+-)
MxN u;Ov-O M N(2)
mu nvCas 27T(- +-) is not a separable Kernel like
M N
mu nvExp [ j2 7T(- +-)] for 2-D DFT, hence the row-
M Ncolumn decomposition method [ 14] of computing a 2-Dtransform from I-D fast Fourier transform algorithm can notbe applied directly in this case ofHT. To take advantage ofthe fast I-D HT algorithms, a separable Hartley liketransform namely CAS-CAS transform T (u, v), is defmed as[12]
M-IN-I 2nmu 21l1'lvT(u, v)= LLx(m,n)cas--cas-- (3)
m;On;O M N
1 M-IN-l 2mnu 21l1'lvX (m, n) = --LLT(u,v)cas--eas--
MxN u;Ov;O M N
(4)
Where
Cas(a)=cos(a)+sin(a) and
1H (u, v) =- [T(u, v) + T (-u, v)+T(u, -v,) - T(-u, -v)]
2
(5)
In this way 2-D Discrete Hartley transform can be computed.
2.1. The algorithmic steps for compression anddecompression using CAS-CAS transform(CCT):
I. Scan the analog Image to get digital image with a size ofpixels (256 x 256) as reference data matrix A. Instead ofparticular size of the digital image data, general· size of theoriginal image matrix A is considered as (M, N).
2. Compute CCT of this matrix A using equation (3) asMatrix B which is of same size A.
3. Modify the CCT coefficients matrix B to reduce its size to(Ml2, NI2), to achieve a compression factor of 4.
Construct a new sequence C (ll, v) from B (u, v) as follows
C(ll,v)=B(ll,v) for u""O, I (M/4)-1
v=O,l (N/4)-1
=B (ll, v+N/2) for u = 0, 1... (M/4)-1
v =(N/4) ... (N/2)-1
=B (u+Ml2, v) for u=Ml4... (MI2)-1
v=0,1...N/4
= B (u+Ml2, V+NI2) for u =Ml4... (MI2)-1
v=N/4... (N12)-1
C (ll, v) is matrix of size (Ml2, NI2); a compression ratio of4 with respect to original size of the image matrix A of size(M, N) is achieved.
4. This reduced CCT coefficient matrix is to be transmittedinstead of the original matrix, so that burden on the channelcan be reduced by a factor of4.
5. At the receiver, after receiving the modified CCT matrixC (u, v), is appended with zeros to make it to original size asfollows.
D (ll, v) = C (ll, v) for u= 0,1... (M/4)-1
v= 0,1... (N/4)-1
=0 for u= 0,1... (MI4)-1
v=N/4 ... (3N/4)-1
= C (u, v-NI2) for u = 0, 1... (MI4)-1
v=3N/4... (N-I)
=0 for u = Ml4 ... (3M14)-1
v=0,1,2 .... (N-I)
= C (u-Ml2, v) for u =3M14 ... (M-I)
v = 0, 1... (N/4)-1
=0 for u =3M14... (M-I)
v=N/4 ... (3N/4)-1
= C (u-Ml2, v-NI2) for u = 3M14... (M-I)
v = 3N/4... (N-l)
6. Multiply each D (u, v) by the square of the compressionfactor.
7. Perform the inverse CCT using equation (4) on thesequence D (ll, v) of size (M, N) to obtain the replica oforiginal image.
3. Data compression and decompression usingdiscrete cosine transform:
The discrete cosine transform relations in 2 -D are
x (ll, v) =
,. ~I~I ( ) 7T(2m + l)u 7T(2n + l)va a L..J L..J x m,n cos cosu v m;On;O 2M 2N
(6)
Where m = 0,1.. .. M-1
And n = 0, 1... N-1
1a =-- foru=O
u .JM
au =/l; foru=l ... M-1
1a =-- forv=O
y .JM
a = rz for v=l ... M-1y '1M
And 2-D inverse discrete cosine Transform is
x (m.n)=
~l~X( ) 1!(2m + l)u 1!(2n + l)vaUay L.J L.J u, V cos cos
u=Ov=O 2M 2N
.... (7)
Where u=O,l M-1
And v=O,l N-1
Equations (6) are forward transform and equation (7) isreverse transform. By using the above two equations discretecosine transform coefficients and inverse discrete cosinetransform coefficients can be computed for the digital image.
3. 1. The algorithmic steps for compression anddecompression using DCT:
The procedure is similar to that of CAS-CAS transform fordata compression and decompression using discrete cosinetransform. In the algorithmic steps given above for CCT is tobe replaced with DCT, for DCT based data compression anddecompression process.
4. Procedure:
4. 1. CCT method
The above algorithmic steps are applied on the originalscanned digital image of size (256,256) pixels shown asfigure 1. For this image data CCT coefficients are computed.Then data is reduced by discarding some of the coefficientsby using the algorithmic step (3). Most of CCT coefficientsare having larger magnitudes in all the four comers of thematrix instead of central region of the matrix. Based on thisprinciple the discarding of small value data points is done.To hring back to the original size matrix, zeros are appendedwhere ever the data points are discarded. Now inverse CCTis applied on this data with a proper multiplier. The resultantreplica of the original image is shown as figure 2. These twoimages figure 1 and figure 2 are looks alike, and there is nonoticeable loss of information. In this a reduction of datapoints to be transmitted is 1I4th of that of original image.
A second example, with a compression ration of 16, isshown as figure 3. The resultant image figure 3 is not up tothe mark, and further compression will completely distortsthe image. The advantage ofCCT is that it is a real transformand hence the storage and transmission is not complex butFFT algorithm ofrow column decomposition can be applied.
4. 2 DCT method
For the sanJe data (figure 1), equation (6) is applied to get theDCT coefficients. DCT coefficients are concentrated nearthe origin. To achieve the compression ratio of 4 the DCTcoefficient matrix is truncated by retaining (M/4, N/4) fromthe origin.
Now zeros are appended to get back the original size of thematrix and with proper multiplier inverse transform equation(7), is applied to get the replica of the original image asshown figure 4. Figures 1 and figure 4 are similar with outappreciable differences. Hence there is a compression ratioof4 is achieved with out distortion. As the compression ratiois increased further say 16, by decreasing the data points tobe transmitted, the recovered image is distorted as shown infigure 5. Figure 6 shows the distribution of magnitudes ofCCT coefficients for the figure 1 data points. It shows thatthe magnitudes are negligibly small in the middle region,which are discarded and zeros are appended for the reversetransform
5. Results and conclusionsCCT ahd DCT methods are tried for different compression
factors. The mean square error estimation with respect to theoriginal image data (Figure I) are presented in table 1.
TabledifferSerialNumb
2
ThesamemethDCTmethsimplcanbanddby usremostatiowith
Fig.4 OCT method Compression factor of 4
Fig. 3 CCT method compression factor of 16
Fig. 2 CCT method Compression factor of 4
We express our sincere thanks to RCMO authorities ofUniversiti Bains Malaysia for providing financial assistancethrough short term grant for this work.
Figure 1. Original Image
The mean square error in both the methods is ahnost thesame for different compression factors. It shows that CCTmethod can be used for image compression process as that ofDCT method. The advantage of CCT method over DCTmethod is that the CCT implementation in two dimensions issimpler and well established techniques available for FFTcan be used for this.. Hence digital image data compressionand decompression process can be effectively implementedby using CCT. RPV continuously taking photogtaphs forremote sensing application, and sending them to groundstation, the band width requirement is a stringent problemwith out compression. By using the compression process thechannel bandwidth reduces drastically with out loss of muchinformation and storage space also reduces.
6. Acknowledgements
Table 1. Mean square error with CCT and FFT methods fordifferent data compression ratiosSerial Compression DCT CCTNumber factor Method Method
Mean MeanSquare error Square Error
1 4 0.0019 0.00182 16 0.0050 0.0048
Fig. 5 DCT method compression factor of 16
Fig. 6 CCT coefficients magnitudes of Fig. 1
7. References
[1] H. W. Musmann, P. Pirsch, and H.G. Grallert," Advances inpicture coding," Proc. IEEE Vol. 73, pp. 523-548, Apr. 1985.[2] W. H. Chen, C. H. Smith, and S. Fralick," A fastcomputational algorithm for the discrete cosine transform," lEEE
Trans. Commun. Vol. com- 25, pp. 1004-1009,Sep.1977[3] "_, Image data compression: A review," Proc. IEEE Vol. 69,
pp.366-406,Mar.1981[4] P.Yip and K. R. Rao." A fast computational algorithm for thediscrete cosine transform," IEEE Trans. Commun., vol. com- 38,pp. 304-307, Feb. 1980[5] K.R. Rao and R. Yip, Discrete cosine transform algorithm,
Advantages and applications, Newyork, academic 1990[6] K. Rose, A. Heiman and I. Dintein, "DCTIDST Alternatetransform image coding", IEEE Trans. Com. Vol. com-38, pp.94-101, Jan. 1990[7] Ziad AIkachouch and Maurice G. Bellengaer," Fast DCTbased spatial domain interpolation of blocks in images", IEEETrans. On image processing, vol. 9 Apr. 2000[8] T.P. O'Rourkr and R.L. Stevenson, "Improved imagedecompression for reduced transform coding artifacts, IEEE Trans.Circuits Systems. Video Technol., Vo1.5, pp. 490-499, Dec. 1995[9] T. K. Truong,L.J. Wang, I.S. Reed, W.S. Hsieh, "Image data
compression using cubic spline interpolation", IEEE Trans. Imageprocessing Vol. 9, Issue 11, Nov. 2000, pp. 1985- 1995[10] Bracewell, R. N.," The Fast Hartley Transform," Proc.IEEE, Vol.72, pp.1010-1018,Aug, 1986[11] Bracewell,R. N." BUIleman, O.,Hao., and Villasenor,1.," Fast two Dimensional Hartley Transform," Proc. IEEE,vol. 74, pp. 1282 - 1283, September 1986..[12] Reddy G R., Sathyanarayana P. & Swamy M. N. S."CAS- CAS Transform for 2-D Signals: A fewapplications," Jomnal Circuits Systems Signal Process, Vol.10, No.2 I991[13]. K.P. Prasad and P. Sathyanarayana," Fast interpolationalgorithm usiug FFT', Electron. Letters, IEE vol. 22, pp.185-187,Feb. 1986
[14] SathyanarayanaP., Reddy, P. S. and Swamy, M. N. S.,"Interpolation of 2-D, signal" IEEE Trans. Circuits andSystems, Vol. 37, pp. 623-625, May 1990.[15] Ioannis pitas and Anestis Karasaridis, "Multi-channelTransforms for signal/image processing," IEEE Trans. onImage processing, vol. 5, No. 10, October 1996.[16] SUIlg Cheol Park, Moon Gi Kang, Segall, C. A. ,
Submissions should include: Katsaggelos, A. K.,"Spatially adaptive high resolution image reconstruction ofOCT based compressed images," IEEE Trans. on ImageProcessing, Vol.13, No.4, pp. 573-585, April 2004
,.
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IMAGE DATA COMPORESSION USING DCT AND INTERPOLAnON AND ALLLIED TOPIC IN DIGITAL
Perbelanjaan Tanggungan Perbelanjaan Jumlah Jumlah Baki Peruntukan
Peruntli<an sehingga semasa Semasa Perbelanjaan Perbelanjaan Semasa
3111m006 2007 2007 2007 Terkumpul 2007(a) (b) (c) (d) (c + d) (b+c+d) (a-(b+c+d)
450.00 0.00 0.00 0.00 0.00 0.00 450.00
4,000.00 2,126.35 0.00 0.00 0.00 2,126.35 1,873.65
394.00 0.00 0.00 0.00 0.00 0.00 394.00
0.00 500.00 0.00 0.00 0.00 500.00 (500.00)
3,000.00 1,947.00 0.00 0.00 0.00 1,947.00 1,053.00
3,350.00 2,500.00 0.00 0.00 0.00 2,500.00 850.00
11,194.00 7,073.35 0.00 0.00 0.60 7,073.35 4,120.65
11,194.00 7,073.35 0.00 0.00 0.00 7,073.35 4,120.65
~ :-
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