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Development of a New Transform:MRT DEVELOPMENT OF A NEW TRANSFORM: MRT A THESIS Submitted by RAJESH CHERIAN ROY for the award of the degree of DOCTOR OF PHILOSOPHY Under the guidance of R. GOPlKAKUMARI DIVISION OF ELECTRONICS ENGINEERING SCHOOL OF ENGINEERING COCHIN UNIVERSITY OF SCIENCE & TECHNOLOGY KOCHI - 682 022, INDIA FEBRUARY 2009 CERTIFICATE This is to certify that the thesis entitled "DEVELOPMENT OF A NEW TRANSFORM: MRT" is a bomifide record of the research work carried out by Rajesh Cherian Roy under my supervision and guidance in the Division of Electronics Engineering, School of Engineering, Cochin University of Science and Technology and that no part thereof has been presented for the award of any other degree. Dr. R. Gopikakumari (Supervising Guide) Head Division of Electronics Engineering Kochi School of Engineering 19 January 2009 Cochin University of Science and Technology DECLARATION I hereby declare that the work presented in the thesis entitled "DEVELOPMENT OF A NEW TRANSFORM: MRT" is based on original research work carried out by me under the supervision and guidance of Dr. R. Gopikakumari in the Division of Electronics Engineering, School of Engineering, Cochin University of Science and Technology and that no part thereof has been presented for the award of any other degree. Kochi ~ 19 Janurary 2009 Rajesh Cherian Ray ABSTRACT Fourier transform methods are employed heavily in digital signal processing. Discrete Fourier Transform (DFT) is among the most commonly used digital signal transforms, The exponential kernel of the DFT has the properties of symmetry and periodicity. Fast Fourier Transform (FFT) methods for fast DFT computation exploit these kernel properties in different ways. In this thesis, an approach of grouping data on the basis of the corresponding phase ofthe exponential kernel of the DFT is exploited to introduce a new digital signal transform, named the M-dimensional Real Transform (MRT), for l-D and 2-D signals. The new transform is developed using number­ theoretic principles as regards its specific features. A few properties of the transform are explored, and an inverse transform presented. A fundamental assumption is that the size of the input signal be even. The transform computation involves only real additions. The MRT is an integer-to-integer transform. There are two kinds ofredundancy, complete redundancy & derived redundancy, in MRT. Redundancy is analyzed and removed to arrive at a more compact version called the Unique MRT (UMRT). l-D UMRT is a non-expansive transform for all signal sizes, while the 2-D UMRT is non-expansive for signal sizes that are powers of 2. The 2-D UMRT is applied in image processing applications like image compression and orientation analysis. The MRT & UMRT, being general transforms, will find potential applications in various fields of signal and image processing. ACKNOWLEDGEMENTS First and foremost I would like to express my profound gratitude to my research guide, Dr. R. Gopikakumari, Professor and Head, Division of Electronics, School of Engineering, Cochin University of Science and Technology, under whose patient supervision and guidance I have been able to complete my research. I wish to thank the Vice Chancellor and Registrar, CUSAT for the opportunity to complete the research work and submit the thesis. I also thank the Principal, School of Engineering, members of the Research Committee and Doctoral Comnittee for guidance and help at various stages of the period of research. I wish to express gratitude to the office staff of various sections of CUSAT for help and assistance. Thanks are also due to the faculty, Division of Electronics, School of Engineering, and office staff at School of Engineering for all help and assistance provided in relation with the research work. I would like to thank ISRO for the opportunity to work in their funded project "Fractal Image Compression using Neural Networks". Sincere thanks are due to Dr. D. Rajaveerappa, Dr. C. Karthikeyan and Dr. S. Ramkumar for the valuable discussions, encouragement and words of advice during various stages of the research work. I am indebted to my friends Mr. Vinu Thomas and Mr. Lakshmana Kumar Krishna Moorthy for their immense support and help throughout the period of research. I thank Mr. Suresh, Librarian, Department of Electronics, Cochin University of Science and Technology, for assistance. I acknowledge the valuable discussions with and support given by Mr. Bhadran V. and Mr. M. S. Anish Kumar, research scholars at Division of Electronics, School of Engineering. I wish to acknowledge my grand parents and parents whose blessings have guided me throughout this period. I also wish to thank my wife, brother, sister, uncles, aunts, cousins and all in-laws for the faith and patience during the period of research. Finally, I wish to remember the late Mr. P. P. Pathrose for his support and encouragement. 11 SYMBOLS AND DEFINITIONS Z Set of integers N Integer, N E Z, mostly used to indicate size of input signal n Signal time index I-D signal sample at n k Frequency index P Phase index y:(p) k I-D MRT coefficient at k, p. Signal spatial index in the vertical direction Signal spatial index in the horizontal direction 2-D signal sample at nI, n2 Frequency index in the vertical direction Frequency index in the horizontal direction 2-D MRT coefficient at k[, k2, p. g(a,b) - Greatest Common Divisor (GCD) of integers a and b alb a divides b ((a))6 Remainder of alb (Modulo operation) q Integer q E Z - q is an element of set Z [a, b] {q E Z : a ~ q ~ b} LaJ Smallest integer lesser than or equal to a ~(N) Totient function of N (A,B) - Inner product between vectors A and B For all values of q Such that III LIST OF TABLES Table 3.1: Time comparison between direct MRT computation and closed-form MRT computation. 33 Table 3.2: Isometric transformations in MRT domain. 39 Table 4.1: Complete redundancy and derived redundancy relations for (a) N= 6, (b) N= 12, (c) N= 18, (d) N= 24, (e) N= 30 ..... 76-77 Table 4.2: Example showing reconstruction of 8-point I-D sequence using inverse I-D 1JMRT............... 85 Table 4.3: Number ofUMRT coefficients involved in inverse I-D UMRT for a few values of N. .. 86 Table 4.4: Proposed mapping between I-D 1JMRT indices and array indices, for N = 8. 87 Table 4.5: Proposed mapping between I-D UMRT indices and array indices, for N= 12.87 Table 5.1: Values of U k evaluated for divisor frequencies k), k2 for N= 8. .. 98 kI' .2 Table 5.2: Number of unique frequencies for various values of N. 112 Table 5.3: Number of unique 2-D MRT coefficients for various values of N. 124 Table 5.4: Totatives for a few integers that are powers of2. 126 Table 5.5: Unique frequencies for N = 8 . 129 Table 6.1: Compression results obtained using UMRT on 5 test images. 142 Table 6.2: Pattern orientations for a few UMRT frequencies, N = 16. 143 IV LIST OF FIGURES Figure 1.1: Transfonn kernel values for 8-point 1-0 WHT. 9 Figure 1.2: Transfonn kernel values for 8-point 1-0 Haar Transfonn. 9 Figure 1.3: Transfonn kernel values for 8-point 1-0 integer OCT. 10 Figure 3.1: An 8 x 8 signal and its MRT matrices. 23 Figure 3.2: Flowchart showing computation method for MRT particular solutions. 30 Figure 3.3: Plot showing 2-D MRT computation time using direct and closed fonn methods. 32 Figure 3.4: (a): A 6 x 6 signal; (b): (a) reversed; (c), (e) & (g): MRT matrices of (a); (d), (t) & (h): MRT matrices of (b) . .. 35 Figure 3.5: (a): A 6 x 6 signal; (b): (a) circularly shifted by 1 row and 1 column; (c), (e) & (g): MRT matrices of (a); (d), (t) & (h): MRT matrices of (b). 37 Figure 4.1: Kernel representation Ak,p,n of 1-D MRT for N = 4. 46 Figure 4.2: Kernel representation Ak,p,n of a few 1-0 MRT coefficients, N = 6. 47 Figure 4.3: Kernel representation Ak,p,n of a few 1-0 MRT coefficients, N = 8. 48 Figure 4.4: (a):Placement ofUMRT coefficients for N= 12; (b): Proposed placement ofMRT & UMRT coefficients for N = 12. , 90 Figure 5.1: Basis images corresponding to UMRT coefficients for N = 8. 130-132 Figure 5.2: Positional details of 8 x 8 2-D UMRT matrix fanned by placement algorithm, specified by values of (kt, k2' p). .. 134 Figure 5.3: Positional details of 8 x 82-0 UMRT matrix fonned from (5.114) - (5.116), specified by values of(kt. k2,p). .. 135 Figure 6.1: Image blocks generated using MRT manipulation from a single image block. 140 Figure 6.2: Original and reconstructed images, 'Lenna'. 141 Figure 6.3: Analysis ofUMRT pattern orientation. 142 (a): Basis images corresponding to MRT coefficients ~\O) ~\I), ~\2) and ~?), (b): Figure 6.4: , , . , . d' MRT ffi' y(O) y(l) y;(2) d yP) B aSIs Images correspon mg to coe lClents 1.0 ' 1,0' 1,0 an 1.0'" 144 Figure 6.5: Global patterns formed by union of: (a) MRT coefficients with frequency index (1,1), (b) MRT coefficients with frequency index (1,0). .. 144 Figure 6.6: Global patterns fonned byUMRT coefficients for N= 16, (a): k2 =1; (b) k2 = 2,4; (c) k2 = 0,8. .. 145 Figure 6.7: Results of fingerprint orientation estimation using MRT coefficients. .. 148 Figure 6.8: Plot showing 2-D MRT computation time using direct method, closed form method, and computation from 2-D UMRT coefficients.
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