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Multidimensional Multirate Systems: Characterization, Design, and Applications MULTIDIMENSIONAL MULTIRATE SYSTEMS: CHARACTERIZATION, DESIGN, AND APPLICATIONS BY JIANPING ZHOU B.S., Xi’an Jiaotong University, 1997 M.E., Peking University, 2000 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2005 Urbana, Illinois c 2005 by Jianping Zhou. All rights reserved. ° ABSTRACT Multidimensional multirate systems have been used widely in signal processing, communications, and computer vision. Traditional multidimensional multirate systems are tensor products of one-dimensional systems. While these systems are easy to implement and design, they are inadequate to represent multidimensional signals since they cannot capture the geometric structure. Therefore, “true” multi- dimensional systems are more suited to multidimensional signals, such as images and videos. This thesis focuses on the characterization, design, and applications of “true” multidimensional multirate systems. One key property of multidimensional mul- tirate systems is perfect reconstruction, which guarantees the original input can be perfectly reconstructed from the outputs. The most popular multidimensional multirate systems are multidimensional filter banks, including critically sampled and oversampled ones. Characterizing and designing multidimensional perfect reconstruction filter banks have been challenging tasks. For critically sampled filter banks, previous one-dimensional theory cannot be extended to the multi- dimensional case due to the lack of a multidimensional factorization theorem. For oversampled filter banks, existing one-dimensional theory does not work in the multidimensional case. We derive complete characterizations of multidimen- sional critically sampled and oversampled filter banks and propose novel design methods for multidimensional filter banks. We illustrate our multidimensional multirate system theory by several image processing applications. iii To my wife Jie and my baby Ruijie. iv ACKNOWLEDGMENTS I would like to thank my advisor, Professor Minh N. Do, for his continuing support and guidance during my three-year Ph.D. research. The discussions and weekly meetings with him inspired many ideas of this work. His enthusiasm and encour- agement kept me working towards the completion of this thesis. I would also thank my collaborators on this work, Professor Jelena Kovaceviˇ c´ and Arthur L. Cunha, for fruitful discussions. I would like to express my grati- tude to my Ph.D. committee members, Professor Richard Blahut, Professor Yoram Bresler, Professor Pierre Moulin, and Professor Andrew Singer, for their valuable advice. I am grateful to my friends and colleagues at the University of Illinois at Urbana-Champaign, with whom I spent wonderful years. Finally, I would give my deepest appreciation to my wife Jie and my parents for their constant love and support. v TABLE OF CONTENTS LISTOFFIGURES.............................. ix LISTOFTABLES .............................. xiii LISTOFABBREVIATIONS. xiv CHAPTER1 INTRODUCTION ...................... 1 1.1 Motivation.............................. 1 1.2 ProblemStatement ......................... 5 1.3 ThesisOutline............................ 5 CHAPTER 2 MULTIDIMENSIONAL PERFECT RECONSTRUCTION FILTERBANKS ............................. 9 2.1 Introduction............................. 9 2.2 Preliminaries ............................ 11 2.2.1 Multidimensional signals and notations . 11 2.2.2 Multidimensional sampling and polyphase representation . 12 2.3 Multidimensional Perfect Reconstruction Filter Banks . ..... 14 2.4 Reduction Characterization . 16 2.5 Special Paraunitary Matrices and Orthogonal Filter Banks .... 23 2.5.1 Special paraunitary matrices . 23 2.5.2 Reduction characterization of orthogonal filter banks . 27 2.6 Conclusion ............................. 28 CHAPTER 3 MULTIDIMENSIONAL ORTHOGONAL FILTER BANKS ANDTHECAYLEYTRANSFORM. 30 3.1 Introduction............................. 30 3.2 CayleyTransform.......................... 33 3.3 Orthogonal IIR Filter Banks . 36 3.3.1 Complete characterization . 36 3.3.2 Design examples of 2-D quincunx IIR filter banks . 38 3.4 Orthogonal FIR Filter Banks . 41 3.5 Two-Channel Orthogonal FIR Filter Banks . 44 3.5.1 Complete characterization . 44 3.5.2 Design of 2-D quincunx orthogonal FIR filter banks . 47 vi 3.5.3 Parameteranalysis . 47 3.5.4 Designexamples . 48 3.6 Special Paraunitary Matrices and Cayley Transform . 54 3.6.1 Cayley transform of special paraunitary matrices . 54 3.6.2 Two-channel special paraunitary matrices . 54 3.6.3 Two-channel special paraunitary FIR matrices . 57 3.6.4 Quincunx orthogonal filter bank design . 59 3.7 Conclusion and Future Work . 62 CHAPTER 4 ORTHOGONAL FILTER BANKS WITH DIRECTIONAL VANISHINGMOMENTS . 64 4.1 Introduction............................. 64 4.2 Orthogonal Filter Banks and Lattice Structure . 65 4.3 Orthogonal Filter Banks with Directional Vanishing Moments . 68 4.3.1 Formulation . 68 4.3.2 Threesimplecases . 69 4.3.3 Reduction and combination characterization . 71 4.4 Closed-Form Expression . 76 4.5 Orthogonal Filter Banks with Higher-Order Directional Vanishing Moments .............................. 80 4.6 Conclusion ............................. 84 CHAPTER 5 MULTIDIMENSIONAL OVERSAMPLED FIR FILTER BANKS .................................. 85 5.1 Introduction............................. 85 5.2 Preliminaries ............................ 86 5.2.1 Problemsetup ....................... 86 5.2.2 Algebraic geometry and Grobnerbases¨ . 89 5.3 Multidimensional Nonsubsampled FIR Filter Banks: Characteri- zation ................................ 90 5.3.1 Introduction . 90 5.3.2 Existence of synthesis filters . 91 5.3.3 Computation of synthesis filters . 96 5.3.4 Characterization of synthesis filters . 98 5.4 Multidimensional Nonsubsampled FIR Filter Banks: Design . 100 5.4.1 Designmethod . 100 5.4.2 Optimal filter design . 102 5.4.3 Two-channel design . 110 5.5 Multidimensional Oversampled FIR Filter Banks . 117 5.5.1 Oversampled filter bank characterization . 117 5.5.2 Oversampled filter bank design . 119 5.6 Conclusion .............................122 vii CHAPTER 6 MULTIDIMENSIONAL MULTICHANNEL FIR DECON- VOLUTION................................123 6.1 Introduction.............................123 6.2 Multichannel FIR Deconvolution Using Grobner¨ Bases . 125 6.2.1 Problem formulation . 125 6.2.2 Complexity and numerical stability . 127 6.3 Optimization of FIR Deconvolution Filters . 127 6.4 Simulations .............................132 6.4.1 Deconvolution filters . 133 6.4.2 Deconvolution with impulsive noise . 134 6.4.3 Deconvolution with independent noise . 139 6.5 Conclusion .............................139 CHAPTER 7 NONSUBSAMPLED CONTOURLET TRANSFORM . 141 7.1 Introduction.............................141 7.2 Construction.............................142 7.2.1 Nonsubsampled pyramids . 142 7.2.2 Nonsubsampled directional filter banks . 143 7.2.3 Fast implementation . 145 7.2.4 Nonsubsampled contourlet transform . 148 7.3 Application: Image Enhancement . 148 7.3.1 Image enhancement algorithm . 148 7.3.2 Numerical experiments . 150 7.4 Conclusion .............................151 CHAPTER 8 CONCLUSIONS AND DIRECTIONS . 154 8.1 Conclusions.............................154 8.2 FutureResearch...........................155 REFERENCES ................................157 AUTHOR’SBIOGRAPHY. 171 viii LIST OF FIGURES Figure Page 1.1 Multidimensional N-channel filter bank: Hi and Gi are MD anal- ysis and synthesis filters, respectively; D is an M M sampling matrix. ...............................× 1 1.2 Comparison between traditional separable systems and new non- separable systems. (a) Original image. (b) Discrete wavelet trans- form. (c) Contourlet transform. 3 1.3 Classification of MD perfect reconstruction filter banks. ...... 4 2.1 Relationship among orthogonal filter banks, paraunitary matrices, and special paraunitary matrices. Orthogonal filter banks are char- acterized by paraunitary matrices in the polyphase domain. Parau- nitary matrices are characterized by special paraunitary matrices andphasefactors. .......................... 10 2.2 (a) Sampling by 2 in two dimensions. (b) Quincunx sampling. 13 2.3 (a) Multidimensional N-channel filter bank: Hi and Gi are MD analysis and synthesis filters, respectively; D is an M M sam- pling matrix with sampling rate P = det D . (b) Polyphase× rep- | | resentation: Hp are MD analysis and synthesis polyphase trans- P 1 form matrices, respectively; lj j=0− is the set of all integer vec- tors in (D).............................{ } 15 N 3.1 One-to-one mapping between paraunitary matrices and para-skew- Hermitian matrices via the Cayley transform. 33 3.2 Magnitude frequency responses of two orthogonal lowpass fil- ters with second-order vanishing moment, obtained by the Cayley transform. (a) Example 3.1. (b) Example 3.2. 40 3.3 Magnitude frequency responses of 24-tap orthogonal lowpass fil- ters with third-order vanishing moment, obtained by the Cayley transform. (a) Two possible solutions, also found by Kovaceviˇ c´ and Vetterli. (b) Four possible solutions. They are new filters. 50 3.4 Tenth iteration of the top two filters in Fig 3.3(b), obtained by the Cayleytransform........................... 52 ix 3.5 Nonlinear approximation comparisons. (a) Original “Barbara” image. Reconstruction images by: (b) an old filter (SNR = 19.7 dB); (c) a new filter (SNR = 20.1 dB). .............. 53 3.6 One-to-one mapping between 2 2 special paraunitary matrices
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