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10 CHAPTER 2. SPARSITY IN IMAGE CODING necessary to transmit a set of signals with prescribed fidelity can be reduced by transmitting a set of linear combinations of the signals, instead of the original signals. Assuming Gaussian signals and RMS or mean-squared error (MSE) distortion, the Karhunen-Loéve transform (KLT) is optimal. Later, Huang and Schultheiss [84, 85] extended this idea to a system that includes a quantizer. Nowadays, PCA has become a ubiquitous technique in multivariate analysis. Its equivalent—KLT—has become a well-known method in the electrical engineering community. To find a transform that will generate even sparser coefficients than most of the existing transform is the ultimate goal of this thesis. Our key idea is to combine several state-of-theart transforms. The combination will introduce redundancy, but at the same time it will provide flexibility and potentially lead to a decomposition that is sparser than one with a single transform. When describing image coding, we cannot avoid mentioning three topics: quantization, entropy coding and predictive coding. Each of the following subsubsections is dedicated to one of these topics. Quantization All quantization schemes can be divided into two categories: scalar quantization and vector quantization. We can think of vector quantization as an extension of scalar quantization in high-dimensional space. Gray and Neuhoff 1998 [76] provides a good review paper about quantization. Two well-written textbooks are Gersho and Gray 1992 [70] and Sayood 1996 [126]. The history of quantization started from Pulse-code modulation (PCM), which was patented in 1938 by Reeves [122]. PCM was the first digital technique for conveying an analog signal over an analog channel. Twenty-five years later, a noteworthy article [37] was written by Reeves and Deloraine. It gave a historical review and an appraisal of the future of PCM. Some predictions in this article turned out to be prophetic, especially the one about the booming of digital technology. Here we give a brief description of scalar quantization. A scalar quantization is a mapping from a real-valued source to a discrete subset; symbolically, Q : R → C,
2.1. IMAGE CODING 11 where Q stands for the quantization operator and set C is a discrete subset of the real number set R: C = {y 1 ,y 2 ,... ,y K }⊂R, y 1
Page 1 and 2: SPARSE IMAGE REPRESENTATION VIA COM
Page 3: I certify that I have read this dis
Page 7 and 8: To find a sparse image representati
Page 9: Abstract We consider sparse image d
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Page 14 and 15: 2.3 Discussion.....................
Page 16 and 17: 6 Simulations 119 6.1 Dictionary...
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Page 21 and 22: List of Figures 2.1 Shannon’s sch
Page 23 and 24: A.3 Edgelet transform of the wood g
Page 25 and 26: Nomenclature Special sets N .......
Page 27 and 28: List of Abbreviations BCR .........
Page 29 and 30: Chapter 1 Introduction 1.1 Overview
Page 31 and 32: Chapter 2 Sparsity in Image Coding
Page 33 and 34: 2.1. IMAGE CODING 5 INFORMATION SOU
Page 35 and 36: 2.1. IMAGE CODING 7 2.1.2 Source an
Page 37: 2.1. IMAGE CODING 9 x ✲ T y ERROR
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Page 43 and 44: 2.2. SPARSITY AND COMPRESSION 15 av
Page 45 and 46: 2.2. SPARSITY AND COMPRESSION 17 wi
Page 47 and 48: 2.2. SPARSITY AND COMPRESSION 19 lo
Page 49 and 50: 2.3. DISCUSSION 21 tail compact. Th
Page 51 and 52: 2.4. PROOF 23 The index l does not
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Page 57 and 58: 3.1. DCT AND HOMOGENEOUS COMPONENTS
Page 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
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3.2. WAVELETS AND POINT SINGULARITI
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3.2. WAVELETS AND POINT SINGULARITI
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3.3. EDGELETS AND LINEAR SINGULARIT
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3.4. OTHER TRANSFORMS 67 uncertaint
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3.4. OTHER TRANSFORMS 69 Chirplets
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3.4. OTHER TRANSFORMS 71 Folding. A
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3.4. OTHER TRANSFORMS 73 We can app
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3.5. DISCUSSION 75 give only a few
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3.7. PROOFS 77 the ijth component o
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3.7. PROOFS 79 Similarly, we have [
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Chapter 4 Combined Image Representa
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4.2. SPARSE DECOMPOSITION 83 interi
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4.3. MINIMUM l 1 NORM SOLUTION 85 l
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4.4. LAGRANGE MULTIPLIERS 87 ρ( x
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4.5. HOW TO CHOOSE ρ AND λ 89 3 (
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4.6. HOMOTOPY 91 A way to interpret
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4.7. NEWTON DIRECTION 93 4.7 Newton
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4.9. ITERATIVE METHODS 95 1. Avoidi
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4.11. DISCUSSION 97 ρ(β) =‖β
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4.12. PROOFS 99 4.12.2 Proof of The
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4.12. PROOFS 101 case of (4.16). Co
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Chapter 5 Iterative Methods This ch
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5.1. OVERVIEW 105 the k-th iteratio
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5.1. OVERVIEW 107 5.1.4 Preconditio
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5.2. LSQR 109 among all the block d
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5.2. LSQR 111 5.2.3 Algorithm LSQR
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5.3. MINRES 113 2. For k =1, 2,...,
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5.3. MINRES 115 using the precondit
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5.4. DISCUSSION 117 From (I + S 1 )
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Chapter 6 Simulations Section 6.1 d
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6.3. DECOMPOSITION 121 10 5 5 5 20
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6.4. DECAY OF COEFFICIENTS 123 10 2
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6.5. COMPARISON WITH MATCHING PURSU
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6.6. SUMMARY OF COMPUTATIONAL EXPER
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Chapter 7 Future Work In the future
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7.2. MODIFYING EDGELET DICTIONARY 1
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7.3. ACCELERATING THE ITERATIVE ALG
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Appendix A Direct Edgelet Transform
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A.2. EXAMPLES 137 edgelet transform
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A.3. DETAILS 139 (a) Stick image (b
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A.3. DETAILS 141 (a) Lenna image (b
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A.3. DETAILS 143 Ordering of Dyadic
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A.3. DETAILS 145 (1,K +1), (1,K +2)
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A.3. DETAILS 147 x 1 , y 1 x 2 , y
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Appendix B Fast Edgelet-like Transf
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B.1. TRANSFORMS FOR 2-D CONTINUOUS
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B.2. DISCRETE ALGORITHM 153 B.2.1 S
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B.2. DISCRETE ALGORITHM 155 extensi
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B.2. DISCRETE ALGORITHM 157 For the
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B.3. ADJOINT OF THE FAST TRANSFORM
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B.4. ANALYSIS 161 above matrix, whi
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B.5. EXAMPLES 163 B.5 Examples B.5.
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B.5. EXAMPLES 165 And so on. Note f
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B.6. MISCELLANEOUS 167 It takes abo
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B.6. MISCELLANEOUS 169 The function
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Bibliography [1] Sensor Data Manage
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BIBLIOGRAPHY 173 [24] C. Victor Che
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BIBLIOGRAPHY 175 [50] David L. Dono
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BIBLIOGRAPHY 177 [75] Vivek K. Goya
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BIBLIOGRAPHY 179 [100] Stéphane Ma
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BIBLIOGRAPHY 181 [127] C. E. Shanno
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