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14 CHAPTER 2. SPARSITY IN IMAGE CODING ❄ IMAGE TRANSFORM ❄ QUANTIZATION/COMPRESSION y BITS ❄ CHANNEL BITS ❄ DECOMPRESSION/RECONSTRUCTION ❄ INVERSE TRANSFORM ỹ IMAGE ❄ Figure 2.4: Diagram of image transmission using transform coding. The sequence (or vector) y is the coefficient sequence (or vector) and ỹ is the recovered coefficient sequence (or vector).
2.2. SPARSITY AND COMPRESSION 15 average length of a bit string that is necessary to code a signal in a quantization-compression scheme. A variety of schemes have been developed in the literature. In different circumstances, some schemes approach this entropy lower bound. The field that includes this research is called vector quantization. Furthermore, if the coefficients after a certain transform are independent and identically distributed (IID), then we can deploy the same scalar quantizer for every coordinate. The scalar quantizer can nearly achieve the informatic lower bound (the lower bound specified in information theory). For example, one can use the Lloyd-Max quantizer. The statistical (or entropic) approach is elegant, but a practical disadvantage is that in image coding we generally do not know the statistical distribution of an image. Images are from an extremely high-dimensional space. Suppose a digital image has 256×256 pixels and each pixel is assigned a real value. The dimensionality of the image is then 65, 536. Viewing every image as a sample from a high-dimensional space, the number of samples is far less than the dimensionality of the space. Think about how many images we can find on the internet. The size of image databases is usually not beyond three digits; for example, the USC image database and the database at the Center for Imaging Science Sensor. It is hard to infer the probability distribution in such a high-dimensional space from such a small sample. This motivates us to find a non-stochastic approach, which is also an empirical approach. By “empirical” we mean that the method depends only on observations. The key in the non-stochastic approach is to measure the empirical sparsity. We focus on the ideas that have been developed in [46, 47]. The measure is called the weak l p norm. The following two subsections answer the following two questions: 1. What is the weak l p norm and what are the scientific reasons to invent such a norm? (Answered in Section 2.2.1.) 2. How is the weak l p norm related to asymptotic efficiency of coding? (Answered in Section 2.2.2.) 2.2.1 Weak l p Norm We start with the definition of the weak l p norm, for 1 ≤ p ≤ 2, then describe its connection with compression number, measure of numerosity and rate of recovery, and finally we introduce the concept of critical index, which plays an important role in the next section (Section 2.2.2).
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 and 38: 2.1. IMAGE CODING 9 x ✲ T y ERROR
Page 39 and 40: 2.1. IMAGE CODING 11 where Q stands
Page 41: 2.2. SPARSITY AND COMPRESSION 13 Pr
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
Page 53 and 54: Chapter 3 Image Transforms and Imag
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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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3.3. 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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