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58 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES HIGH PASS FILTER DOWN SAMPLER SIGNAL HIGH PASS FILTER DOWN SAMPLER LOW PASS FILTER DOWN SAMPLER LOW PASS FILTER DOWN SAMPLER HIGH PASS FILTER LOW PASS FILTER DOWN SAMPLER DOWN SAMPLER Figure 3.3: Illustration of a filter bank for forward orthonormal wavelet transform. 3.2.3 Discrete Algorithm Nowadays, computers are widely used in scientific computing, and most of a signal processing job is done by a variety of chips. All chips use digital signal processing (DSP) technology. It is not an exaggeration to say that a technique is crippled if it does not have a corresponding discrete algorithm. A nice feature of wavelets is that it has a fast discrete algorithm. For length N signal, the order of computational complexity is O(N). It can be formulated as an orthogonal transform. Each of the wavelets can have finite support. To explain how the discrete wavelet transform (DWT) works, we need to introduce some notation. Consider a function f ∈ L 2 . Let α j k denote a coefficient of a transform of f; αj k corresponds to the kth scaling function at the jth scale: ∫ α j k = f(x)φ(2 j x − k)dx. Let β j k also denote a coefficient of a transform of f corresponding to the k ′ th wavelet ′ function at the jth scale: ∫ β j k = f(x)ψ(2 j x − k ′ )dx. ′ Note that coefficients α j k and βj k are at the same scale. The following two relations can be ′
3.2. WAVELETS AND POINT SINGULARITIES 59 derived from the two-scale relationships in (3.21) and (3.25): ∫ α j k = f(x)φ(2 j x − k)dx (3.21) = ∑ ∫ h(n) f(x)φ(2 j+1 x − 2k − n)dx n∈Z = ∑ n∈Z h(n)α j+1 n+2k ; (3.31) ∫ β j k = f(x)ψ(2 j x − k)dx (3.25) = ∑ ∫ g(n) f(x)φ(2 j+1 x − 2k − n)dx n∈Z = ∑ n∈Z g(n)α j+1 n+2k . (3.32) These two relations, (3.31) and (3.32), determine the two-scale relationship in the discrete algorithm. Suppose that function f resides in a subspace V j : f ∈V j , j ∈ N. By definition, there must exist a sequence of coefficients α j = {α j k ,k ∈ Z}, such that f = ∑ n∈Z α j k φ(2j x − k). The subspace V j can be divided as V j = V 0 ⊕W 0 ⊕ ...⊕W j−1 . We want to know the decompositions of function f in subspaces V 0 , W 0 , W 1 , ... ,and W j−1 .Letβ l = {βk l ,k ∈ Z} denote the set of coefficients that correspond to the wavelets in the subspace W l ,1≤ l
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SPARSE IMAGE REPRESENTATION VIA COM
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I certify that I have read this dis
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To find a sparse image representati
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Abstract We consider sparse image d
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xii
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2.3 Discussion.....................
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6 Simulations 119 6.1 Dictionary...
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xviii
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List of Figures 2.1 Shannon’s sch
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A.3 Edgelet transform of the wood g
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Nomenclature Special sets N .......
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List of Abbreviations BCR .........
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Chapter 1 Introduction 1.1 Overview
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Chapter 2 Sparsity in Image Coding
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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 and 42: 2.2. SPARSITY AND COMPRESSION 13 Pr
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
Page 53 and 54: Chapter 3 Image Transforms and Imag
Page 55 and 56: 27 Some of the figures show the bas
Page 57 and 58: 3.1. DCT AND HOMOGENEOUS COMPONENTS
Page 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
Page 85: 3.2. WAVELETS AND POINT SINGULARITI
Page 93 and 94: 3.3. EDGELETS AND LINEAR SINGULARIT
Page 95 and 96: 3.4. OTHER TRANSFORMS 67 uncertaint
Page 97 and 98: 3.4. OTHER TRANSFORMS 69 Chirplets
Page 99 and 100: 3.4. OTHER TRANSFORMS 71 Folding. A
Page 101 and 102: 3.4. OTHER TRANSFORMS 73 We can app
Page 103 and 104: 3.5. DISCUSSION 75 give only a few
Page 105 and 106: 3.7. PROOFS 77 the ijth component o
Page 107 and 108: 3.7. PROOFS 79 Similarly, we have [
Page 109 and 110: Chapter 4 Combined Image Representa
Page 111 and 112: 4.2. SPARSE DECOMPOSITION 83 interi
Page 113 and 114: 4.3. MINIMUM l 1 NORM SOLUTION 85 l
Page 115 and 116: 4.4. LAGRANGE MULTIPLIERS 87 ρ( x
Page 117 and 118: 4.5. HOW TO CHOOSE ρ AND λ 89 3 (
Page 119 and 120: 4.6. HOMOTOPY 91 A way to interpret
Page 121 and 122: 4.7. NEWTON DIRECTION 93 4.7 Newton
Page 123 and 124: 4.9. ITERATIVE METHODS 95 1. Avoidi
Page 125 and 126: 4.11. DISCUSSION 97 ρ(β) =‖β
Page 127 and 128: 4.12. PROOFS 99 4.12.2 Proof of The
Page 129 and 130: 4.12. PROOFS 101 case of (4.16). Co
Page 131 and 132: Chapter 5 Iterative Methods This ch
Page 133 and 134: 5.1. OVERVIEW 105 the k-th iteratio
Page 135 and 136: 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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