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90 CHAPTER 4. COMBINED IMAGE REPRESENTATION 0.2 0.18 0.16 0.14 0.12 160 80 40 20 0.1 0.08 10 0.06 0.04 0.02 0 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Figure 4.3: Function ¯ρ at a neighborhood of origin with different values of γ, γ = 10, 20, 40, 80, 160. The dashed line is the absolute value.
4.6. HOMOTOPY 91 A way to interpret the above result is that for the residual r = y − Φˆx, the maximum amplitude of the analysis transform of the residual ‖Φ T r‖ ∞ is upper bounded by λ 2 . Hence if λ is small enough and if Φ T is norm preserving—each column of Φ has almost the same l 2 norm—then the deviation of the reconstruction based on ̂x, Φ̂x, from the image y is upper bounded by a small quantity (literally λ 2 ) at each direction given by the columns of Φ. So if we choose a small λ, the corresponding reconstruction cannot be much different from the original image. After choosing ρ and λ, we have the Hessian and gradient of the objective function (later denoted by f(x)). For fixed vector x k =(x k 1 ,xk 2 ,... ,xk N )T ∈ R N , the gradient at x k is ⎛ g(x k )=−2Φ T y +2Φ T Φx k + λ ⎜ ⎝ ¯ρ ′ (x k 1 ) ⎞ . ⎟ ⎠ , (4.7) ¯ρ ′ (x k N ) where x k i is the i-th element of vector x k , i =1, 2,... ,N, and the Hessian is the matrix ⎛ H(x k )=2Φ T Φ+λ ⎜ ⎝ ¯ρ ′′ (x k 1 ) ⎞ . .. ⎟ ⎠ , (4.8) ¯ρ ′′ (x k N ) where ¯ρ ′′ is the second derivative of function ¯ρ. 4.6 Homotopy Based on the previous choice of ρ and ¯ρ, when the parameter γ goes to +∞, ρ(x, γ) = ∑ N i=1 ¯ρ(x i,γ) goes to function ‖x‖ 1 . Note our ultimate goal is to solve the minimum l 1 norm problem (P 1 ). Considering the Lagrangian multiplier method, for a fixed λ, wesolve minimize ‖y − Φx‖ 2 x 2 + λ‖x‖ 1 . (4.9) When γ goes to +∞, will the solution of the problem in (4.2) converge to the solution of the problem in (4.9)? We prove the convergence under the following two assumptions. The first one is easy to satisfy. The second one seems too rigorous and may not be true for many situations that we
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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
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2.1. IMAGE CODING 7 2.1.2 Source an
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2.1. IMAGE CODING 9 x ✲ T y ERROR
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2.1. IMAGE CODING 11 where Q stands
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2.2. SPARSITY AND COMPRESSION 13 Pr
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2.2. SPARSITY AND COMPRESSION 15 av
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2.2. SPARSITY AND COMPRESSION 17 wi
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2.2. SPARSITY AND COMPRESSION 19 lo
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2.3. DISCUSSION 21 tail compact. Th
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2.4. PROOF 23 The index l does not
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Chapter 3 Image Transforms and Imag
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27 Some of the figures show the bas
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3.1. DCT AND HOMOGENEOUS COMPONENTS
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Page 67 and 68: 3.1. DCT AND HOMOGENEOUS COMPONENTS
Page 81 and 82: 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: 4.5. HOW TO CHOOSE ρ AND λ 89 3 (
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
Page 137 and 138: 5.2. LSQR 109 among all the block d
Page 139 and 140: 5.2. LSQR 111 5.2.3 Algorithm LSQR
Page 141 and 142: 5.3. MINRES 113 2. For k =1, 2,...,
Page 143 and 144: 5.3. MINRES 115 using the precondit
Page 145 and 146: 5.4. DISCUSSION 117 From (I + S 1 )
Page 147 and 148: Chapter 6 Simulations Section 6.1 d
Page 149 and 150: 6.3. DECOMPOSITION 121 10 5 5 5 20
Page 151 and 152: 6.4. DECAY OF COEFFICIENTS 123 10 2
Page 153 and 154: 6.5. COMPARISON WITH MATCHING PURSU
Page 155 and 156: 6.6. SUMMARY OF COMPUTATIONAL EXPER
Page 157 and 158: Chapter 7 Future Work In the future
Page 159 and 160: 7.2. MODIFYING EDGELET DICTIONARY 1
Page 161 and 162: 7.3. ACCELERATING THE ITERATIVE ALG
Page 163 and 164: Appendix A Direct Edgelet Transform
Page 165 and 166: A.2. EXAMPLES 137 edgelet transform
Page 167 and 168: 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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