sparse image representation via combined transforms - Convex ...

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114 CHAPTER 5. ITERATIVE METHODS (c) Compute p k−1 =[q k −T (k−1,k)p k−2 −T (k−2,k)p k−3 ]/T (k, k), where undefined terms are zeros for k
5.3. MINRES 115 using the preconditioned Lanczos algorithm: Set v k = Aw k − β k−1 q k−1 , α k = 〈v k ,w k 〉, v k ← v k − α k q k , Solve M ˜w k+1 = v k , Set q k+1 = v k /β k and w k+1 = ˜w k+1 /β k , where β k = 〈v k , ˜w k+1 〉 1/2 . (b) Work on the last column of T , that is: If k>2, then ( ) ( )( T (k − 2,k) ck−2 s k−2 0 ← T (k − 1,k) −¯s k−2 c k−2 T (k − 1,k) ) . If k>1, then ( T (k − 1,k) T (k, k) ) ( )( ck−1 s k−1 T (k − 1,k) ← c k−1 T (k, k) −¯s k−1 ) . Set a = |T (k, k)|/ (|T (k, k)| + |T (k +1,k)|) , b = 1− a, c k = a/ √ a 2 + b 2 , ¯s k = c k T (k +1,k)/T (k, k). Apply the k-th rotation to ξ and to the last column of T : ( ) ( )( ) ξ(k) ck s k ξ(k) ← , ξ(k +1) −¯s k c k 0 T (k, k) ← c k T (k, k)+s k T (k +1,k), T (k +1,k) ← 0.
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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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3.2. WAVELETS AND POINT SINGULARITI
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Page 91 and 92: 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
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: 5.3. MINRES 113 2. For k =1, 2,...,
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
Page 169 and 170: A.3. DETAILS 141 (a) Lenna image (b
Page 171 and 172: A.3. DETAILS 143 Ordering of Dyadic
Page 173 and 174: A.3. DETAILS 145 (1,K +1), (1,K +2)
Page 175 and 176: A.3. DETAILS 147 x 1 , y 1 x 2 , y
Page 177 and 178: Appendix B Fast Edgelet-like Transf
Page 179 and 180: B.1. TRANSFORMS FOR 2-D CONTINUOUS
Page 181 and 182: B.2. DISCRETE ALGORITHM 153 B.2.1 S
Page 183 and 184: B.2. DISCRETE ALGORITHM 155 extensi
Page 185 and 186: B.2. DISCRETE ALGORITHM 157 For the
Page 187 and 188: B.3. ADJOINT OF THE FAST TRANSFORM
Page 189 and 190: B.4. ANALYSIS 161 above matrix, whi
Page 191 and 192: 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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