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120 CHAPTER 6. SIMULATIONS defined by the identity, ˜λ e [I] = ∑ k 1 ,k 2 I k1 ,k 2 ˜ψe (k 1 ,k 2 ), gives a collection of “thickened edgelets”, i.e., line segments digitally sampled and of thickness a few pixels wide. Our “edgelet dictionary” is precisely this collection of representers. We will call this an edgelet dictionary even though edgelets have been previously defined in Section 3.3.1 as line segments in R 2 rather than vectors { ˜ψ e (k 1 ,k 2 ):0≤ k 1 ,k 2
6.3. DECOMPOSITION 121 10 5 5 5 20 10 10 10 30 15 15 15 40 20 20 20 50 25 25 25 60 30 30 30 10 20 30 40 50 60 Car: Lichtenstein 5 10 15 20 25 30 Pentagon 5 10 15 20 25 30 Overlapped singularities 5 10 15 20 25 30 Separate singularities Figure 6.1: Four test images. [2] They are simple but sufficient to examine whether our basic assumption is true— that different transforms will automatically represent the corresponding features in a sparse image decomposition. 6.3 Decomposition A way to test whether this approach works is to see how it decomposes the image into parts associated with included transforms. Our approach presumably provides a global sparse representation. Because of the global sparsity, if we reconstruct part of an image by using only coefficients associated with a certain transform, then this partial reconstruction should be a superposition of a few atoms (from the dictionary made by representers of the transform). So we expect to see features attributable to the associated transforms in these partial reconstructions. We consider decomposing an image into two parts—wavelet part and edgelet part. In principle, we expect to see points and patches in the wavelet part and lines in the edgelet part. Figure 6.2 shows decompositions of the four testing images. The first row is for Car. The second, third and fourth row are for Pentagon, Overlap and Separate respectively. In each row, from the left, the first squared sub-image is the original, the second is the wavelet part of the image, the third is the edgelet part of the image, and the last is the superposition of the wavelet part and the edgelet part. With appropriate parameter λ, the
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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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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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3.3. EDGELETS AND LINEAR SINGULARIT
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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 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: Chapter 6 Simulations Section 6.1 d
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.
Page 193 and 194: B.5. EXAMPLES 165 And so on. Note f
Page 195 and 196: B.6. MISCELLANEOUS 167 It takes abo
Page 197 and 198: 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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