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164 APPENDIX B. FAST EDGELET-LIKE TRANSFORM (a) Image (b) Image (c) Image (d) Image 10 10 10 10 20 20 20 20 30 10 20 30 30 10 20 30 30 10 20 30 30 10 20 30 (e) MSET (2−5) (f) MSET (2−5) (g) MSET (2−5) (h) MSET (2−5) 20 20 20 20 40 40 40 40 60 60 60 60 80 80 80 80 100 100 100 100 120 120 120 120 20 40 60 20 40 60 20 40 60 20 40 60 Figure B.4: Multiscale fast edgelet-like transform of artificial needle-like images. B.5.2 Edgelet-like Transform for Some Artificial Images Since this algorithm is designed to capture the linear features in an image, it will be interesting to see how it works on some artificial images made by a single linear singularity. The first row of Figure B.4 shows some needle-like images. The second row is their multiscale fast edgelet-like transforms. To explain the images of the coefficients, we need to explain how we arrange the output of our algorithm. Suppose we only do the transform for the whole image (scale-0 transform). Then as in B.2.4, the input I is mapped to a coefficient matrix with two components [E1 1,E2 1 ]. When we divide the image into 2 × 2 block images (scale-1 transform), each subimage is mapped to a coefficient sub-matrix with two components: [ ] [ ] I11 I 12 E 1 → 11 E11 2 E12 1 E12 2 . I 21 I 22 E21 1 E21 2 E22 1 E22 2
B.5. EXAMPLES 165 And so on. Note for a fixed scale, the output matrix has the same number of rows, but the number of columns expands by 2. The second row in Figure B.4 is illustrations of output by taking four scales. So the number of rows are expanded by 4. The dark points are associated with coefficients with large amplitudes. Obviously there are few coefficients with significantly large amplitudes and all the rest are small. This matches our original goal to design this transformation. B.5.3 Edgelet-like Transform for Some Real Images (a) Woodgrain image 100 200 300 400 500 100 200 300 400 500 (scale=2) (scale=1) (scale=0) 1600 2500 4000 1400 1200 1000 800 2000 1500 1000 3000 2000 1000 200 400 600 800 1000 200 400 600 800 1000 200 400 600 800 1000 Coeff. Matrix (scale=2) Coeff. Matrix (scale=1) Coeff. Matrix (scale=0) Figure B.5: Fast edgelet-like transform of wood grain image. Here real means that they are from some other independent sources and are not made intentionally for our method. In these images, the linear features are embedded, and we see that our transform still captures them. Figure B.5 is a wood grain image. There are many needle-like objects in the image, and they are pretty much at the the same scale—having the same length—and along the same direction. We do our transform at three scales (0, 1, 2 corresponding to no division of 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
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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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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
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: B.5. EXAMPLES 163 B.5 Examples B.5.
Page 195 and 196: B.6. MISCELLANEOUS 167 It takes abo
Page 197 and 198: B.6. MISCELLANEOUS 169 The function
Page 199 and 200: Bibliography [1] Sensor Data Manage
Page 201 and 202: BIBLIOGRAPHY 173 [24] C. Victor Che
Page 203 and 204: BIBLIOGRAPHY 175 [50] David L. Dono
Page 205 and 206: BIBLIOGRAPHY 177 [75] Vivek K. Goya
Page 207 and 208: BIBLIOGRAPHY 179 [100] Stéphane Ma
Page 209 and 210: BIBLIOGRAPHY 181 [127] C. E. Shanno
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