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170 APPENDIX B. FAST EDGELET-LIKE TRANSFORM
Bibliography [1] Sensor Data Management System web site. http://www.mbvlab.wpafb.af.mil/public/ sdms/. [2] USC-SIPI Image Database. http://sipi.usc.edu/services/database/Database.html. [3] H. K. Aghajan. Subspace Techniques for Image Understanding and Computer Vision. PhD thesis, Stanford University, March 1995. [4] O. Axelsson. Incomplete block matrix factorization preconditioning methods: the ultimate answer? J. Comput. Appl. Math., 12:3–18, 1985. [5] O. Axelsson. A general incomplete block-matrix factorization method. Linear Algebra Appl., 74:179–90, 1986. [6] David H. Bailey and Paul N. Swarztrauber. The fractional Fourier transform and applications. SIAM Rev., 33(3):389–404, 1991. [7] Richard G. Baraniuk and Douglas L. Jones. Shear madness: new orthonormal bases and frames using chirp functions. IEEE Transactions on Signal Processing, 41(12):3543–9, December 1993. [8] R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition. SIAM, Philadelphia, PA, 1994. [9] Michele Benzi, Carl D. Meyer, and Miroslav ˚Tuma. A <strong>sparse</strong> approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput., 17(5):1135– 1149, 1996. 171
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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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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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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 )
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.
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: B.6. MISCELLANEOUS 169 The function
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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