sparse image representation via combined transforms - Convex ...

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180 BIBLIOGRAPHY [113] Yurii Nesterov and Arkadii Nemirovskii. Interior-point Polynomial Algorithms in Convex Programming, volume13ofSIAM Studies in Applied Mathematics. SIAM, Philadelphia, PA, 1994. [114] Henry J. Nussbaumer. Fast Fourier Transform and Convolution Algorithms. Springer- Verlag, 1982. [115] Kramer H. P. and Mathews M. V. A linear coding from transmitting a set of correlated signals. IRE Trans. Inform. Theory, 2:41–46, September 1956. [116] C. C. Paige and M. A. Saunders. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal., 12(4):617–29, 1975. [117] Christopher C. Paige and Michael A. Saunders. LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software, 8(1):43–71, 1982. [118] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C: The Art of Scientific Computing. Cambridge, second edition, 1995. [119] A. G. Ramm and A. I. Katsevich. Radon Transform and Local Tomography. CRC Press, 1996. [120] B. D. Rao. Signal processing with the sparseness constraint. In Proceedings of ICASSP, pages III–1861–4, 1998. [121] K. R. Rao and P. Yip. Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic Press, 1990. [122] A. H. Reeves. French Patent No. 852,183, October 3 1938. [123] R.T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [124] S. Sardy, A. Bruce, and P. Tseng. Block coordinate relaxation methods for nonparametric signal denoising with wavelet dictionaries. Web page, October 1998. http://www.math.washington.edu/˜tseng/papers.html. [125] S. Sardy, A. G. Bruce, and P. Tseng. Block coordinate relaxation methods for nonparametric signal de-noising. Received from E. Candès, October 1998. [126] K. Sayood. Introduction to Data Compression. Morgan Kaufmann Publishers, 1996.
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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 )
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Chapter 6 Simulations Section 6.1 d
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6.3. DECOMPOSITION 121 10 5 5 5 20
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6.4. DECAY OF COEFFICIENTS 123 10 2
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6.5. COMPARISON WITH MATCHING PURSU
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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
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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
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: BIBLIOGRAPHY 179 [100] Stéphane Ma
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