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3.4 Illustration of the discrete algorithm for forward orthonormal wavelet transform on a finite-length discrete signal. The upper graph is for cases having 3 layers. The width of each block is proportional to the length of the corresponding subsequence in the discrete signal. The bottom one is a symbolic versionforgeneralcases. ............................ 61 3.5 Multiresolution analysis of a point singularity with Haar wavelets. . . . . . 62 3.6 Two-dimensional wavelet basis functions. These are 32 × 32 images. The upper left one is a tensor product of two scaling functions. The bottom right 2 by 2 images and the (2, 2)th image are tensor products of two wavelets. The remaining images are tensor products of a scaling function with a wavelet. 63 3.7 Idealized tiling on the time-frequency plane for (a) sampling in time domain (Shannon), (b) Fourier transform, (c) Gabor analysis, (d) orthogonal wavelet transform, (e) chirplet, (f) orthonormal fan basis, (g) cosine packets, and (h) wavelet packets. . . . . .............................. 67 4.1 Quasi-sparsityanddistortioncurve. ...................... 87 4.2 Function ¯ρ (as ρ in figures) and its first and second derivatives. ¯ρ, ¯ρ ′ and ¯ρ ′′ are solid curves. The dashed curve in figure (a) is the absolute value. The dashed curve in figure (b) is the signum function. . .............. 89 4.3 Function ¯ρ at a neighborhood of origin with different values of γ, γ = 10, 20, 40, 80, 160. The dashed line is the absolute value. . .......... 90 6.1 Four test images. . . . .............................. 121 6.2 Decompositions. ................................. 123 6.3 Decaying amplitudes of coefficients. . ...................... 124 6.4 Aglobalalgorithmvs.MatchingPursuit. ................... 126 A.1 Edgelet transform of the Chinese character “Huo”: (a) is the original; (d) is the sorted coefficients; (b), (c), (e) and (f) are reconstructions based on the largest 200, 400, 800, 1600 coefficients, respectively. .............. 138 A.2 Edgelet transform of the sticky image: (a) is the original; (b) is the filtered image; (c) is the sorted coefficients; (d), (e) and (f) are the reconstructions based on the largest 100, 300 and 500 coefficients, respectively. . . . . . . . 139 xxii
A.3 Edgelet transform of the wood grain image: (a) is the original; (d) is the sorted coefficients; (b), (c), (e) and (f) are reconstructions based on the largest 1 × 10 4 ,2× 10 4 ,4× 10 4 ,8× 10 4 coefficients, respectively. . . . . . . 140 A.4 Edgelet transform of Lenna image: (a) is the original Lenna image; (b) is the filtered version; (c) is the sorted largest 5, 000 coefficients out of 428032. (d), (e) and (f) are the reconstructions based on the largest 1000, 2000 and 4000 coefficients, respectively. . . . .......................... 141 A.5 Vertices at scale j, fora8× 8 image with l = 1. The arrows shows the trend of ordering. Integers outside are the labels of vertices. . . .......... 144 A.6 Weights of pixels for one edgelet coefficient. .................. 147 B.1 X-interpolation................................... 156 B.2 Effectiveregion. ................................. 162 B.3 Basic elements of the fast edgelet-like transform. . .............. 163 B.4 Multiscale fast edgelet-like transform of artificial needle-like images. . . . . 164 B.5 Fast edgelet-like transform of wood grain image. . .............. 165 B.6 Fast edgelet-like transform of wood grain image. . .............. 166 B.7 Sinc function in (a) and its Fourier transform—blocky function in (b). . . . 167 B.8 Raised cosine function in (b) and its counterpart in time domain in (a). . . 168 B.9 Fifty percent tapered window function in (b) and its counterpart in time domain in (a). . . . . . .............................. 169 xxiii
Page 1 and 2: SPARSE IMAGE REPRESENTATION VIA COM
Page 3: I certify that I have read this dis
Page 7 and 8: To find a sparse image representati
Page 9: Abstract We consider sparse image d
Page 12 and 13: xii
Page 14 and 15: 2.3 Discussion.....................
Page 16 and 17: 6 Simulations 119 6.1 Dictionary...
Page 18 and 19: xviii
Page 21: List of Figures 2.1 Shannon’s sch
Page 25 and 26: Nomenclature Special sets N .......
Page 27 and 28: List of Abbreviations BCR .........
Page 29 and 30: Chapter 1 Introduction 1.1 Overview
Page 31 and 32: Chapter 2 Sparsity in Image Coding
Page 33 and 34: 2.1. IMAGE CODING 5 INFORMATION SOU
Page 35 and 36: 2.1. IMAGE CODING 7 2.1.2 Source an
Page 37 and 38: 2.1. IMAGE CODING 9 x ✲ T y ERROR
Page 39 and 40: 2.1. IMAGE CODING 11 where Q stands
Page 41 and 42: 2.2. SPARSITY AND COMPRESSION 13 Pr
Page 43 and 44: 2.2. SPARSITY AND COMPRESSION 15 av
Page 45 and 46: 2.2. SPARSITY AND COMPRESSION 17 wi
Page 47 and 48: 2.2. SPARSITY AND COMPRESSION 19 lo
Page 49 and 50: 2.3. DISCUSSION 21 tail compact. Th
Page 51 and 52: 2.4. PROOF 23 The index l does not
Page 53 and 54: Chapter 3 Image Transforms and Imag
Page 55 and 56: 27 Some of the figures show the bas
Page 57 and 58: 3.1. DCT AND HOMOGENEOUS COMPONENTS
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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
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Chapter 7 Future Work In the future
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7.2. MODIFYING EDGELET DICTIONARY 1
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7.3. ACCELERATING THE ITERATIVE ALG
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Appendix A Direct Edgelet Transform
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A.2. EXAMPLES 137 edgelet transform
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A.3. DETAILS 139 (a) Stick image (b
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A.3. DETAILS 141 (a) Lenna image (b
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A.3. DETAILS 143 Ordering of Dyadic
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A.3. DETAILS 145 (1,K +1), (1,K +2)
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A.3. DETAILS 147 x 1 , y 1 x 2 , y
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Appendix B Fast Edgelet-like Transf
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B.1. TRANSFORMS FOR 2-D CONTINUOUS
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B.2. DISCRETE ALGORITHM 153 B.2.1 S
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B.2. DISCRETE ALGORITHM 155 extensi
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B.2. DISCRETE ALGORITHM 157 For the
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B.3. ADJOINT OF THE FAST TRANSFORM
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B.4. ANALYSIS 161 above matrix, whi
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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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