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162 APPENDIX B. FAST EDGELET-LIKE TRANSFORM 28 times the effort of doing the 2-D fast Fourier transform for an N by N matrix. The storage needed for the previous algorithm is proportional to N 2 . B.4.2 Effective Region The way we sample in the 2-D Fourier domain is actually equivalent to sampling in the region surrounded by a dashed curve in Figure B.2. A column of the output of the fast algorithm is an equally-spaced sample of every straight line passing through the origin within this region. The dashed curve is made by four half circles. The reason that the region looks like this is simple: dilation in the Fourier domain is equivalent to shrinkage in the time domain with the same factor. 1 Effective region 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure B.2: Effective region. B.4.3 Ill-conditioning of the Fast X-interpolation Transform The fast X-interpolation transform can be divided into three steps: if we only consider the first half of the matrix, we have 2-D Fourier transform ← orthogonal ⇓ fractional Fourier transform (FRFT) for each row ⇓ inverse 1-D Fourier transform for each column ← orthogonal Note the matrix associated with the fractional Fourier transform may have a large condition number. We leave further discussion as future research.
B.5. EXAMPLES 163 B.5 Examples B.5.1 Basic Elements for the Fast Edgelet-like Transform For any linear transform, we can regard the coefficients as inner products of the input signal with given basic elements. If it is an isometric transform, then these basic elements form an orthonormal basis. Note our fast edgelet-like transform is redundant. Hence the corresponding set of basic elements does not form a basis. In Figure B.3, we plot some basic elements of our transform. Note we actually apply our transform for square images with different size (scale). In the first (second, third) row, the squared images have sides 1 4 ( 1 2 , 1) of the side of the original image. We intentionally renormalize the basic elements so that each of them should have l 2 norm roughly equal to 1. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Figure B.3: Basic elements of the fast edgelet-like transform.
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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
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 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
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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: B.4. ANALYSIS 161 above matrix, whi
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 and 208: BIBLIOGRAPHY 179 [100] Stéphane Ma
Page 209 and 210: BIBLIOGRAPHY 181 [127] C. E. Shanno
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