sparse image representation via combined transforms - Convex ...

More documents

Recommendations

Info

48 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES The basic idea is that if the covariance matrix can be diagonalized by a type of DCT, then the covariance matrix must be written as a summation of two matrices. Moreover, the two matrices must have some particular properties. One matrix must be Toeplitz and symmetric, or such a matrix multiplied by some diagonal matrices. The other matrix must be Hankel and counter symmetric, or counter antisymmetric, or a matrix obtained by shifting such a matrix and then multiplying it with a diagonal matrix. Furthermore, for an N × N covariance matrix, the dimensionality (or the degrees of freedom) of the matrix is no higher than N. To be more clear, we have to use some notation. Let N denote the length of the signal, so the size of the covariance matrix is N × N. Letb k ,k =0, 1,... ,N − 1 be the sequence defined in (3.12). Let B be the diagonal matrix ⎛ ⎞ b 0 b 1 B = ⎜ . . (3.14) .. ⎟ ⎝ ⎠ b N−1 Let {c 0 ,c 1 ,c 2 ,... ,c 2N−1 } and {c ′ 0 ,c′ 1 ,c′ 2 ,... ,c′ 2N−1 } be two finite real-valued sequences whose elements c i and c ′ i satisfy a symmetric or an antisymmetric condition: { ci = c 2N−i i =0, 1, 2,... ,2N − 1, c ′ i = −c′ 2N−i i =0, 1, 2,... ,2N − 1. Note that when i = N, the second condition implies c ′ N = 0. Suppose C 1 and C 1 ′ are the Toeplitz and symmetric matrices ⎛ ⎞ ⎛ ⎞ c 0 c 1 ... c N−1 c ′ 0 c ′ 1 ... c ′ N−1 c 1 c 0 c N−2 c ′ 1 c ′ 0 c ′ N−2 C 1 = ⎜ . , and C ⎝ . .. . ⎟ 2 = ⎜ . .(3.15) ⎠ ⎝ . .. . ⎟ ⎠ c N−1 c N−2 ... c 0 c ′ N−1 c ′ N−2 ... c ′ 0
3.1. DCT AND HOMOGENEOUS COMPONENTS 49 Suppose C 2 and C ′ 2 are the two Hankel matrices ⎛ C 2 = ⎜ ⎝ ⎞ c 1 c 2 ... c N−1 c N c 2 c 3 ... c N c N−1 . . . . , (3.16) c N−1 c N ... c 3 c 2 ⎟ ⎠ c N c N−1 ... c 2 c 1 . .. ⎛ C 2 ′ = ⎜ ⎝ c ′ 1 c ′ 2 ... c ′ N−1 0 c ′ 2 c ′ 3 ... 0 −c ′ N−1 . . . .. . . c ′ N−1 0 ... −c ′ 3 −c ′ 2 0 −c ′ N−1 ... −c ′ 2 −c ′ 1 ⎞ . (3.17) ⎟ ⎠ Note that C 2 is counter symmetric (if a {i,j} is the {i, j}th element of the matrix, then a {i,j} = a {N+1−i,N+1−j} ,for i, j =1, 2,... ,N)andC 2 ′ is counter antisymmetric (if a {i,j} is the {i, j}th element of the matrix, then a {i,j} = −a {N+1−i,N+1−j} ,fori, j =1, 2,... ,N). Let D denote a down-right-shift operator on matrix, such that ⎛ D(C 2 )= ⎜ ⎝ ⎞ c 0 c 1 ... c N−2 c N−1 c 1 c 2 ... c N−1 c N . . . . , (3.18) c N−2 c N−1 ... c 4 c 3 ⎟ ⎠ c N−1 c N ... c 3 c 2 . .. and ⎛ D(C 2)= ′ ⎜ ⎝ c ′ 0 c ′ 1 ... c ′ N−2 c ′ N−1 c ′ 1 c ′ 2 ... c ′ N−1 0 . . . .. . . c ′ N−2 c ′ N−1 ... −c ′ 4 −c ′ 3 c ′ N−1 0 ... −c ′ 3 −c ′ 2 ⎞ . (3.19) ⎟ ⎠ Note that D(C 2 )andD(C 2 ′ ) are still Hankel matrices.
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 and 22:
List of Figures 2.1 Shannon’s sch
Page 23 and 24:
A.3 Edgelet transform of the wood g
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
Page 75: 3.1. DCT AND HOMOGENEOUS COMPONENTS
Page 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
Page 93 and 94: 3.3. EDGELETS AND LINEAR SINGULARIT
Page 95 and 96: 3.4. OTHER TRANSFORMS 67 uncertaint
Page 97 and 98: 3.4. OTHER TRANSFORMS 69 Chirplets
Page 99 and 100: 3.4. OTHER TRANSFORMS 71 Folding. A
Page 101 and 102: 3.4. OTHER TRANSFORMS 73 We can app
Page 103 and 104: 3.5. DISCUSSION 75 give only a few
Page 105 and 106: 3.7. PROOFS 77 the ijth component o
Page 107 and 108: 3.7. PROOFS 79 Similarly, we have [
Page 109 and 110: Chapter 4 Combined Image Representa
Page 111 and 112: 4.2. SPARSE DECOMPOSITION 83 interi
Page 113 and 114: 4.3. MINIMUM l 1 NORM SOLUTION 85 l
Page 115 and 116: 4.4. LAGRANGE MULTIPLIERS 87 ρ( x
Page 117 and 118: 4.5. HOW TO CHOOSE ρ AND λ 89 3 (
Page 119 and 120: 4.6. HOMOTOPY 91 A way to interpret
Page 121 and 122: 4.7. NEWTON DIRECTION 93 4.7 Newton
Page 123 and 124: 4.9. ITERATIVE METHODS 95 1. Avoidi
Page 125 and 126: 4.11. DISCUSSION 97 ρ(β) =‖β
Page 127 and 128:
4.12. PROOFS 99 4.12.2 Proof of The
Page 129 and 130:
4.12. PROOFS 101 case of (4.16). Co
Page 131 and 132:
Chapter 5 Iterative Methods This ch
Page 133 and 134:
5.1. OVERVIEW 105 the k-th iteratio
Page 135 and 136:
5.1. OVERVIEW 107 5.1.4 Preconditio
Page 137 and 138:
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
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 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
show all

sparse image representation via combined transforms - Convex ...

Create successful ePaper yourself

Delete template?

Save as template?