sparse image representation via combined transforms - Convex ...

More documents

Recommendations

Info

40 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES 5 I III I III I III I III I II VI II VI II VI II VI II 4 I III I III I III I III I 3 II I II VI III VI II I II VI III VI II I II VI III VI II I II VI III VI II I II second index 2 I III I III I III I III I II VI II VI II VI II VI II 1 I III I III I III I III I 1 2 3 4 5 first index Figure 3.1: Sampling of four types of DCT at the index domain. For example, both the first index and the second index for the type-I DCT take integral values.
3.1. DCT AND HOMOGENEOUS COMPONENTS 41 elements in the 8N-D vector are zero. Equivalently, for {X[0],X[1],... ,X[N − 1]} ∈R N , I N→8N ({X[0],X[1],... ,X[N − 1]}) = {0,X[0], 0,X[1], 0,... ,0,X[N − 1], 0,... ,0}∈R 8N . } {{ } 6N Let symbol S 8N→N denote the inverse operator of I N→8N . The operator S 8N→N is actually a downsampler: S 8N→N ({•,X[0], •,X[1], •,... ,•,X[N − 1], •,... ,•}) ={X[0],X[1],... ,X[N − 1]}, } {{ } 6N where “•” represents any complex number. If DFT 8N denotes the 8N-point DFT operator and DCT IV N denotes the N-point type-IV DCT operator, then the following is true (note the index of DFT starts at zero): DCT IV N = R◦S 8N→N ◦ DFT 8N ◦I N→8N . In order to do an N-point type-IV DCT, we can first insert the sequence into an eight-times expanded sequence, and then apply the 8N-point DFT to the expanded vector, and then downsample the output of DFT, and then take the real part of the downsampled result. Note in this section, we always assume that the input is a real sequence or a real vector. The reason that we need a much higher dimension (8 times) in DFT to implement DCT is because DCT could be sampled at half integer points (multipliers of 1/2). Summary of Development in Fast Algorithms We summarize the development of fast algorithms for DCTs. Since the type-II DCT has been used in JPEG—an international standard for <strong>image</strong> coding, compression and transmission—research in this area has been very active. The 2-D DCT on small blocks— for example, an 8 × 8 block or an 16 × 16 block—is of particular importance. JPEG uses 2-D DCT on an 8 × 8block. Let’s start with the 1-D DCT. Table 3.3 summarizes some key results. (The table is originally from Antonio Ortega for a talk given at ISL, Stanford University.) By definition, the 1-D N-point DCT takes N 2 multiplications and N(N − 1) additions; when N =8, it is equivalent to 64 multiplications and 56 additions. In 1977 [28], Chen, Smith and
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 67: 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?