sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
132 CHAPTER 7. FUTURE WORK of block coordinate relaxation is the following: consider minimizing an objective function that is a sum of a residual sum of square and a l 1 penalty term on coefficients x, (B1) minimize x ‖y − Φx‖ 2 2 + λ‖x‖ 1 . Suppose the matrix Φ can be split into several orthogonal and square submatrices, Φ = [Φ 1 , Φ 2 ,... ,Φ k ]; and x can be split into some subvectors accordingly, x =(x T 1 ,xT 2 ,... ,xT k )T . Problem (B1) is equivalent to (B2) minimize x 1 ,x 2 ,... ,x k k∑ k∑ ‖y − Φ i x i ‖ 2 2 + λ ‖x i ‖ 1 . i=1 i=1 When x 1 ,... ,x i−1 ,x i+1 ,... ,x k are fixed but not x i , the minimizer ˜x i of (B2) for x i is ˜x i = η λ/2 ⎛ ⎜ ⎝Φ T i ⎛ ⎜ ⎝y − ⎞⎞ k∑ ⎟⎟ Φ l x l ⎠⎠ , l=1 l≠i where η λ/2 is a soft-thresholding operator (for x ∈ R): { sign(x)(|x|−λ/2), if |x| ≥λ/2, η λ/2 (x) = 0, otherwise. We may iteratively solve problem (B2) and at each iteration, we rotate the soft-thresholding scheme through all subsystems. (Each subsystem is associated with a pair (Φ i ,x i ).) This method is called block coordinate relaxation (BCR) in [124, 125]. Bruce, Sardy and Tseng [124] report that in their experiments, BCR is faster than the interior point method proposed by Chen, Donoho and Saunders [27]. They also note that if some subsystems are not orthogonal, then BCR does not apply. Motivated by BCR, we may split our original optimization problem into several subproblems. (Recall that our matrix Φ is also a combination of submatrices associated with some image transforms.) We can develop another iterative method. In each iteration, we solve each subproblem one by one by assuming that the coefficients associated with other subsystems are fixed. If a subproblem corresponds to an orthogonal matrix, then the solution is simply a result of soft-thresholding of the analysis transform of the residual image. If a subproblem does not correspond to an orthogonal matrix, moreover, if it corresponds to a
7.3. ACCELERATING THE ITERATIVE ALGORITHM 133 overcomplete system, then we use our approach (that uses Newton method and LSQR) to solve it. Compared with the original problem, each subproblem has a smaller size, so hopefully this splitting approach will give us faster convergence than our previously presented global approach (that minimizes the objective function as whole). Some experiments with BCR and its comparison with our approach is an interesting topic for future research.
- 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: 7.2. MODIFYING EDGELET DICTIONARY 1
- 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
7.3. ACCELERATING THE ITERATIVE ALGORITHM 133<br />
overcomplete system, then we use our approach (that uses Newton method and LSQR) to<br />
solve it. Compared with the original problem, each subproblem has a smaller size, so hopefully<br />
this splitting approach will give us faster convergence than our previously presented<br />
global approach (that minimizes the objective function as whole).<br />
Some experiments with BCR and its comparison with our approach is an interesting<br />
topic for future research.
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