sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
116 CHAPTER 5. ITERATIVE METHODS (c) Compute p k−1 =[w k −T (k−1,k)p k−2 −T (k−2,k)p k−3 ]/T (k, k), where undefined terms are zeros for k
5.4. DISCUSSION 117 From (I + S 1 ) 1/2 = T 1 (I + D 1 ) 1/2 T T 1 , (I + S 1 ) −1/2 = T 1 (I + D 1 ) −1/2 T T 1 , and there are fast algorithms to implement matrix-vector multiplication with matrix T 1 , T1 T ,(I + D 1) −1/2 and (I + D 1 ) 1/2 , so there are fast algorithms to multiply with a 11 and a 12 . But for a 22 ,noneofT 1 , inverse of T 1 , T 2 and inverse of T 2 can simultaneously diagonalize I + S 2 and (I + S 1 ) −1 . Hence there is no trivial fast algorithm to multiply with matrix a 22 . In general, a complete Cholesky factorization will destroy the structure of matrices from which we can have fast algorithms. The fast algorithms of matrix-vector multiplication are so vital for solving large-scale problems with iterative methods (and an intrinsic property of our problem is that the size of data is huge) that we do not want to sacrifice the existence of fast algorithms. Preconditioning may reduce the number of iterations, but the amount of computation within each iteration is increased significantly, so overall, the total amount of computing may increase. Because of this philosophy, we stop plodding in the direction of complete Cholesky factorization. 5.4.2 Sparse Approximate Inverse The idea of a sparse approximate inverse (SAI) is that if we can find an approximate eigendecomposition of the inverse matrix, then we can use it to precondition the linear system. More precisely, suppose Z is an orthonormal matrix and at the same time the columns of Z, denoted by z i , i =1, 2,... ,N,areÃ-conjugate orthogonal to each other: Z =[z 1 ,z 2 ,... ,z N ], and ZÃZT = D, where D is a diagonal matrix. We have à = ZT DZ and Ã−1 = Z T D −1 Z. If we can find such a Z, then (D −1/2Z)Ã(D−1/2 Z) T ≈ I, so(D −1/2 Z) is a good preconditioner. We now consider its block matrix analogue. Actually in the previous subsection, we have already argued that when we have the block matrix version of a preconditioner, each element of the block matrix should be able to be associated with a fast algorithm that is based on matrix multiplication with matrices T 1 ,T 2 ,... ,T m and matrix-vector multiplication with diagonal matrices. Unfortunately, a preconditioner derived by using the idea of
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- 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
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- 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
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- Page 125 and 126: 4.11. DISCUSSION 97 ρ(β) =‖β
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- Page 129 and 130: 4.12. PROOFS 101 case of (4.16). Co
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- Page 139 and 140: 5.2. LSQR 111 5.2.3 Algorithm LSQR
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- 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
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- 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
116 CHAPTER 5. ITERATIVE METHODS<br />
(c) Compute p k−1 =[w k −T (k−1,k)p k−2 −T (k−2,k)p k−3 ]/T (k, k), where<br />
undefined terms are zeros for k
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