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34 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES and ⎛ P 2 ⎜ ⎝ x 0 x 1 . x mn−1 ⎛ ⎞ ⎟ = ⎠ mn×1 ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ x 0 x ṃ . . x (n−1)m x 1 x 1+m . x 1+(n−1)m x (m−1) ⎞ ⎟ ⎠ • • • x (m−1)+m . ⎞ ⎟ ⎠ n×1 n×1 ⎞ ⎟ ⎠ ⎞ . (3.4) ⎟ ⎠ x (m−1)+(n−1)m n×1 Obviously, we have P 1 P1 T = I mn and P 2 P2 T = I mn , where I mn is an mn × mn identity matrix. Like all permutation matrices, P 1 and P 2 are orthogonal matrices with only one nonzero element in each row and column, and these nonzero elements are equal to one. Let Ω mn denote a diagonal matrix, and let I n be an n × n identity matrix. For ω mn = 2π −i e mn , the matrix Ω mn is ⎡ Ω mn = ⎢ ⎣ I n ⎛ 1 ⎜ ⎝ ω 1×1 mn . .. ω (n−1)×1 mn ⎞ ⎟ ⎠ . .. ⎛ 1 ⎜ ⎝ ω 1×(m−1) mn . .. ω (n−1)×(m−1) mn ⎞ ⎟ ⎠ ⎤ . (3.5) ⎥ ⎦
3.1. DCT AND HOMOGENEOUS COMPONENTS 35 Armed with these definitions, we are now able to state the Butterfly Theorem. The following is a formal statement. Theorem 3.2 (Butterfly Theorem) Consider any two positive integers m and n. Let DFT mn , DFT m and DFT n be the mn-, m- andn- point discrete Fourier transform matrices respectively. The matrices P 1 , P 2 and Ω mn are defined in (3.3), (3.4) and (3.5), respectively. The following equality is true: ⎡ ⎤ ⎡ ⎤ DFT m DFT n DFT mn = P1 T . .. P 2 Ω mn . .. P2 T . (3.6) ⎢ ⎥ ⎢ ⎥ ⎣ DFT m ⎦ ⎣ DFT n ⎦ } {{ } } {{ } n m The proof would be lengthy and we omit it. At the same time, it is not difficult to find a standard proof in the literature; see, for example, [136] and [134]. Why does Theorem 3.2 lead to a fast DFT? To explain this, we first use (3.6) in the following way. Suppose N is even. Letting m = N/2 andn =2,wehave DFT N = P T 1 ⎡ ⎤ ⎡ ⎤ DFT 2 DFT N/2 ⎢ ⎥ ⎣ DFT N/2 ⎦ P 2Ω N . .. P2 T . (3.7) } {{ } ⎢ ⎣ DFT 2 ⎥ ⎦ 2 } {{ } N/2 Let C(N) denote the number of operations needed to multiply a vector with matrix DFT N . Operations include shifting, scalar multiplication and scalar addition. Since P 1 and P 2 are permutation matrices, it takes N shifting operations to do vector-matrix-multiplication with P1 T , P 2 and P2 T . Note that DFT 2 is a 2 × 2 matrix. Obviously, it takes 3N operations (2N
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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
Page 12 and 13: xii
Page 14 and 15: 2.3 Discussion.....................
Page 16 and 17: 6 Simulations 119 6.1 Dictionary...
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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 61: 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
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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
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5.2. LSQR 111 5.2.3 Algorithm LSQR
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5.3. MINRES 113 2. For k =1, 2,...,
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5.3. MINRES 115 using the precondit
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5.4. DISCUSSION 117 From (I + S 1 )
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Chapter 6 Simulations Section 6.1 d
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6.3. DECOMPOSITION 121 10 5 5 5 20
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6.4. DECAY OF COEFFICIENTS 123 10 2
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6.5. COMPARISON WITH MATCHING PURSU
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6.6. SUMMARY OF COMPUTATIONAL EXPER
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Chapter 7 Future Work In the future
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7.2. MODIFYING EDGELET DICTIONARY 1
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7.3. ACCELERATING THE ITERATIVE ALG
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Appendix A Direct Edgelet Transform
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A.2. EXAMPLES 137 edgelet transform
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A.3. DETAILS 139 (a) Stick image (b
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A.3. DETAILS 141 (a) Lenna image (b
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A.3. DETAILS 143 Ordering of Dyadic
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A.3. DETAILS 145 (1,K +1), (1,K +2)
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A.3. DETAILS 147 x 1 , y 1 x 2 , y
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Appendix B Fast Edgelet-like Transf
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B.1. TRANSFORMS FOR 2-D CONTINUOUS
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B.2. DISCRETE ALGORITHM 153 B.2.1 S
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B.2. DISCRETE ALGORITHM 155 extensi
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B.2. DISCRETE ALGORITHM 157 For the
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B.3. ADJOINT OF THE FAST TRANSFORM
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B.4. ANALYSIS 161 above matrix, whi
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B.5. EXAMPLES 163 B.5 Examples B.5.
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B.5. EXAMPLES 165 And so on. Note f
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B.6. MISCELLANEOUS 167 It takes abo
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B.6. MISCELLANEOUS 169 The function
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Bibliography [1] Sensor Data Manage
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BIBLIOGRAPHY 173 [24] C. Victor Che
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BIBLIOGRAPHY 175 [50] David L. Dono
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BIBLIOGRAPHY 177 [75] Vivek K. Goya
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BIBLIOGRAPHY 179 [100] Stéphane Ma
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BIBLIOGRAPHY 181 [127] C. E. Shanno
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