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66 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES of this transform on images are given in Appendix B, together with detailed discussion on algorithm design issues and analysis. 3.4 Other Transforms In this section, we review some other widely known transforms. The first subsection is for one-dimensional (1-D) transforms. The second subsection is for 2-D transforms. We intend to be comprehensive, but sacrifice depth and mathematical rigor. The motivation for writing this section is to provide us with some starting points in this field. 3.4.1 Transforms for 1-D Signals We briefly describe some 1-D transforms. Note that all the transforms we mention are decomposition schemes. As mentioned at the beginning of this chapter, there is another popular idea in designing the 1-D transform. This idea is to map the 1-D signal into a 2-D function, which is sometimes called a distribution. This idea is called the idea of distribution. Typically the mapping is a bilinear transform; for example, the Wigner-Ville distribution. In this thesis, we are not going to talk about the distribution idea. Readers can learn more about it in Flandrin’s book [68]. We focus on the idea of time-frequency decomposition. Suppose every signal is a superposition of some elementary atoms. Each of these elementary atoms is associated with a 2-D function on a plane, in which the horizontal axis indicates the time and the vertical axis indicates the frequency. This two-dimensional plane is called a time frequency plane (TFP). The 2-D function is called a time frequency representation (TFR) of the atom. Usually the TFR of an atom is a bilinear transform of the atom. Accordingly, the same transform of a signal is the TFR of the signal. We pick atoms such that the TFR of every atom resides in a small cell on the TFP. We further assume that these previously mentioned cells cover the entire TFP. The collection of these cells is called a time frequency tiling. When a signal is decomposed as a linear combination of these atoms, the decomposition can also be viewed as a decomposition of a function on the TFP, so we can call the decomposition a time-frequency decomposition. Based on Heisenberg’s uncertainty principle, these cells, which correspond to the elementary atoms, cannot be too small. There are many ways to present the Heisenberg’s
3.4. OTHER TRANSFORMS 67 uncertainty principle. One way is to say that for any signal (here a signal corresponds to a function in L 2 ), the multiplication of the variances in both the time domain and the frequency domain are lower bounded by a constant. From a time-frequency decomposition point of view, the previous statement means that the area of a cell for each atom can not be smaller than a certain constant. A further result from Slepian, Pollak and Landau claims that a signal can never have both finite time duration and finite bandwidth simultaneously. For a careful description about them, readers are referred to [68]. (a) Shannon (b) Fourier (c) Gabor (d) WaveLet 1 1 1 1 Frequency Frequency Frequency Frequency 0 Time 1 0 Time 1 0 Time 1 0 Time 1 (e) Chirplet (f) fan (g) Cosine Packet (h) Wavelet Packet 1 1 1 1 Frequency Frequency Frequency Frequency 0 Time 1 0 Time 1 0 Time 1 0 Time 1 Figure 3.7: Idealized tiling on the time-frequency plane for (a) sampling in time domain (Shannon), (b) Fourier transform, (c) Gabor analysis, (d) orthogonal wavelet transform, (e) chirplet, (f) orthonormal fan basis, (g) cosine packets, and (h) wavelet packets. The way to choose these cells determines the time-frequency decomposition scheme. Two basic principles are generally followed: (1) the tiling on TFP does not overlap, and (2) the union of these cells cover the entire TFP. Time-frequency decomposition is a powerful idea, because we can idealize different transforms as different methods of tiling on the TFP. Figure 3.7 gives a pictorial tour of idealized tiling for some transforms.
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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
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xii
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2.3 Discussion.....................
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6 Simulations 119 6.1 Dictionary...
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xviii
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List of Figures 2.1 Shannon’s sch
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A.3 Edgelet transform of the wood g
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Nomenclature Special sets N .......
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List of Abbreviations BCR .........
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Chapter 1 Introduction 1.1 Overview
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Chapter 2 Sparsity in Image Coding
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2.1. IMAGE CODING 5 INFORMATION SOU
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2.1. IMAGE CODING 7 2.1.2 Source an
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2.1. IMAGE CODING 9 x ✲ T y ERROR
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2.1. IMAGE CODING 11 where Q stands
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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 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
Page 93: 3.3. EDGELETS AND LINEAR SINGULARIT
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
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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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