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64 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES 3.3 Edgelets and Linear Singularities We describe the edgelet system and its associated transforms. They are designed for linear features in images. Detailed description of edgelet system, edgelet transform and fast approximate edgelet transform are topics of Appendix A and Appendix B. 3.3.1 Edgelet System The edgelet system, defined in [50], is a finite dyadically-organized collection of line segments in the unit square, occupying a range of dyadic locations and scales, and occurring at a range of orientations. It is a low cardinality system. This system has a nice property: any line segment in an image (in the system or not) can be approximated well by afewelements from this system. More precisely, it takes at most 8 log 2 (N) edgelets to approximate any line segment within distance 1/N + δ, where N is the size of the image and δ is a constant. The so-called edgelet transform takes integrals along these line segments. The edgelet system is constructed as follows: [E1] Partition the unit square into dyadic sub-squares. The sides of subsquares are 1/2, 1/4, 1/8, .... [E2] On each subsquare, put equally-spaced vertices on the boundary. The inter-distance is prefixed and dyadic. [E3] If a line segment connecting any two vertices is not on a boundary, then it is an edgelet. [E4] The edgelet system is a collection of all the edgelets as defined in [E3]. The size of the edgelet system is O(N 2 log 2 N). 3.3.2 Edgelet Transform Edgelets are not functions and do not make a basis; instead, they can be viewed as geometric objects—line segments in the square. We can associate these line segments with linear functionals: for a line segment e and a smooth function f(x 1 ,x 2 ), let e[f] = ∫ e f. Then the edgelet transform can be defined as the mapping: f →{e[f] :e ∈E n }, where E n is the collection of edgelets, and e[f] is, as above, the linear functional ∫ e f. Some examples of the edgelet transform on images are given in Appendix A.
3.3. EDGELETS AND LINEAR SINGULARITIES 65 3.3.3 Edgelet Features Suppose we have a digital image I and we calculate f = ϕ ∗ I, the filtering of image I by ϕ according to semi-discrete convolution: f(x 1 ,x 2 )= ∑ k 1 ,k 2 ϕ(x 1 − k 1 ,x 2 − k 2 )I k1 ,k 2 . Assuming ϕ is smooth, so is f, and we may define the collection of edgelet features of I via T [I] ={e[f] :e ∈E n }. That is, the edgelet transform of the smoothed image f = ϕ ∗ I. In terms of the unsmoothed image Ĩ = ∑ k 1 ,k 2 I k1 ,k 2 δ k1 ,k 2 with δ k1 ,k 2 the Dirac mass, we may write (T [I]) e = ∫ ψ e (x 1 ,x 2 )Ĩ(x 1,x 2 )dx 1 dx 2 , where ψ e (x 1 ,x 2 )= ∫ 1 0 ϕ(x 1−e 1 (t),x 2 −e 2 (t))dt, where t → (e 1 (t),e 2 (t)) is a unit-speed parameterized path along edgelet e. Inshort,ψ e is a kind of “smoothed-out edgelet” with smoothing ϕ. Since edgelets are features with various locations, scales and orientations, the associated transform—the edgelet transform—is well-suited to processing linear features. 3.3.4 Fast Approximate Edgelet Transform The edgelet transform (defined in previous sections) is an O(N 3 log 2 N) algorithm. The order of complexity can be reduced to O(N 2 log 2 N), if we allow a slight modification of the original edgelet system. Recently, a fast algorithm for calculating a discrete version of the Radon transform has been developed. The Radon transform of a 2-D continuous function is simply a projection of the 2-D function onto lines passing through the origin. In discrete case, people can utilize the idea from the Fourier Slice Theorem (see Appendix B) to design a fast algorithm. It takes the following three steps: image → 2-D FFT → X-interpolation → 1-D inverse FFT → output (using fractional FT) The so-called fast approximate edgelet transform is essentially the discrete Radon transform of an image at various scales: we partition a unit square into subsquares having various widths, and then apply the discrete version of the Radon transform on images limited within each of these subsquares. This method was described in [48] and will be described in detail in Appendix B. As we can see, the existence of this fast algorithm relies on the existence of FFT and fractional FT. The Riesz representers of the fast approximate edgelet transform are close to edgelets but not exactly the same. Illustrations of some of these representers and some examples
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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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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
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 81 and 82: 3.2. WAVELETS AND POINT SINGULARITI
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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
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,...,
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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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