sparse image representation via combined transforms - Convex ...

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LPF ....................................... LS ......................................... LTI ........................................ MAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MOF ...................................... MOFDN ................................... MP ........................................ MPDN ..................................... MPEG ..................................... MRA ...................................... MRF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MSE ....................................... OMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCA ....................................... RF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RMS ....................................... SAI ........................................ SIRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TFR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TVDN ..................................... p.d.f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Pass Filter Least Square Linear and Time Invariant Mean of the Absolute Deviation Method of Frames Method-of-Frames De-Noising Matching Pursuit Matching Pursuit De-Noising Moving Picture Experts Group Multiresolution Analysis Markov Random Field Mean-Square Error Orthogonal Matching Pursuit Principal Component Analysis Random Field Residual Mean Square Sparse Approximate Inverse Sparse Image Representation Pursuit Time Frequency Plane Time Frequency Representation Total Variation De-Noising probability density function xxviii
Chapter 1 Introduction 1.1 Overview Recently, many new methods of image representation have been proposed, including wavelets, cosine packets, brushlets, edgelets, ridgelets, and so on. Typically each of these is good for a specific class of features, but not good for others. We propose a method of combining image representations, more particularly, a method based on the 2-D wavelet transform and the edgelet-like transform. The 2-D wavelet transform is good at capturing point singularities, while the newly proposed edgelet-like transform is good at capturing linear singularities (edges). Both transforms have fast algorithms for digital images. Wavelets and edgelet-like features (together) form an overcomplete dictionary. To find a sparse representation, we have had success by minimizing the objective function: ‖y − Φx‖ 2 2 + λρ(x), where y is the image, Φ is the matrix whose columns are all the basis vectors in all the image representations, x is the vector of coefficients and ρ(x) is a penalty function that is convex, l 1 -like, and separable. Sparsity of representation is achieved by adding the penalty term: λρ(x) to balance the goodness of fit measure ‖y − Φx‖. To develop an efficient algorithm to solve this problem, we utilize insights from convex optimization and the LSQR solver from computational linear algebra. These methods combine to give a fast algorithm for our problem. Numerical experiments provide promising results. 1
Page 1 and 2: SPARSE IMAGE REPRESENTATION VIA COM
Page 3: I certify that I have read this dis
Page 7 and 8: To find a sparse image representati
Page 9: 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...
Page 18 and 19: xviii
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: List of Abbreviations BCR .........
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
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3.1. DCT AND HOMOGENEOUS COMPONENTS
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3.2. WAVELETS AND POINT SINGULARITI
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3.3. EDGELETS AND LINEAR SINGULARIT
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3.4. OTHER TRANSFORMS 67 uncertaint
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3.4. OTHER TRANSFORMS 69 Chirplets
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3.4. OTHER TRANSFORMS 71 Folding. A
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3.4. OTHER TRANSFORMS 73 We can app
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3.5. DISCUSSION 75 give only a few
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3.7. PROOFS 77 the ijth component o
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3.7. PROOFS 79 Similarly, we have [
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Chapter 4 Combined Image Representa
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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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