A FAST AND ROBUST FRAMEWORK FOR IMAGE FUSION AND ...

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a: Original b: Noisy c: Reconstruction using Tikhonov d: Reconstruction using TV Error 0.5 0.4 0.3 0.2 0.1 Tikhonov TV Bilateral−TV 0 0 10 20 30 40 50 Iteration 60 70 80 90 100 e: Reconstruction using BTV f: Error in gray-level value estimation Figure 2.4: a-e: Simulation results of denoising using different regularization methods. f: Error in graylevel value estimation of the pixel indicated by arrow in (a) versus the iteration number in Tikhonov (solid line), TV (dotted line), and Bilateral TV (broken line) denoising. to reconstruct the blurred noisy image. Figure 2.5 shows the results of our experiment. Figure 2.5(a) shows the original 27
image(X). Figure 2.5(b) is the corrupted Y = HX + V , where V is the additive noise. Figure 2.5(c) is the result of reconstruction with Tikhonov regularization (Υ(X) =�ΛX�2 2 ), where Λ in (2.15) was replaced by the Laplacian kernel (2.19) and λ =0.03. Figure 2.5(d) shows the result of applying our regularization criterion (Υ(X) = �P l,m=−P α|m|+|l| �X − Sl xSm y X�1) with the following parameters: α =0.7, λ =0.17 and P =2. The best Mean Square Error7 (MSE) achieved by Tikhonov regularization was 313 versus 215 for the proposed regularization. The superior edge preserving property of the bilateral prior is apparent in this example. 2.2.4 Robust Super-Resolution Implementation In this subsection, based on the material developed in sections 2.2.2 and 2.2.3, a solution for the robust super-resolution problem will be proposed. Combining the ideas presented thus far, we propose a robust solution of the super-resolution problem as follows ⎡ N� ⎣ �D(k)H(k)F (k)X − Y (k)�1 + λ P� α |m|+|l| �X − S l xS m ⎤ y X�1⎦ . �X=ArgMin X k=1 l,m=−P We use steepest descent to find the solution to this minimization problem: ⎧ �X n+1 = � ⎪⎨ N� Xn− β F ⎪⎩ k=1 T (k)H T (k)D T (k)sign(D(k)H(k)F (k) � X n− Y (k)) +λ P� l,m=−P α |m|+|l| [I − S −m y S −l (2.21) x ]sign( � Xn − S l xS m y � ⎫ ⎬ Xn) , (2.22) ⎭ where β is a scalar defining the step size in the direction of the gradient. S−l x the transposes of matrices Sl x and Sm y directions as S l x and Sm y . and S−m y define , respectively and have a shifting effect in the opposite Simulation results in Section 2.3 will show the strength of the proposed algorithm. 7 Mean square error of an estimate is defined as MSE =E{( X� 2 − X) }, where E is the expected value operator, �X is the estimate, and X is the true value of the vector to be estimated. 28
Page 1 and 2: UNIVERSITY OF CALIFORNIA SANTA CRUZ
Page 3 and 4: Contents List of Figures vi List of
Page 5 and 6: Appendix F Appendix: Affine Motion
Page 7 and 8: 2.3 Effect of upsampling D T matrix
Page 9 and 10: 3.11 Multi-frame color super-resolu
Page 11 and 12: 4.5 A sequence of 250 low-resolutio
Page 13 and 14: List of Tables 2.1 The true motion
Page 15 and 16: show that using such constraints wi
Page 17 and 18: I thank all the handsome gentlemen
Page 19 and 20: It’s the robustness, stupid! xix
Page 21 and 22: quality. We develop the theory and
Page 23 and 24: That is, ⎧ ⎪⎨ ⎪⎩ y1 = h1.
Page 25 and 26: problem of super-resolution is line
Page 27 and 28: accomplished by way of Lagrangian t
Page 29 and 30: which addresses the artifacts resul
Page 31 and 32: noise and Y is the resulting discre
Page 33 and 34: ods, where the regularization-like
Page 35 and 36: 2.2 Robust Super-Resolution 2.2.1 R
Page 37 and 38: and horizontal directions and subsa
Page 39 and 40: Note that if p =2then (2.8) will be
Page 41 and 42: Figure 2.3: Effect of upsampling D
Page 43 and 44: where λ, the regularization parame
Page 45: Although a relatively large regular
Page 49 and 50: of high-resolution frame � Xn. Bl
Page 51 and 52: as conjugate gradient is not a stra
Page 53 and 54: of minimization problem (2.23) can
Page 55 and 56: norm is not robust to motion error,
Page 57 and 58: e one. Figure 2.12(f) is the result
Page 59 and 60: 50 100 150 200 250 300 350 50 100 1
Page 61 and 62: a: One of 8 LR Frames b: Cubic Spli
Page 63 and 64: a: Frame 1 of 55 LR Frames b: Frame
Page 65 and 66: Chapter 3 Multi-Frame Demosaicing a
Page 67 and 68: chrominance layers are separated fr
Page 69 and 70: Other examples of popular demosaici
Page 71 and 72: a: Original b: Down-sampled c: Blur
Page 73 and 74: 3.3 Mathematical Model and Solution
Page 75 and 76: epresents the down-sampling operato
Page 77 and 78: image (Spatial Luminance Penalty Te
Page 79 and 80: Acquire a Set of LR Color Filtered
Page 81 and 82: espect to the other color channels.
Page 83 and 84: in different color bands. Our propo
Page 85 and 86: the robust super-resolution method
Page 87 and 88: We used the method described in [48
Page 89 and 90: the color artifacts of a set of low
Page 91 and 92: a: Reconst. with lumin. regul. b: R
Page 93 and 94: a: Reconst. with all terms. Figure
Page 95 and 96: a: LR b: Shift-and-Add c: SR [4] on
Page 97 and 98:
a b c d e f Figure 3.14: Multi-fram
Page 99 and 100:
a b c d Figure 3.16: Multi-frame co
Page 101 and 102:
a b c d e f Figure 3.18: Multi-fram
Page 103 and 104:
ates from them in several important
Page 105 and 106:
order. The current image X(t) is of
Page 107 and 108:
With this alternative definition of
Page 109 and 110:
GS a 0 0 0 b 0 0 0 c D T GSD =⇒
Page 111 and 112:
Figure 4.2: Block diagram represent
Page 113 and 114:
ˆZ(t), necessitating a further joi
Page 115 and 116:
Figure 4.3: Block diagram represent
Page 117 and 118:
4.3 Simultaneous Deblurring and Int
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Y (t) Input CFA Data Y R (t) Y G (t
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4.5(b) & 4.5(f) show frames #50 and
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a b c d e f g h Figure 4.6: A seque
Page 125 and 126:
dB 24 22 20 18 16 14 12 10 8 0 50 1
Page 127 and 128:
a b c d e f g h Figure 4.10: A sequ
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Chapter 5 Constrained, Globally Opt
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In this chapter, we study such prio
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(a) (b) Figure 5.2: The consistent
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vector field δi,j is spatially con
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5.3.3 Robust Multi-Frame Registrati
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MSE 10 −1 10 −2 10 −3 10 0 10
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(a) (b) (c) (d) Figure 5.6: Experim
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we advocated the use of the L1 norm
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y the user. - The user is able to s
Page 147 and 148:
There is need for more research on
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than the corresponding single-frame
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In [111], Elad proved that such fil
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all iterations, which means estimat
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Appendix C Noise Modeling Based on
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Appendix D Error Modeling Experimen
Page 159 and 160:
Appendix E Derivation of the Inter-
Page 161 and 162:
Appendix F Appendix: Affine Motion
Page 163 and 164:
Bibliography [1] A. Zomet, A. Rav-A
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[26] H. Ur and D. Gross, “Improve
Page 167 and 168:
[55] K. Hirakawa and T. Parks, “A
Page 169 and 170:
[84] S. Farsiu, D. Robinson, M. Ela
Page 171:
[112] S. M. Kay, Fundamentals of st
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