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

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minimization family as they are not convex functions). In the square 4 or under-determined cases, there is only one measurement available for each high-resolution pixel. As median and mean operators for one or two measurements give the same result, L1 and L2 norm minimizations will result in identical answers. Also in the under-determined cases certain pixel locations will have no estimate at all. For these cases, it is essential for the estimator to have an extra term, called the regularization term, to remove outliers. The next section discusses different regularization terms and introduces a robust and convenient regularization term. 2.2.3 Robust Regularization As mentioned in Chapter 1, super-resolution is an ill-posed problem [17], [43]. For the under-determined cases (i.e. when fewer than r 2 non-redundant frames are available), there exist an infinite number of solutions which satisfy (2.2). The solution for square and over- determined 5 cases is not stable, which means small amounts of noise in the measurements will result in large perturbations in the final solution. Therefore, considering regularization in super-resolution as a means for picking a stable solution is indeed necessary. Also, regularization can help the algorithm to remove artifacts from the final answer and improve the rate of convergence. Of the many possible regularization terms, we desire one which results in high- resolution images with sharp edges and is easy to implement. A regularization term compensates the missing measurement information with some general prior information about the desirable high-resolution solution, and is usually imple- mented as a penalty factor in the generalized minimization cost function (5.5): � N� � �X = ArgMin ρ(Y (k),D(k)H(k)F (k)X)+λΥ(X) , (2.14) X k=1 sources of outliers as Laplacian probability density function (PDF) rather than Gaussian PDF. 4 where the number of non-redundant low-resolution frames is equal to the square of resolution enhancement factor. 5 where the number of non-redundant low-resolution frames is larger than the square of resolution enhancement factor. 23
where λ, the regularization parameter, is a scalar for properly weighting the first term (similarity cost) against the second term (regularization cost) and Υ is the regularization cost function. tion [11], [17]: One of the most widely used regularization cost functions is the Tikhonov cost func- ΥT (X) =�ΛX� 2 2 , (2.15) where Λ is usually a high-pass operator such as derivative, Laplacian, or even identity matrix. The intuition behind this regularization method is to limit the total energy of the image (when Λ is the identity matrix) or forcing spatial smoothness (for derivative or Laplacian choices of Λ). As the noisy and edge pixels both contain high-frequency energy, they will be removed in the regularization process and the resulting reconstructed image will not contain sharp edges. Certain types of regularization cost functions work effectively for some special types of images but are not suitable for general images. For instance Maximum Entropy regularization produces sharp reconstructions of point objects, such as star fields in astronomical images [16], however it is not applicable to natural images. One of the most successful regularization methods for denoising and deblurring is the total variation (TV) method [44]. The total variation criterion penalizes the total amount of change in the image as measured by the L1 norm of the magnitude of the gradient and is loosely defined as: ΥTV(X) =�∇X�1, where ∇ is the gradient operator. The most useful property of total variation is that it tends to preserve edges in the reconstruction [16], [44], [45], as it does not severely penalize steep local gradients. Based on the spirit of the total variation criterion, and a related technique called the bilateral filter (Appendix A), we introduce our robust regularizer called Bilateral Total Variation (BTV), which is computationally cheap to implement, and preserves edges. The regularizing 24
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: Figure 2.3: Effect of upsampling D
Page 45 and 46: Although a relatively large regular
Page 47 and 48: image(X). Figure 2.5(b) is the corr
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
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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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