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

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The practitioners of super-resolution usually explicitly or implicitly (e.g. the projec- tion onto convex sets (POCS) based methods [10]) define a cost function to estimate X in an iterative fashion.This type of cost function assures a certain fidelity or closeness of the final solution to the measured data. Historically, the construction of such a cost function has been motivated from either an algebraic or a statistical perspective. Perhaps the cost function most common to both perspectives is the least-squares (LS) cost function, which minimizes the L2 norm of the residual vector, ˆX = ArgMin X J(X) =ArgMin �Y − MX� X 2 2 . (1.3) For the case where the noise V is additive white, zero mean Gaussian, this approach has the interpretation of providing the Maximum Likelihood estimate of X [11]. We shall show in this thesis that such a cost function is not necessarily adequate for super-resolution. An inherent difficulty with inverse problems is the challenge of inverting the forward model without amplifying the effect of noise in the measured data. In the linear model, this results from the very high, possibly infinite, condition number for the model matrix M. Solving the inverse problem, as the name suggests, requires inverting the effects of the system matrix M. At best, this system matrix is ill-conditioned, presenting the challenge of inverting the matrix in a numerically stable fashion [12]. Furthermore, finding the minimizer of (1.3) would amplify the random noise V in the direction of the singular vectors (in the super-resolution case these are the high spatial frequencies), making the solution highly sensitive to measurement noise. In many real scenarios, the problem is exacerbated by the fact that the system matrix M is singular. For a singular model matrix M, there is an infinite space of solutions minimizing (1.3). Thus, for the problem of super-resolution, some form of regularization must be included in the cost function to stabilize the problem or constrain the space of solutions. Needless to say, the choice of regularization plays a vital role in the performance of any super-resolution algorithm. Traditionally, regularization has been described from both the algebraic and statistical perspectives. In both cases, regularization takes the form of soft constraints on the space of possible solutions often independent of the measured data. This is 7
accomplished by way of Lagrangian type penalty terms as in J(X) =�Y − MX� 2 2 + λΥ(X) . (1.4) The function Υ(X) places a penalty on the unknown X to direct it to a better formed solution. The coefficient λ dictates the strength with which this penalty is enforced. Generally speak- ing, choosing λ could be either done manually, using visual inspection, or automatically using methods like Generalized Cross-Validation [13, 14], L-curve [15], and other techniques. Tikhonov regularization 4 [11, 16, 17] is a widely employed form of regularization, which has been motivated from an analytic standpoint to justify certain mathematical properties of the estimated solution. Often, little attention, however, is given to the effects of such simple regularization on the super-resolution results. For instance, the regularization often penalizes energy in the higher frequencies of the solution, opting for a smooth and hence blurry solution. From a statistical perspective, regularization is incorporated as a priori knowledge about the solution. Thus, using the Maximum A-Posteriori (MAP) estimator, a much richer class of regularization functions emerges, enabling us to capture the specifics of the particular application (e.g. in [18] the piecewise-constant property of natural images are captured by modeling them as Huber-Markov random field data). Such robust methods, unlike the traditional Tikhonov penalty terms, are capable of performing adaptive smoothing based on the local structure of the image. For instance, in Chapter 2, we offer a penalty term capable of preserving the high frequency edge structures commonly found in images. In summary, an efficient solution to the multi-frame imaging inverse problem should 1. define a forward model describing all the components of the imaging channel (such as probability density function (PDF) of additive noise, blur point spread function (PSF), relative motion vectors,...). 2. adopt proper prior information to turn the ill-posed inverse problem to a well-posed problem (regularization) 4 Tikhonov regularization is often implemented by penalizing a high-pass filtered image by L2 norm as formu- lated and explained in details in Section 2.2.3. 8
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: problem of super-resolution is line
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 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
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image (Spatial Luminance Penalty Te
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Acquire a Set of LR Color Filtered
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espect to the other color channels.
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in different color bands. Our propo
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the robust super-resolution method
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We used the method described in [48
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the color artifacts of a set of low
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a: Reconst. with lumin. regul. b: R
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a: Reconst. with all terms. Figure
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a: LR b: Shift-and-Add c: SR [4] on
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a b c d e f Figure 3.14: Multi-fram
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a b c d Figure 3.16: Multi-frame co
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a b c d e f Figure 3.18: Multi-fram
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ates from them in several important
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order. The current image X(t) is of
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With this alternative definition of
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GS a 0 0 0 b 0 0 0 c D T GSD =⇒
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Figure 4.2: Block diagram represent
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ˆZ(t), necessitating a further joi
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Figure 4.3: Block diagram represent
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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
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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
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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
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Appendix E Derivation of the Inter-
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Appendix F Appendix: Affine Motion
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Bibliography [1] A. Zomet, A. Rav-A
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[26] H. Ur and D. Gross, “Improve
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[55] K. Hirakawa and T. Parks, “A
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[84] S. Farsiu, D. Robinson, M. Ela
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[112] S. M. Kay, Fundamentals of st
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A FAST AND ROBUST FRAMEWORK FOR IMAGE FUSION AND ...

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