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Variational Pairing of Image Segmentation and Blind Restoration Leah Bar Nir Sochen* Nahum Kiryati School of Electrical Engineering *Dept. of Applied Mathematics Tel Aviv University

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Segmentation: images meet concepts Borrowed from Georges Koepfler

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Segmentation: images meet concepts Borrowed from Georges Koepfler Formalization (Mumford & Shah) min [(fidelity to image) + β (gradients within segments) + α (total edge length)] Segmentation by minimizing a functional

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Segmentation: images meet concepts Borrowed from Georges Koepfler Formalization (Mumford & Shah) min [(fidelity to image) + β (gradients within segments) + α (total edge length)] Segmentation by minimizing a functional Calculus of Variations PDE ’ s Numerical Techniques Linear Systems of Equations

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Mumford-Shah Segmentation fidelity to imagegradients within segmentstotal edge length Ω: image domain K: edge set f : segmented image g : observed image

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Mumford-Shah Segmentation fidelity to imagegradients within segmentstotal edge length Ω: image domain K: edge set f : segmented image g : observed image Problem: Discontinuities in the domains ( Ω/ K, K ) make minimization difficult

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Mumford-Shah Segmentation fidelity to imagegradients within segmentstotal edge length Ω: image domain K: edge set f : segmented image g : observed image Problem: Discontinuities in the domains ( Ω/ K, K ) make minimization difficult Solution: Continuous approximation of F(f,K) (Gamma-convergence framework)

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Mumford-Shah Segmentation fidelity to imagegradients within segmentstotal edge length Ω: image domain K: edge set f : segmented image g : observed image Problem: Discontinuities in the domains ( Ω/ K, K ) make minimization difficult Solution: Continuous approximation of F(f,K) (Gamma-convergence framework) fidelity to imagegradients in segmentstotal edge length v(x): smooth function v(x)~0 at edges v(x)~1 otherwise (in segments) (Ambrosio & Tortorelli, 1990)

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In a blurred image, edges are degraded and segmentation is difficult.

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Image Restoration Given the image g and the blur kernel h, restore the original image f. 1. Brute force... ill posed. 2. Tikhonov regularization Minimize... oversmoothing. 3. Total Variation (TV) regularization Minimize... better edge preservation.

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Blind Image Restoration Given the image g, restore the original image f (and the blur-kernel h ). - Ill posed (1): sensitivity to small changes in g. - Ill posed (2): maybe the original image was already blurred?

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Blind Image Restoration Given the image g, restore the original image f (and the blur-kernel h ). - Ill posed (1): sensitivity to small changes in g. Chan & Wong (1998) Minimize - Ill posed (2): maybe the original image was already blurred? TV-regularization with respect to both the image and the kernel. - The restored image is very sensitive to the recovered kernel - The recovered kernel depends on the contents of the image (bad news)

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Chan & Wong - The recovered kernel depends on the contents of the image. source image recovered kernel isotropic blur blind restoration

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Chan & Wong (1998) - Performance original blurredrestored isotropic gaussian kernel, =2.1 recovered kernel - The restored image is very sensitive to the recovered kernel. - The recovered kernel depends on the contents of the image.

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In blind image restoration, one can’t get it all. Borrowed from Mickey Mouse (The Sorcerer’s Apprentice)

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Vogel & Oman, 1998 You & Kaveh, 1996 Carasso, 2001 Chan & Wong, 1998 Mumford & Shah, 1989 Rudin, Osher & Fatemi, 1992 Chambolle, 1995 Hewer et al, 1998 Kim et al, 2002 Ambrosio & Tortorelli, 1992 Aubert & Kornprobst, 2002 Tikhonov & Arsenin, 1977 Some related work... (Blind) Restoration Segmentation Mathematics, Foundations

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The suggested approach Why? What? How? - Segmentation is hard, but easier if the image is sharp - Blind restoration is hard, but easier if the edges are known Blind restoration and segmentation as mutually supporting processes Unified variational framework, iterative algorithm

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Combined objective functional Mumford-Shah segmentation + blind restoration Make it work: Use the Γ-convergence approximation Make it work well: Use a parametric blur-kernel Reminder v(x): smooth function v(x)~0 at edges v(x)~1 otherwise (in segments) fidelity, parametric blur gradients in segments “ total edge length ”“ smooth v ” “ wide kernel ”

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Minimizing the functional Iterate Minimize with respect to v (segmentation / edge detection) Minimize with respect to f (image restoration) Minimize with respect to σ (blur-kernel recovery)

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Iterative Minimization Equations Minimization with respect to v (Euler equation)

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Iterative Minimization Equations Minimization with respect to v (Euler equation) Minimization with respect to f (Euler equation)

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Iterative Minimization Equations Minimization with respect to v (Euler equation) Minimization with respect to f (Euler equation) Minimization with respect to σ (derivative)

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Iterative Minimization Equations Minimization with respect to v (Euler equation) Minimization with respect to f (Euler equation) Minimization with respect to σ (derivative) Calculus of Variations PDE ’ s Numerical Techniques Linear Systems of Equations

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Frequently Asked Questions What are the initial values? What is the stopping condition? Does it converge? To a global optimum? We use f=g (output=input), v=1 (no edges) and σ = ε (small blur). We stop when the radius σ of the recovered kernel has converged. Nice theoretical properties Excellent experimental behavior Additional analytic work in progress Typical convergence: σ vs. iteration number

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Experimental Results (1): Known Blur Kernel Blurred

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Experimental Results (1): Known Blur Kernel Blurred Lucy-Richardson restoration

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Experimental Results (1): Known Blur Kernel Blurred Lucy-Richardson restoration Suggested restoration

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Experimental Results (1): Known Blur Kernel Blurred Lucy-Richardson restoration Suggested restorationSuggested edges ( v function)

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Experimental Results (2): Blind Blurred

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Experimental Results (2): Blind BlurredChan-Wong restoration

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Experimental Results (2): Blind BlurredChan-Wong restoration Suggested restoration

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Experimental Results (2): Blind BlurredChan-Wong restoration Suggested restorationSuggested edges ( v function)

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Experimental Results (3): Blind Blurred

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Experimental Results (3): Blind Blurred Chan-Wong restoration

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Experimental Results (3): Blind Blurred Chan-Wong restoration Suggested restoration

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Experimental Results (3): Blind Blurred Chan-Wong restoration Suggested restorationSuggested edges ( v function)

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Experimental Results (4): Blind Blurred

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Experimental Results (4): Blind Blurred Chan-Wong restoration

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Experimental Results (4): Blind Blurred Chan-Wong restoration Suggested restoration

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Experimental Results (4): Blind BlurredChan-Wong restoration Suggested restorationSuggested edges ( v function)

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Conclusions Image segmentation and (blind) restoration, sont les mots qui vont tres bien ensemble*. Blind restoration is easier if you can use a parametric blur model. *these are words that go together well. The whole is larger than the sum of its parts (in this case).

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