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IMAGE INPAINTING AS A CAUCHY PROBLEM Anis THELJANI Moez KALLEL Maher MOAKHER ENIT-LAMSIN, Tunisia IPEIT&ENIT-LAMSIN, Tunisia ENIT-LAMSIN, Tunisia [email protected] [email protected] [email protected] 1 Introduction • Inpainting is to reconstruct (or complete) a damaged (or in- complete) image by filling in the missing information in the damaged regions. • Some applications image reconstruction Retouching an ancient text document (in the museum). • Many variational models for this problem have appeared in the literature. Among them we mention the Total Variation (TV). These models give rise to partial differential diffusion equations (PDEs). • TV-inpainting: minimize the level line lengths in D. The principle of the TV-inpainting method. . { ∇·[k (|∇u| 2 )∇u]=0, in D, u = f, on ∂D. where k (x)= 1 √ x+ε 2 and ε is a small parameter. • Two cases can be modeled: (a) (b). • Case (a): f is known on the hole boundary ∂D. • Case (b): f is unknown on the boundary Γ c . • We note that in the later case, classical methods use the ho- mogeneous Neumann condition on Γ i . • The use of both Dirichlet and Neumann boundary condi- tions on Γ c . The later condition can be computed from in- formation available in Ω\D. • The inpainting problem is formulated as a nonlinear boundary value inverse problem. • Solve a Cauchy Problem. ∇·[k (|∇u| 2 )∇u]=0, in D, u = f, on Γ c , k (|∇u| 2 )∇u · n = φ, on Γ c . 2 Approximate control formula- tion Extending the work in [1] in the linear case, the idea consist for (η , τ ) ∈ L 2 (Γ i ) × L 2 (Γ i ), minimize J (η , τ )= 1 2 ∫ Γ c (k (|∇u 1 (η )| 2 )∇u 1 (η ) · n − Φ) 2 dΓ c + 1 2 ∫ Γ i (u 1 (η ) − u 2 (τ )) 2 dΓ i s.t. u 1 and u 2 are the solutions of ∇.k (|∇u 1 | 2 )∇u 1 =0 in D u 1 = f on Γ c k (|∇u 1 | 2 )∇u 1 · n = η on Γ i and ∇.k (|∇u 2 | 2 )∇u 2 =0 in D k (|∇u 2 | 2 )∇u 2 · n = Φ on Γ c u 2 = τ on Γ i 2.1 Algorithm Step 1 Given f ∈ L 2 (Γ c ) and Φ ∈ L 2 (Γ c ), choose an arbitrary initial guess (ϕ p ,t p ) ∈ L 2 (Γ i ) × H 1 2 (Γ i ), if p =0. Step 2 Solve with the fixed point method two well-posed mixed forward nonlinear problems ∇.k (|∇u p 1 | 2 )∇u p 1 =0, in D, u p 1 = f, on Γ c , k (|∇u p 1 | 2 )∇u p 1 · n = ϕ p , on Γ i , and ∇.k (|∇u p 2 | 2 )∇u p 2 =0, in D, k (|∇u p 2 | 2 )∇u p 2 · n =Φ, on Γ c , u p 2 = t p , on Γ i . Step 3. Solve the adjoint problem ∫ Ω k (|∇u p 1 | 2 )∇σ ·∇λ p 1 − k 0 (|∇u p 1 | 2 )(∇σ ·∇u p 1 )(∇u p 1 ·∇λ p 1 ) |∇u p 1 | dΓ c = ∫ Γ i (u p 1 − u p 2 )σdΓ i + ∫ Γ c (k (|∇u p 1 | 2 )∇u p 1 · n − φ)(k (|∇u p 1 | 2 )∇σ · n − k 0 (|∇u p 1 | 2 )(∇σ ·∇u p 1 )(∇u p 1 · n) |∇u p 1 | )dΓ c and evaluate the gradient ∇J (ϕ p ,t p )= −(λ p 1 | Γ i ,u p 1 | Γ i − u p 2 | Γ i ). = ⇒ (ϕ p+1 ,t p+1 )=(ϕ p ,t p ) − α∇J (ϕ p ,t p ) Step 4. Having obtained ϕ p+1 and t p+1 for p ≥ 0, set p = p +1 and repeat step 2 and step 3 until a prescribed stopping crite- rion is satisfied. As a stopping criterion we choose the first p such that, ku p 2 | Γ c − f σ k≤ δ, where f σ is a perturbation of the data f . 3 Numerical experiments • Test 1: noisy data (noise=3%) The original image (left) and the incomplete image (right). Image reconstructed by the TV method [3] (left) and image recon- structed by our approach (right). ———————————————————- • Test 2 The original image (left) and incomplete image (right). Image reconstructed by the TV method [3] (left) and image recon- structed by our approach (right). ———————————————————- The numerical solution based on the error functional de- scribed in Section 2 was obtained by the finite-element method which was implemented in the Freefem++ Software environment. We present the results of the inpainting prob- lem with our method and CCD and TV methods [3,4] which was implemented under MATLAB. 4 Conclusions • A Cauchy problem for the nonlinear elliptic equation was formulated when a Dirichlet boundary condition was un- known on a part on the boundary ∂D . • Numerical tests on Case (b) and comparison with others PDEs methods were performed and showed the advantage of the proposed method.. • In process: the theoretical study (existence, uniqueness, etc). References [1] R. Aboula¨ ıch, A. Ben Abda and M. Kallel. Missing boundary data reconstruction via an approximate optimal control, Inverse Problems and Imaging, 2, p. 411-426 (2008). [2] J. Hadamard. Lectures on Cauchy’s Problem in Linear Partial Differential Equation, Dover, New York USA (1953). [3] T. Chan and J. Shen, Mathematical models for local non-texture inpainting, SIAM Journal on Applied Mathematics, 62(3), pp. 1019–1043, 1992. [4] T. Chan and J. Shen, Nontexture Inpainting by Curvature-Driven Diffusions, Journal of Visual Communication and Image Representation, 12(4), pp. 436–449, 2001. Inverse Problems, Control and Shape Optimization, April 2 - 4 , 2012, Ecole Polytechnique, Palaiseau, France
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