IMAGE INPAINTING AS A CAUCHYPROBLEM
Anis THELJANI Moez KALLEL Maher MOAKHERENIT-LAMSIN, Tunisia IPEIT&ENIT-LAMSIN, Tunisia ENIT-LAMSIN, [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 thedamaged regions.
• Some applications
image reconstruction
Retouching an ancient text document (in the museum).
• Many variational models for this problem have appeared inthe literature. Among them we mention the Total Variation(TV). These models give rise to partial differential diffusionequations (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+ε2
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 Γc.
• We note that in the later case, classical methods use the ho-mogeneous Neumann condition on Γi.
• The use of both Dirichlet and Neumann boundary condi-tions on Γc. The later condition can be computed from in-formation available in Ω\D.
• The inpainting problem is formulated as a nonlinearboundary value inverse problem.
• Solve a Cauchy Problem.∇·[k(|∇u|2)∇u] = 0, in D,
u = f, on Γc,
k(|∇u|2)∇u · n = φ, on Γc.
2 Approximate control formula-tion
Extending the work in [1] in the linear case, the idea consist
for (η, τ ) ∈ L2(Γi) × L2(Γi),minimize
J(η, τ ) = 12
∫Γc
(k(|∇u1(η)|2)∇u1(η) · n − Φ)2dΓc
+12
∫Γi
(u1(η) − u2(τ ))2dΓi
s.t. u1 and u2 are the solutions of
∇.k(|∇u1|2)∇u1 = 0 in D
u1 = f on Γc
k(|∇u1|2)∇u1 · n = η on Γi
and
∇.k(|∇u2|2)∇u2 = 0 in D
k(|∇u2|2)∇u2 · n = Φ on Γc
u2 = τ on Γi
2.1 Algorithm
Step 1 Given f ∈ L2(Γc) and Φ ∈ L2(Γc), choose an arbitraryinitial guess
(ϕp, tp) ∈ L2(Γi) × H12(Γi), if p = 0.
Step 2 Solve with the fixed point method two well-posedmixed forward nonlinear problems
∇.k(|∇up1|
2)∇up1 = 0, in D,
up1 = f, on Γc,
k(|∇up1|
2)∇up1 · n = ϕp, on Γi,
and ∇.k(|∇u
p2|
2)∇up2 = 0, in D,
k(|∇up2|
2)∇up2 · n = Φ, on Γc,
up2 = tp, on Γi.
Step 3. Solve the adjoint problem∫Ω
k(|∇up1|
2)∇σ ·∇λp1−
k′(|∇up1|
2)(∇σ · ∇up1)(∇u
p1 · ∇λ
p1)
|∇up1|
dΓc =∫Γi
(up1 − u
p2)σdΓi +
∫Γc
(k(|∇up1|
2)∇up1 · n − φ)(k(|∇u
p1|
2)∇σ · n
−k′(|∇u
p1|
2)(∇σ · ∇up1)(∇u
p1 · n)
|∇up1|
)dΓc
and evaluate the gradient
∇J(ϕp, tp) = −(λp1|Γi
, up1|Γi
− up2|Γi
).
=⇒ (ϕp+1, tp+1) = (ϕp, tp) − α∇J(ϕp, tp)
Step 4. Having obtained ϕp+1 and tp+1 for p ≥ 0, set p = p + 1and repeat step 2 and step 3 until a prescribed stopping crite-rion is satisfied. As a stopping criterion we choose the first psuch that,
‖up2|Γc
− fσ‖ ≤ δ,
where fσ 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).
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• 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).
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The numerical solution based on the error functional de-scribed in Section 2 was obtained by the finite-elementmethod which was implemented in the Freefem++ Softwareenvironment. We present the results of the inpainting prob-lem with our method and CCD and TV methods [3,4] whichwas 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 othersPDEs methods were performed and showed the advantageof the proposed method..
• In process: the theoretical study (existence, uniqueness, etc).
References[1] R. Aboulaıch, 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
