A generalization of Cahn-Hilliard inpainting for grayvalue images

A generalization of Cahn-Hilliard inpainting for grayvalue images

Martin Burger∗, Lin He∗∗, Peter Markowich∗∗∗, and Carola-Bibiane Schonlieb3†

1 Institut fur Numerische und Angewandte Mathematik, Westfalische Wilhelms Universitat Munster, Einsteinstrasse 62,D-48149, Munster, Germany.

2 RICAM, Altenbergerstrasse 69 A-4040, Linz, Austria.3 Faculty of Mathematics, Univ. of Vienna, Nordbergstrasse 15, A-1090, Vienna, Austria.

The Cahn-Hilliard equation has its origin in material sciences and serves as a model for phase separation and phase coarseningin binary alloys. A new approach in the class of fourth order inpainting algorithms is inpainting of binary images usingthe Cahn-Hilliard equation. We will present a generalization of this fourth order approach for grayvalue images. This isrealized by using subgradients of the total variation functional within the flow, which leads to structure inpainting with smoothcurvature of level sets. We will present some numerical examples for this approach and analytic results concerning existenceand convergence of solutions.

1 Introduction

Second order variational inpainting methods, like total variation inpainting [1], have drawbacks as in the connection of edgesover large distances or the continuous propagation of level lines into the damaged domain. In an attempt to solve both theconnectivity principle and the so called staircasing effect resulting from second order image diffusions, a number of third andfourth order diffusions have been suggested for image inpainting.One of the most important works in this direction is the algorithm of Chan, Kang and Shen [2] based on Eulers elastica energy.Their approach leads to a continuous connection of level lines also over large distances. Another new approach in the class offourth order inpainting algorithms is inpainting of binary images using the Cahn-Hilliard equation [3], [4]. Let f be a givenimage in a domain Ω with inpainting domain D ⊂ Ω. The result of the Cahn-Hilliard inpainting approach u evolves in timeto become a fully inpainted version of f under the following equation

∂u

∂t= ∆(−ε∆u +

1

εF ′(u)) + λ(f − u), (1)

where F is a so called double-well potential, e.g. F (u) = u2(u − 1)2, and λ(x) = λ0 in Ω \ D, λ(x) = 0 in D.In the following a generalization of this fourth-order inpainting approach for grayvalue images is shown. Starting with

an iterative scheme to solve (1) we take the Γ limit for ε → 0 to obtain a scheme with subgradients of the total variationfunctional. A similar form of this Γ-limit already appeared in the context of decomposition and restoration for grayvalueimages, see for example [9] and [6]. Motivated by these works a fourth order TV-inpainting approach is proposed and firstnumerical examples are presented.

2 TV − H−1 inpainting

We consider the following iterative scheme for (1)

uk+1 = arg minJε

k(u) (2)

with

Jε

k(u) =

∫Ω

(ε

2|∇u|

2+

1

εF (u)

)dx +

1

2τ

∥∥∇∆−1(u − uk)∥∥2

+λ0

2

∥∥∥∥∇∆−1(u −λ

λ0

f − (1 −λ

λ0

)uk)

∥∥∥∥2

,

and uk = u(tk), tk+1 − tk = τ . Our first result is stated in the following theorem and shows that problem (2) attains a uniquesolution in H1(Ω).

Theorem 2.1 The optimization problem (2) attains a unique minimum in H1(Ω).

∗ E-mail: [email protected]∗∗ E-mail: [email protected]∗∗∗ E-mail: [email protected]† Corresponding author E-mail: [email protected], Phone: +43 1 4277 50717, Fax: +43 1 4277 50620

PAMM · Proc. Appl. Math. Mech. 7, 1041905–1041906 (2007) / DOI 10.1002/pamm.200700377

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Fig. 1 TV − H−1 inpainting: u(1500) with λ = 10

3

To build the connection to the inpainting of grayvalue images we will show (cf. [5]) that minimizers of (2) Γ-converge tosolutions of an optimization problem regularized with the TV norm. In fact Modica and Mortola have shown in [7] and [8]that the sequence of Cahn-Hilliard functionals CH(u) =

∫Ω( ε

2|∇u|

2+ 1

εF (u)) dx Γ-converges in the topology L1(Ω) to

TV (u) =

C0

∫Ω|∇u| dx if |u(x)| = 1 a.e. in Ω

+∞ otherwise

as ε → 0, where C0 = 2∫ 1

−1

√F (s) ds. Motivated by this Γ-limit we consider the minimization problem for the following

sequence of functionals

Jk(u) := TV (u) +1

2τ||u − uk||

2−1 +

λ0

2||u −

λ

λ0

f − (1 −λ

λ0

)uk||2−1, (3)

with

TV (u) =

|u|

BVif − 1 ≤ u(x) ≤ 1 a.e. in Ω

+∞ otherwise,(4)

for f ∈ BV (Ω), |f | ≤ 1 is the given grayvalue image destroyed inside the inpainting domain D. We obtain the followingtheorem,

Theorem 2.2 The minimization problem for (3) attains a unique solution in BV (Ω).

Further we can prove that minimizers of Jk(u) converge to a steady state.

Theorem 2.3 Minimizers uk+1 of (3) weakly converge in H−1 for k → ∞ to solutions u of the equation

−1(λ(u − f)) = p, (5)

where p ∈ ∂TV (u) and u is a stationary solution of the evolution equation

−∆−1ut = p − ∆−1(λ(u − f)). (6)

A detailed description of the stated results can be found in [5].1

References

[1] L.I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, Vol.60(1-4), .259-268(1992).

[2] T.F. Chan, S.H. Kang, and J. Shen, Euler’s elastica and curvature-based inpainting, SIAM J. Appl. Math., Vol.63(2), 564–592 (2002).[3] A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of Binary Images Using the Cahn-Hilliard Equation, IEEE Trans. Image Proc.

Vol.16(1), 285-291 (2007).[4] A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting, accepted in Multiscale

Modeling and Simulation.[5] M. Burger, L. He, P. Markowich, C.-B. Schonlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, in preparation.[6] L. Lieu and L. Vese, Image restoration and decompostion via bounded total variation and negative Hilbert-Sobolev spaces, UCLA

CAM Report 05-33, May 2005.[7] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici, Boll. Unione Mat. Ital., V. Ser., A 14,

pp. 526-529, 1977.[8] L. Modica, S. Mortola, Un esempio di Γ

−-convergenza, Boll. Unione Mat. Ital., V. Ser., B 14, pp. 285-299, 1977.[9] S. Osher, A. Sole, and L. Vese. Image decomposition and restoration using total variation minimization and the H -1 norm, UCLA

CAM Report, 02(57), 2002.

1 This work was partially supported by the WWTF (Wiener Wissenschafts-, Forschungs- und Technologiefonds) project nr.CI06 003 and by the FWF(Fonds zur Forderung der wissenschaftlichen Forschung) Wittgenstein award of Peter Markowich, project nr. Z-50 MAT


ICIAM07 Minisymposia – 04 Partial Differential Equations (linear and non-linear) 1041906

A generalization of Cahn-Hilliard inpainting for grayvalue images

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