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Unconditionally stable schemes for higher order inpainting. (English) Zbl 1216.94016

Summary: Higher order equations, when applied to image inpainting, have certain advantages over second order equations, such as continuation of both edge and intensity information over larger distances. Discretizing a fourth order evolution equation with a brute force method may restrict the time steps to a size up to order \(\Delta x^4\) where \(\Delta x\) denotes the step size of the spatial grid. In this work we present efficient semi-implicit schemes that are guaranteed to be unconditionally stable. We explain the main idea of these schemes and present applications in image processing for inpainting with the Cahn-Hilliard equation, TV-H\(^{-1}\) inpainting, and inpainting with LCIS (low curvature image simplifiers).

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
35G25 Initial value problems for nonlinear higher-order PDEs
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
49M25 Discrete approximations in optimal control
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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