Schönlieb, Carola-Bibiane; Bertozzi, Andrea Unconditionally stable schemes for higher order inpainting. (English) Zbl 1216.94016 Commun. Math. Sci. 9, No. 2, 413-457 (2011). 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). Cited in 42 Documents 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 Keywords:image inpainting; higher order equations; numerical schemes PDFBibTeX XMLCite \textit{C.-B. Schönlieb} and \textit{A. Bertozzi}, Commun. Math. Sci. 9, No. 2, 413--457 (2011; Zbl 1216.94016) Full Text: DOI Euclid