A Well-Posedness Framework for Inpainting Based on Coherence Transport

Image inpainting is the process of touching-up damaged or unwanted portions of a picture and is an important task in image processing. For this purpose Bornemann and März [J. Math. Imaging Vis. , 28 (2007), pp. 259– 278] introduced a very efficient method called Image Inpainting Based on Coherence Transport which fills the missing region by advecting the image information along integral curves of a coherence vector field from the boundary towards the interior of the hole. The mathematical model behind this method is a first-order functional advection PDE posed on a compact domain with all inflow boundary. We show that this problem is well-posed under certain conditions. 1. Introduction. As image processing has become an active field of mathematical research [15, 5], the task of digital image inpainting has also been approached by mathematical methods in the last decades. Image inpainting serves the purpose of touching-up damaged or unwanted portions of a picture. In mathematical image processing, images are considered to be functions w : Ω 0 → R defined on a typically rectangular image domain Ω 0 ⊂ R 2. The value w(x) ∈ R often represents an intensity of light which is perceptible as a gray color. Technically, image inpainting is a problem of data interpolation: apart from Ω 0 we are given, as part of the problem formulation, an inpainting domain, i.e., a subdomain Ω ⊂ Ω 0 which marks the damage or the portion which has to be touched-up. And the " good " part of the image, which is to be kept, is given as a function u 0 : Ω 0 \ Ω → R defined on the data domain Ω 0 \ Ω (u 0 is undefined on Ω). Now, the task is to find a function u : Ω → R, defined on the missing part Ω, which interpolates the data u 0. The very important side condition on an acceptable solution u is that the completed image ¯ u := u 0 · 1 Ω 0 \Ω + u · 1 Ω , now defined on the whole image domain Ω 0 , should look nice, i.e., u should interpolate the data u 0 in a visually plausible manner. Since there is no ubiquitous mathematical definition of the terms " nice " and " visually plausible " such a definition is part of the task. Consequently, many different approaches to …