A Well-Posedness Framework for Inpainting Based on Coherence Transport

Abstract

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.

Guidefill: GPU Accelerated, Artist Guided Geometric Inpainting for 3D Conversion of Film

“…3: Creation of shocks by Algorithm 1: When Algorithm 1 is used to inpaint problems with incompatible boundary conditions, such as the problem illustrated in (a) of inpainting a stripe that is red on one end and green on the other, the result may contain shocks as in (b). These shocks can be understood by adopting the framework proposed in [7,28], where the output of Algorithm 1 under a high resolution and vanishing viscosity limit is shown to be equivalent to the solution of a first order transport equation on D Σ, where Σ is a set of measure zero containing any potential shocks. Ballester et al [4] suggested overcoming this problem by adding a diffusive term − ∆u to the transport equation and taking → 0.…”

“…See Figure 3(c), where we solve the resulting nonsymmetric linear system with = 10 −7 using GMRES (the Generalized Minimum RESidual method for nonsymmetric linear systems) [36]. In a series of papers [7,27,28] März took a different approach and instead showed that (3.2) is well posed on D Σ, where Σ is a set of measure zero containing any potential shocks, and related to a distance map prescribing the order in which pixels are filled. In our case this issue is less significant as we only specify boundary data on ∂ active D h ⊂ ∂D h .…”

Analysis of Artifacts in Shell-Based Image Inpainting: Why They Occur and How to Eliminate Them

“…However, they were puzzled by some of the artifacts they were seeing, most notably kinking artifacts. After a literature review it became clear that the existing theory [7,31,32] was enough to explain some, but not all, of what they observed. This paper is an attempt to fill the gap.…”

“…Although both are perfectly valid mathematically, numerical experiments indicate that the high resolution viscosity limit gives a good approximation of the behaviour of Algorithm 1 in practice only when r 1, whereas our fixed-ratio limit gives a good approximation even when r is a small integer, as it typically is in practice (see Remark 4.3 in our previous work [27] for an explanation of why this is). There has also been significant work in studying the well-posedness of the high resolution and vanishing viscosity limit of Algorithm 1, both in [7] and especially in [32]. See Fig.…”

“…Our contributions are both theoretical and practical, aimed at a deeper understanding of the mathematical properties of both the direct and semi-implicit versions of Algorithm 1, and through that understanding, a better grasp of the underlying causes of the artifacts described above and what, if anything, can be done about them. Our main targets are "kinking" and "blurring" artifacts, the others having already been thoroughly analyzed in [7,32] and well understood. Broadly speaking, we have three main contributions:…”