Thomas Holding

Thomas Holding, Ioannis Lestas

In part I we considered the problem of convergence to a saddle point of a concave-convex function via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of subgradient dynamics that provide a restriction in an arbitrary convex domain. We show that despite the nonlinear and non-smooth character of these dynamics their $ω$-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.

L. Robert Hocking, Thomas Holding, Carola-Bibiane Schoenlieb

In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging, and the neighborhood that is averaged over. Varying these degrees of freedom leads to different algorithms, and indeed the literature contains several methods falling into this general class. All of them are very fast, but at the same time all of them leave undesirable artifacts such as "kinking" (bending) or blurring of extrapolated isophotes. Our objective in this paper is to build a theoretical model, based on a continuum limit and a connection to stopped random walks, in order to understand why these artifacts occur and what, if anything, can be done about them. At the same time, we consider a semi-implicit extension in which pixels in a given shell are solved for simultaneously by solving a linear system. We prove (within the continuum limit) that this extension is able to completely eliminate kinking artifacts, which we also prove must always be present in the direct method. Although our analysis makes the strong assumption of a square inpainting domain, it makes weak smoothness assumptions and is thus applicable to the low regularity inherent in images.