MULTIGRID METHOD FOR A MODIFIED CURVATURE DRIVEN DIFFUSION MODEL FOR IMAGE INPAINTING

Digital inpainting is a fundamental problem in image processing and many variationalmodels for this problem have appeared recently in the literature.Among them are the verysuccessfully Total Variation (TV) model [11] designed for local inpainting and its improvedversion for large scale inpainting:the Curvature-Driven Diffusion (CDD) model [10].Forthe above two models,their associated Euler Lagrange equations are highly nonlinear partialdifferential equations.For the TV model there exists a relatively fast and easy toimplement fixed point method,so adapting the multigrid method of [24] to here is immediate.For the CDD model however,so far only the well known but usually very slow explicittime marching method has been reported and we explain why the implementation of afixed point method for the CDD model is not straightforward.Consequently the multigridmethod as in [Savage and Chen,Int.J.Comput.Math.,82 (2005),pp.1001-1015] willnot work here.This fact represents a strong limitation to the range of applications of thismodel since usually fast solutions are expected.In this paper,we introduce a modificationdesigned to enable a fixed point method to work and to preserve the features of the originalCDD model.As a result,a fast and efficient multigrid method is developed for themodified model.Numerical experiments are presented to show the very good performanceof the fast algorithm.