The nonlinear diffusion model introduced by Perona and Malik (1990 IEEE
Trans. Pattern Anal. Mach. Intell. 12 629–39) is well suited to preserve salient
edges while restoring noisy images. This model overcomes well-known edge
smearing effects of the heat equation by using a gradient dependent diffusion
function. Despite providing better denoizing results, the analysis of the PM
scheme is difficult due to the forward-backward nature of the diffusion flow.
We study a related adaptive forward-backward diffusion equation which uses a
mollified inverse gradient term engrafted in the diffusion term of a general
nonlinear parabolic equation. We prove a series of existence, uniqueness and
regularity results for viscosity, weak and dissipative solutions for such forward-
backward diffusion flows. In particular, we introduce a novel functional
framework for wellposedness of flows of total variation type. A set of synthetic
and real image processing examples are used to illustrate the properties
and advantages of the proposed adaptive forward-backward diffusion flows.
author = "V. B. S. Prasath and J. M. Urbano and D. Vorotnikov",
title = "Analysis of adaptive forward-backward diffusion flows with applications in image processing",
year = 2015,
journal = "Inverse Problems",
volume = 31,
number = 105008,
pages = "30pp",
keywords = "restoration, anisotropic diffusion",
doi = "10.1088/0266-5611/31/10/105008"
V. B. S. Prasath, J. M. Urbano, and D. Vorotnikov. Analysis of adaptive forward-backward diffusion flows with applications in image processing. Inverse Problems, volume 31, issue 105008, pages 30pp, 2015.