Analysis of adaptive forward-backward diffusion flows with applications in image processing. (24th September 2015)

• Record Type: Journal Article
• Authors: Prasath, V B Surya; Urbano, José Miguel; Vorotnikov, Dmitry
• Is Part Of: Inverse problems. Volume 31:Number 10(2015:Oct.)
• Publication Date: 2015-09-24
• Subjects: anisotropic diffusion -- regularization -- image restoration -- forward backward diffusion -- wellposedness -- total variation flow; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/10/105008
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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.