Nonlinear regularizations of TV based PDEs for image processing

We consider both second order and fourth order TV-based PDEs for image processing in one space dimension. A general class of nonlinear regularizations of the TV functional result in well-posed uniformly parabolic equations in two dimensions. However the fourth order analogue (Osher et. al. Multiscale Methods and Simulation 1(3) 2003) based on a total variation minimization in an H−1 norm, has very different properties. In particular, nonlinear regularizations should have special structure in order to guarantee that the regularized PDE does not produce finite time singularities. 1991 Mathematics Subject Classification. Primary 35G25; Secondary 68U10, 94A18, 35Q80. AB and JG are supported by ONR grants N000140410078 and N000140410054, NSF grants ACI-0321917 and DMS-0244498, and ARO grant DAAD19-02-1-0055. This paper is based upon work supported by the National Science Foundation and the intelligence community through the joint “Approaches to Combat Terrorism” program (NSF Grant DMS-0345602). We thank Tony Chan for useful comments. 1 2 ANDREA BERTOZZI, JOHN GREER, STANLEY OSHER, AND KEVIN VIXIE Nonlinear PDEs are now commonly used in image processing for issues ranging from edge detection, denoising, and image inpainting, to texture decomposition. Second order PDEs for image denoising and boundary or edge sharpening date back to the seminal works of Rudin-Osher-Fatemi [14], and Perona-Malik [13]. All of these methods have some common features; they are based on a nonlinear version of the heat equation (0.1) ut = ∇ · ((g(|∇u|)∇u) in which the ‘thresholding function’ g is small in regions of sharp gradients. A number of mathematical issues arise with these equations and their use. For example, Perona-Malik, suggest using a smooth g that decays fast enough for large ∇u so that significant diffusion only takes place in regions of small change in the image, i.e. away from edge boundaries. A typical choice might be (0.2) g(s) = k2/(k2 + s2). However, this and similar choices result in a PDE that is linearly ill-posed in regions of high gradients and the ensuing dynamics results in a characteristic “staircase” instability. A particular class of denoising algorithms are the TV (total variation) methods introduced by Rudin, Osher and Fatemi [14]. The technique minimizes the total variation norm of the image. The TV functional is defined as (0.3) TV (u) = Z Ω |∇u|. The TV functional does not penalize discontinuities in u and thus allows one to recover the edges of the original image. The restoration problem can be written as