Well-posed anisotropic diffusion for image denoising

A nonlinear iterative smoothing filter based on a second-order partial differential equation is introduced. It smooths out the image according to an anisotropic diffusion process. The approach is based on a smooth approximation of the total variation (TV) functional which overcomes the non-differentiability of the TV functional at the origin. In particular, the authors perform linear smoothing over smooth areas but selective smoothing over candidate edges. By relating the smoothing parameter to the time step, they arrive at a CFL condition which guarantees the causality of the discrete scheme. This allows the adoption of higher time discretisation steps, while ensuring the absence of artefacts deriving from the non-smooth behaviour of the TV functional at the origin. In particular, it is shown that the proposed approach avoids the typical staircase effects in smooth areas which occur in the standard time-marching TV scheme.