Fourth order diffusions for image processing

A number of fourth order diffusion equations have recently been introduced for image smoothing and denoising. Although discrete implementations of these methods produce impressive results, very little is known about the mathematical properties of the equations themselves. We prove some of the first results regarding a few of these nonlinear diffusions. In particular, we use energy methods to prove that a class of H1 diffusions for image processing is well posed. We use similar methods to show that the 'Low Curvature Image Simplifier' (LCIS) equation of Tumblin and Turk (SIGGRAPH, August, 1999) has smooth solutions locally in time in R2 , and globally in time in R . We devise a new finite difference discretization of the LCIS equation that ensures the discrete Laplacian of the image intensity remains bounded. We also introduce new model advection-diffusion equations for which we use topology and dynamical system theory to prove analytical results that give insight into the behavior of some of these new diffusions. We study traveling wave solutions of the model equations and provide numerical evidence complementing our results.