Analysis of a Discrete, 4th Order PDE from Image Processing

Catherine Dupuis
University of Michigan

Abstract: We study the discrete version of the ill-posed, 4th order, nonlinear diffusion equation proposed by You and Kaveh as an edge preserving image denoising and segmentation model. It is the high order analogue of the Perona-Malik model and was intended to address some of its well-known drawbacks such as staircasing. Like the Perona-Malik model, the continuum version of the You-Kaveh equation violates parabolicity and lacks well-posedness theory. We prove a weak upper bound on the coarsening rate of the discrete-in-space version of this equation in any dimensions. Our bounds are obtained by following a recent technique of Kohn and Otto. They constitute a rigorous step towards understanding how important parameters in the model should be chosen.