Journal of Mathematical Imaging and Vision 14: 195–209, 2001 c  2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Diffusions and Confusions in Signal and Image Processing N. SOCHEN Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel sochen@math.tau.ac.il R. KIMMEL AND A.M. BRUCKSTEIN Department of Computer Science, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel ron@cs.technion.ac.il freddy@cs.technion.ac.il Abstract. In this paper we link, through simple examples, between three basic approaches for signal and image denoising and segmentation: (1) PDE axiomatics, (2) energy minimization and (3) adaptive filtering. We show the relation between PDE’s that are derived from a master energy functional, i.e. the Polyakov harmonic action, and non-linear filters of robust statistics. This relation gives a simple and intuitive way of understanding geometric differential filters like the Beltrami flow. The relation between PDE’s and filters is mediated through the short time kernel. Keywords: Anisotropic diffusion, selective smoothing, geometric filtering 1. Introduction Averaging is a standard procedure for smoothing noisy data and summarizing information, but it can have rather dangerous and misleading results. Outliers, even if rare by definition, can distort the results consider- ably if they are given similar weights to “typical” data. Such concerns led to the development of so-called ro- bust estimation procedures in statistical data analysis, procedures that data-adaptively determine the “influ- ence” each data point will have on the results. Only recently were such ideas and methods imported to sig- nal and image processing and analysis [3, 6, 11, 16]. The application of the robust statistics ideas in signal and image analysis lead to the introduction of various non-linear filters. To fix ideas and get a perspective on the serious problems that must be addressed we shall consider below a series of simple examples. A seemingly different approach to denoising and segmentation is based on geometric properties of signals. The filtering is done, in this approach, by solv- ing a non-linear Partial Differential Equation (PDE). The derivation of the PDE is based either on axioms and requirements, such as invariance, separability [1, 15, 17] etc., or as a by product of a minimization pro- cess for an energy functional [15, 18, 21]. In this paper we discuss, through simple examples, the intimate connection between the above-mentioned signal and image processing methodologies: (1) PDE axiomatics, (2) energy minimization and (3) adaptive smoothing filters. We show the relation between PDE’s that are derived from a master energy functional and non-linear filters. This relation gives a simple and intu- itive way of understanding the Beltrami flow, and con- nects between geometric differential filters and classi- cal linear and non-linear filters. We show that the Beltrami flow, which results from the minimization process of the Polyakov action, is related to non-linear filters of special type upon choos- ing a L γ induced metric and after discretization of the