Images and PDE's

In this paper I will present an overview of some results, concerning EDP's and Image Processing, obtained by the author of this job with the collaboration of the following researchers: J.Esclarfn, F.Guichard, P.L.Lions, L.Mazorra, F.Morales, J.M.Morel and F.Santana that belong to the CEREMADE laboratory of the University of Paris IX and the Computer Science Department of the University of Las Palmas (The Canary Islands). This job is the result of several research projects together with both universities and has given raise to different publications such as: [ALM92], [AGLM92c], [AGLM92b], [AGLM92a], [AGLM93], [AM94a], [AM94c], [AM94d], [AM94b], [AE94]. I am going to focus my attention in the results and experiences that I have accumulated in the last years. So I am not go to present a general overview of the field which is impossible in a single paper. In particular, I will not quote, a great number of interesting aspects and results introduced for other authors. I think that the theoretical background of the results that I'm going to present here are very well exposed in the above references. Rather than to repeat in this paper the theoretical presentation of the mathematical results, I am going to try, in this paper, to give an unified overview of the numerical analysis aspects associated to these equations. Of course, I would like to say again, that this presentation is based in my experiences and in my particular point of view. We will study some applications of partial differential equations to image processing. For us, an image is a function f : ~n _.4 N where in general n = 2, though in some cases n = 1 or n = 3 also. The starting point in most applications of PDE's to image processing is to take the original image f as the initial datum of a PDE of the form: