Nonlinear operators in image restoration Pierre Kornprobst, Rachid Deriche INRIA 2004 route des Lucioles, BP 93 F-06902 Sophia-Antipolis Cedex , France e-mail: {pkornp,der}@sophia.inria.fr Gilles Aubert Laboratoire J.A. Dieudonne ´ UMR n 6621 du CNRS 06108 Nice Cedex 2, France e-mail: aubert@orque.unice.fr Abstract We firstly present a variational approach such that dur- ing image restoration, edges detected in the original im- age are being preserved, and then we compare in a sec- ond part, the mathematical foundation of this method with respect to some of the well known methods recently pro- posed in the literature within the class of PDE based al- gorithms (anisotropic diffusion, mean curvature motion, min/max flow technique,...). The performance of our approach is carefully examined and compared to the classical methods. Experimental re- sults on synthetic and real images will illustrate the capa- bilities of all the studied approaches. 1. Introduction Numerous PDE based algorithms have been proposed re- cently to tackle the problems of noise removal, image en- hancement and image restoration in real images. These methods are based on evolving nonlinear partial differen- tial equations (PDE’s) (e.g. Perona & Malik [9], Nordstro ¨m [8], Shah [14], Osher & Rudin [11, 12], Proesman et al.[10], Cottet and Germain, Alvarez et al [1, 2], Cohen [4], Weick- ert, Malladi & Sethian [7], Aubert et al. [3], You et al.[15], Sapiro et al.[13] , ...). Hence, due this large number of approaches, it clearly ap- pears that there is a strong need to better compare and quan- tify the experimental results of the most promising tech- niques. In this article, three different approaches are proposed and a comparison is made in terms of theoretical back- ground, quality of results and efficiency with the most promising techniques already proposed in literature [9, 1, 7, 12]. 2. Minimizing a functional 2.1. An edge preserving regularization The method we propose here is based on minimizing in the space , the space of bounded variations in (the domain of the image), the following energy: (1) where and are two constants, is the noisy image, and is a function still to be defined. Notice that if , we recognize the Tikhonov regularization term. This method is well known to smooth the image isotropically without pre- serving discontinuities in intensity. What about changing this quadratic term ? The key idea is that for low gradients, isotropic smoothing should be performed, and for high gra- dient, smoothing should only be applied in the direction of the isophote and not across it [15, 5, 3]. If we assume that the diffusion operator can be written , where we note , and the normal vector to , we can for- mulate preceding conditions by: (2) and (3) In the case of problem (1), the Euler-Lagrange equation is: (4) with neuman conditions at the boundary and we can decom- pose the diffusion operator: