Multigrid Convergence and Surface Area Estimation David Coeurjolly 1 , Fr´ ed´ eric Flin 2 , Olivier Teytaud 3 , and Laure Tougne 1 1 Laboratoire ERIC 5, avenue Pierre-Mendes-France, F-69676 Bron Cedex {dcoeurjo,ltougne}@eric.univ-lyon2.fr 2 Centre d’Etudes de la Neige 1441, rue de la Piscine, Domaine Universitaire, F-38406, Saint Martin d’H` eres Cedex Frederic.Flin@meteo.fr 3 Artelys 215, rue Jean-Jacques Rousseau, F-92136 Issy-les-Moulineaux Cedex olivier.teytaud@artelys.com Abstract. Surface area of discrete objects is an important feature for model-based image analysis. In this article, we present a theoretical framework in order to prove multigrid convergence of surface area es- timators based on discrete normal vector field integration. The paper details an algorithm which is optimal in time and multigrid convergent to estimate the surface area and a very efficient algorithm based on a lo- cal but adaptive computation. 1 Introduction In three-dimensional shape analysis, the surface area is one of the important features. Its definition and calculation are well known in classical mathematics. Problemsarisewhen we want to define such a measure on discrete data. Recently, many papers have proposed both theoretical analysis and algorithmic aspects for the surface area estimation of discrete surfaces. When an estimation of an Euclidean measure is proposed on the discrete model, a way to formally evaluate this estimator is to consider its multigrid con- vergence [17]: we assume a multigrid digitization of a family of Euclidean shapes and we prove that the proposed estimator will converge to the Euclidean mea- sure when the grid resolution increases. An important interest of this property is the soundness of such an estimator in multi-scale object analysis processes. In the literature, two main approaches exist, the first one consists in polyhe- dral approximation of the discrete volume. In that approach, Klette et al. [16] use a digital plane segmentation process but no proof of multigrid convergence is given. Sloboda et al. [27] introduce the notion of relative convex hull of the discrete objet, this approach is multigrid convergent but no algorithm exists. Another approach consists in local approximations given a neighborhood. More precisely, Mullikin et al. [23] consider a finite set of voxels configurations in a given neighborhood on a discrete surface and they associate a weight to each T. Asano et al. (Eds): Geometry, Morphology, . . . 2002, LNCS 2616, pp. 101–119, 2003. c  Springer-Verlag Berlin Heidelberg 2003