Approximation of a variable density cloud of points by shrinking a discrete membrane Jordi Esteve Pere Brunet Àlvar Vinacua e-mail: [jesteve, brunet, alvar]@lsi.upc.es Universitat Politècnica de Catalunya. Barcelona Abstract This paper describes a method to obtain a closed surface that approximates a general 3D data point set with non-uniform density. Aside from the positions of the initial data points, no other information is used. Particularly, neither the topological relations between the points nor the normal to the surface at the data points are needed. The reconstructed surface does not exactly interpolate the initial data points, but approximates them with a bounded maximum distance. The method allows to reconstruct closed surfaces with arbitrary genus and closed surfaces with disconnected shells. ACM CSS: I.3.5 Computational Geometry and Object Modeling. Keywords: Scattered data points, surface approximation, voxelization, discrete geometry. 1. Introduction Scattered data points obtained from real objects with opti- cal, ultrasonic, tactile or other sensors are frequently used as data sources. Geometric modeling applications must process these scattered data points to obtain a surface that approxi- mates the data point set. In this way, • The generated surface will be more compact than the ini- tial data point set. • A more realistic visualization can be obtained from the surface. • Standard geometric modeling operations (surface evalua- tion and editing, surface-surface intersection, etc.) will be feasible. A large diversity of algorithms that approximate scattered data points have been published. There are many valid so- lutions approximating a cloud of points and each algorithm provides a solution with its own “aesthetic”. It is not possible to detail all published papers dealing with this problem here. The reader can consult some of the exist- ing State-of-the-Art reports 1, 2 . Some papers are mentioned below. They are classified in four blocks according to the taxonomy used in the Mencl and Müller report 1 : • Spatial subdivision. The space is decomposed in cells, then the cells that are stabbed by the final surface (sur- face oriented algorithms) or the cells that do not belong to the volume bounded by the surface (oriented to vol- ume) are determined and, from them, the final surface is computed. Some surface oriented algorithms use a regular voxelization (Algorri et al. 3 and Hoppe et al. 4 ), others de- compose the space in tetrahedrals (Bajaj et al. 5 , α-shapes of Edelsbrunner et al. 6 , Amenta et al. 7 and the Cocone al- gorithm of Amenta et al. 8 ). Examples of volume oriented algorithms are Veltkamp 9 , Boissonnat 10 and Isselhard et al. 11 (their strategy is based on computing a Delaunay tetrahedrization of the convex-hull of the data point set and eliminating successively the tetrahedrals carrying out some properties). • Distance function. The distance function calculates the minimum distance from any point of the space to the final surface. The distance function can give positive or neg- ative values if the surface is closed. The final surface is implicitly determined by the distance function. Examples of algorithms using a distance function are Hoppe et al. 4 ,