Digital geometry processing with topological guarantees

Abstract

We describe novel approaches to compute reliable solutions for many non-linear geometric problems that arise in geometric modeling, computer graphics and robotics. Specifically, we focus on problems that can be formulated as surface extraction problems. These include Boolean operations and Minkowski sums of polyhedral or higher models as well as reliable polygonization of general implicit surfaces. All these problems reduce to computing a topology preserving isosurface from a volumetric grid, i.e. the zero set of a scalar field. A common way of representing a scalar field is to discretize the continuous scalar field into discrete samples - to compute the value of the scalar field at the vertices of a volumetric grid. We refer to this step as a sampling of the scalar field. The grid is an approximate representation of the scalar field; the accuracy of the approximate representation depends on the rate of sampling - the resolution of the grid. An explicit boundary representation of the implicit surface can be obtained by extracting the zero-level isosurface using Marching Cubes or any of its variants. We refer to these isosurface extraction algorithms collectively as MC-like algorithms. The output of an MC-like algorithm is an approximation - usually a polygonal approximation - of the implicit surface. We refer to this step as reconstruction of the implicit surface. © 2008 Springer-Verlag Berlin Heidelberg.