Non-convex hull surfaces

Abstract

We present a new algorithm to reconstruct approximating watertight surfaces from finite oriented point clouds. The Convex Hull (CH) of an arbitrary set of points, constructed as the intersection of all the supporting linear half spaces, is a piecewise linear watertight surface, but usually a poor approximation of the sampled surface. We introduce the Non-Convex Hull (NCH) of an oriented point cloud as the intersection of complementary supporting spherical half spaces; one per point. The boundary surface of this set is a piecewise quadratic interpolating surface, which can also be described as the zero level set of the NCH Signed Distance function. We evaluate the NCH Signed Distance function on the vertices of a volumetric mesh, regular or adaptive, and generate an approximating polygonal mesh for the NCH Surface using an isosurface algorithm. Despite its simplicity, this simple algorithm produces high quality polygon meshes competitive with those generated by state-of-the-art algorithms. The relation to the Medial Axis Transform is described.