Recovering the polyhedral geometry of fragments

Abstract: Not only is the geometry of rock fragments often well approximated by ideal convex polyhedra having few faces and vertices, but these numbers carry vital geophysical information on the fragmentation process. Despite their significance, the identification of the ideal polyhedron has so far been carried out through visual inspection. Here, we present an algorithm capable of performing this task in a reliable manner. The input is a 3D scan of the fragment which is essentially a triangulated polyhedron, which however has often large number of faces and vertices. Our algorithm performs a systematic simplification using the following steps: - Gaussian smoothing is performed on the spherical histogram of the 3D scans faces to identify the most important face orientations. - Planes carrying the faces of the ideal polyhedron are identified as maxima of the smoothed histogram - Polygon is reconstructed using the identified planes - Small faces are removed in a systematic manner We present two versions of the algorithm that we benchmarked the algorithm against a dataset of human measurements on 132 fragments. Beyond identifying the ideal polyhedral approximation for fragments, our method is also capable of tracing backward the shape evolution of rounded pebbles to their origins.