Probabilistic triangles for point set surfaces

In this paper, we propose a novel statistical surface representation for point data sets, called probabilistic triangles (PT). This framework has a unique characteristic such that it is statistical method based on the triangulation compared with the other geometry representations and surface reconstruction algorithms. Given a point set, its PT can be considered as an average or expected surface of all possible meshes that interpolate the input point set. To construct a PT from the point set, we compute the weight of each triangle candidate which is a spatially close 3-combination of points. A set of combinations of triangles and their associated weights defines the PT. To perform ray shooting or directional query on the PT, which is often required by many graphical applications, we find the triangles intersected with the ray, and, using the weights of the triangles, calculate the probability of each intersected triangle that may correspond to the unknown underlying surface, from which the input data sets are sampled. We also provide a novel algorithm to efficiently render the PT based on the notion of expected position and normal. We also show that the rendering can be effectively implemented on GPUs, and the results are visually pleasing compared to other alternatives, especially for noisy point data sets. (C) 2015 Elsevier Ltd. All rights reserved.