Three-dimensional point cloud recognition via distributions of geometric distances

A geometric framework for the recognition of three-dimensional objects represented by point clouds is introduced in this paper. The proposed approach is based on comparing distributions of intrinsic measurements on the point cloud. In particular, intrinsic distances are exploited as signatures for representing the point clouds. The first signature we introduce is the histogram of pairwise diffusion distances between all points on the shape surface. These distances represent the probability of traveling from one point to another in a fixed number of random steps, the average intrinsic distances of all possible paths of a given number of steps between the two points. This signature is augmented by the histogram of the actual pairwise geodesic distances in the point cloud, the distribution of the ratio between these two distances, as well as the distribution of the number of times each point lies on the shortest paths between other points. These signatures are not only geometric but also invariant to bends. We further augment these signatures by the distribution of a curvature function and the distribution of a curvature weighted distance. These histograms are compared using the χ² or other common distance metrics for distributions. The presentation of the framework is accompanied by theoretical and geometric justification and state-of-the-art experimental results with the standard Princeton 3D shape benchmark, ISDB, and nonrigid 3D datasets. We also present a detailed analysis of the particular relevance of each one of the different proposed histogram-based signatures. Finally, we briefly discuss a more local approach where the histograms are computed for a number of overlapping patches from the object rather than the whole shape, thereby opening the door to partial shape comparisons.