Towards Multi-scale Heat Kernel Signatures for Point Cloud Models of Engineering Artifacts

Point clouds are becoming ubiquitous thanks to the recent advances in low-cost depth camera hardware. However, many key shape analysis techniques require the availability of a global mesh model and exploit its intrinsic connectivity information. We seek to develop methods for global and local geometric similarity, feature recognition, and segmentation that operate directly on point clouds corresponding to engineering artifacts. Specifically, we are investigating Heat Kernel Signatures (HKS) of point clouds that use a Point Cloud Data Laplace (PCL) estimation, and explore feature vectors that are well suited to the unique challenges posed by objects with sharp features, as are often encountered in engineered objects. We also discuss promising research directions aimed at providing robust shape segmentation and classification algorithms for point cloud data corresponding to engineering artifacts. 1 The Point Cloud Data Laplacian and the Heat Kernel Signature We consider global and local similarity between point cloud data of engineering artifacts, such as individual parts or assemblies that are in (nearly) isometric invariant configurations. Furthermore, the relatively noisy point clouds, such as those captured by depth sensors, require a signature that is relatively stable under small perturbations. One such signature that was recently developed for mesh models is the Heat Kernel Signature [1] that is defined in terms of a discrete version of the Laplace-Beltrami operator. It has been shown that HKS possesses key properties such as isometric invariance, informativeness, multi-scale description, and stability under small perturbations (e.g., noise). Moreover, a careful selection of the HKS maxima that are being tracked can even provide matching between incomplete or partial models [2], providing utility in cases where, for example, occlusion is unavoidable. The Heat Kernel Signature is one of many “spectral” techniques for shape analysis that relies on the eigensystem (also called the spectrum) of the Laplace-Beltrami operator of the underlying manifold. The Laplace-Beltrami operator of a smooth Riemannian manifold is the manifold geometry analogue of the Laplace operator, i.e., the divergence of the gradient of a function on a smooth Riemannian manifold (M,μ). The eigensystem of this Laplacian operator is thus the spectrum of the manifold under examination and its eigenfunctions form a natural basis for functions on the manifold. The Heat Kernel Signature relies on this basis and has been shown to be the basis of the heat kernel itself [1], which, in turn is an intrinsic solution to the heat equation [3]. This heat kernel intuitively describes how an initial heat distribution evolves over time on surfaces, a process which is dependent on the geometry and topology of the surface. It has been shown in [1] that, in order to construct ∗Corresponding author