Reconstruction of sampled surfaces with boundary via Morse theory

Abstract

We study the perception problem for garments (e.g. a pair of pants) using tools from computational topology: the identification of their geometry and position from point-cloud samples, as obtained e.g. with 3D scanners. We present a reconstruction algorithm based on Morse theory that proceeds directly from the point-cloud to obtain a cellular decomposition of the surface derived via a Morse function. No intermediate triangulation or local implicit equations are used, avoiding reconstruction-induced artifices. The results are a piecewise parametrization of the surface as a union of Morse cells, suitable for tasks such as noise-filtering or mesh-independent reparametrization, and a cell complex of small rank determining the surface topology. This algorithm can be applied to smooth surfaces with or without boundary, embedded in an ambient space of any dimension.