Topological reconstruction of sampled surfaces via Morse theory

TL;DR

This paper introduces a topology-based surface reconstruction algorithm from point-cloud data that avoids triangulation, works without prior knowledge of the surface, and produces a cellular decomposition suitable for various applications.

Contribution

The novel contribution is a Morse theory-based reconstruction method that directly derives a cellular decomposition from point samples without triangulation or implicit equations.

Findings

Successfully applied to real and synthetic surfaces

Works with surfaces with or without boundary

Applicable in any ambient space dimension

Abstract

In this work, we study the perception problem for sampled surfaces (possibly with boundary) using tools from computational topology, specifically, how to identify their underlying topology starting from point-cloud samples in space, such as those obtained with 3D scanners. We present a reconstruction algorithm based on a careful topological study of the point sample that allows us to obtain a cellular decomposition of it using a Morse function. No triangulation or local implicit equations are used as intermediate steps, avoiding in this way reconstruction-induced artifices. The algorithm can be run without any prior knowledge of the surface topology, density or regularity of the point-sample. The results consist of a piece-wise decomposition of the given surface as a union of Morse cells (i.e. topological disks), suitable for tasks such as mesh-independent reparametrization or noise-filtering, and a small-rank cellular complex determining the topology of the surface. The algorithm, which we test with several real and synthetic surfaces, can be applied to smooth surfaces with or without boundary, embedded in an ambient space of any dimension.