Topological reconstruction of sampled surfaces via Morse theory

TL;DR

This work presents a reconstruction algorithm based on a careful topological study of the point sample that allows for a cellular decomposition of it using a Morse function, avoiding in this way reconstruction-induced artifices.

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.

Figures (15)

• Figure 1: Critical points of the Morse-Smale function $f(x,y,z) = z$ on an example surface. The cellular decomposition is achieved by attaching two $2$-cells containing $d$ and $c$ (the maxima) along the curve passing through $b$ (the saddle point) and $a$ (the minimum).
• Figure 2: Three types of critical points $p$ (from top to bottom: minima, saddles and maxima) and their Morse data $(A(p),B(p))$ for surfaces without boundary.
• Figure 3: Critical points of the Morse-Smale function $f(x,y,z) = z$ on an example surface with boundary. Notice that now we have a boundary maximum $b'$ and a boundary minimum $a'$ since we have removed a disk.
• Figure 4: Typical problems associated to k-nearest neighbors (left) and Voronoi neighbors (right). On the left, most closest points to $v$ (in bold) are clustered at one side of it. On the right, vertices that are too far apart from each other belong to neighboring cells.
• Figure 5: In principle, a boundary point can be identified easily because after projecting it and its neighbors on the tangent plane, they cluster in a semi-space of $\mathbb{R}^2$ (left panel). Nevertheless, this is not always the case for every boundary point (middle panel). To overcome this, we declare a point as being on the boundary when none of the plane projections of it and its neighbors in all their respective tangent planes enclose the point (right panel).

Theorems & Definitions (13)

• Definition 1: Morse data
• Theorem 1: Main theorem of Morse theory
• Definition 2: Morse function
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition 3: Tangential and normal data
• Remark 6