Surface Reconstruction Using Rotation Systems

TL;DR

This work advances surface reconstruction from point clouds by leveraging rotation systems and Euler operators to construct a genus-0 polygonization from a spanning tree, then incrementally insert edges to refine faces and optionally add handles to raise genus. The approach uses a topology test to preserve planarity and a geometry test to avoid local intersections, enabling explicit topology control (e.g., genus-0 cortical surfaces) while preserving most input points. It supports reconstruction from noisy data through tangent-plane projections and robust neighbor filtering, and it demonstrates competitive performance on synthetic and real-scanned datasets compared with established baselines. The method offers precise topology control and robustness to outliers, providing a scalable combinatorial alternative to volumetric methods with potential for parallelization and further refinement.

Abstract

Inspired by the seminal result that a graph and an associated rotation system uniquely determine the topology of a closed manifold, we propose a combinatorial method for reconstruction of surfaces from points. Our method constructs a spanning tree and a rotation system. Since the tree is trivially a planar graph, its rotation system determines a genus zero surface with a single face which we proceed to incrementally refine by inserting edges to split faces and thus merging them. In order to raise the genus, special handles are added by inserting edges between different faces and thus merging them. We apply our method to a wide range of input point clouds in order to investigate its effectiveness, and we compare our method to several other surface reconstruction methods. We find that our method offers better control over outlier classification, i.e. which points to include in the reconstructed surface, and also more control over the topology of the reconstructed surface.

Figures (28)

• Figure 1: Reconstruction of a 2D point cloud. From left to right, the top row shows the input point cloud, the graph formed by connecting each point to its nearest neighbors within a given radius, and the minimum spanning tree of this graph. The bottom row shows the process of adding edges to the minimum spanning tree. The edges are added in order of increasing length, and edges are only added if they do not violate planarity. From left to right, the bottom row images show the MST with 10, 100, and all possible edges added.
• Figure 2: This figure shows a partial graph consisting of vertices, edges, and halfedges (ordered pairs of vertices) indicated as fat arrows. The functions $\rho$, $\iota$, and $\tau = \rho \circ \iota$ map halfedges to halfedges. The thin arrows show the actions of the functions. Orbits of the functions are shown in red (faces), green (edges), and blue (one ring).
• Figure 3: Before (top) and after (below) illustration of the edge insertion operation both in the case (left) where a face $f$ is split into two faces and in the case (right) where two faces, $f_1$ and $f_2$, are merged into a single face $f$. Note that the new face, $f$, is a tube that connects the boundaries of $f_1$ and $f_2$ which is hard to convey adequately in a 2D drawing.
• Figure 4: $RS$ of an example neighborhood. Taking $v_2$ as an example, it is projected on the disk. Then an angle $\alpha_2$ is calculated as the radian of vertex $v_2$.
• Figure 5: Pipeline overview on synthetic letter point clouds. General High-genus reconstruction follows the bottom pipeline, where $g$ stands for the genus number. Given the prior knowledge of a genus-0 shape, the shorter pipeline depicted at the top is employed. In the pipeline shown at the bottom, the red edges in the third inset denote the newly added handles. Concurrently, the path formed by the blue edges represents the shortest path between the two end vertices before the red edge is introduced.