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Feature-aware manifold meshing and remeshing of point clouds and polyhedral surfaces with guaranteed smallest edge length

Henriette Lipschütz, Ulrich Reitebuch, Konrad Polthier, Martin Skrodzki

TL;DR

The paper addresses the challenge of reconstructing high-quality, manifold triangle meshes from unstructured point clouds and polyhedral surfaces while guaranteeing a smallest edge length. It introduces a feature-aware, sphere-packing-based framework that produces near-uniform edge lengths in a single greedy sweep, without requiring surface parametrization. The method accommodates feature ridges, remeshing of polyhedral surfaces, and robust performance under moderate noise, underpinned by theoretical guarantees that rely on assumptions about the underlying manifold's reach $\rho$ and sampling. A Box Grid data structure, windowed border traversal, and adaptive splat sizing support efficient implementation, and extensive experiments on 20 real models and CAD data demonstrate superior or competitive mesh quality (high $Q_{avg}$, low $Q_{RMS}$) while preserving mesh topology. The work advances practical, feature-preserving remeshing pipelines with potential extensions to boundaries and volumetric meshing, balancing edge-length guarantees with fidelity to sharp features.

Abstract

Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online: https://ms-math-computer.science/projects/guaranteed-smallest-edge-length-manifold-meshing.html

Feature-aware manifold meshing and remeshing of point clouds and polyhedral surfaces with guaranteed smallest edge length

TL;DR

The paper addresses the challenge of reconstructing high-quality, manifold triangle meshes from unstructured point clouds and polyhedral surfaces while guaranteeing a smallest edge length. It introduces a feature-aware, sphere-packing-based framework that produces near-uniform edge lengths in a single greedy sweep, without requiring surface parametrization. The method accommodates feature ridges, remeshing of polyhedral surfaces, and robust performance under moderate noise, underpinned by theoretical guarantees that rely on assumptions about the underlying manifold's reach and sampling. A Box Grid data structure, windowed border traversal, and adaptive splat sizing support efficient implementation, and extensive experiments on 20 real models and CAD data demonstrate superior or competitive mesh quality (high , low ) while preserving mesh topology. The work advances practical, feature-preserving remeshing pipelines with potential extensions to boundaries and volumetric meshing, balancing edge-length guarantees with fidelity to sharp features.

Abstract

Point clouds and polygonal meshes are widely used when modeling real-world scenarios. Here, point clouds arise, for instance, from acquisition processes applied in various surroundings, such as reverse engineering, rapid prototyping, or cultural preservation. Based on these raw data, polygonal meshes are created to, for example, run various simulations. For such applications, the utilized meshes must be of high quality. This paper presents an algorithm to derive triangle meshes from unstructured point clouds. The occurring edges have a close to uniform length and their lengths are bounded from below. Theoretical results guarantee the output to be manifold, provided suitable input and parameter choices. Further, the paper presents several experiments establishing that the algorithms can compete with widely used competitors in terms of quality of the output and timing and the output is stable under moderate levels of noise. Additionally, we expand the algorithm to detect and respect features on point clouds as well as to remesh polyhedral surfaces, possibly with features. Supplementary material, an extended preprint, a link to a previously published version of the article, utilized models, and implementation details are made available online: https://ms-math-computer.science/projects/guaranteed-smallest-edge-length-manifold-meshing.html
Paper Structure (76 sections, 1 theorem, 13 equations, 91 figures, 52 tables, 1 algorithm)

This paper contains 76 sections, 1 theorem, 13 equations, 91 figures, 52 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Theory and Methodology
  4. Assumptions and Theory
  5. Methodology
  6. Initialization
  7. Disk Growing to Add Vertices of$~\mathcal{G}$
  8. Prioritizing of Vertex Candidates
  9. Creating a New Vertex
  10. Triangulating the Resulting Regions
  11. Implementation
  12. Box Grid Data Structure
  13. Window Size
  14. Discussion of Splat Size
  15. Processing Vertex Candidates
  16. ...and 61 more sections

Key Result

Lemma 3.1

Let $p \in \mathcal{M}$ be a point with normal $\textcolor{black}{n}_p$. Then, for $r < \rho$, the image of $B_r(p) \cap \mathcal{M}$ under the projection $\pi$ in direction of $\textcolor{black}{n}_p$ to the tangent plane $T_p \mathcal{M}$ is a convex set.

Figures (91)

  • Figure 1: Illustration of Lemma \ref{['lem:projectionConvex']}. Intersection of a saddle-shaped surface with a sphere (blue). Six points on the surface are marked as well as their projections to the tangent plane belonging to the center of the sphere.
  • Figure 2: Illustration of uniform splat size (\ref{['fig:SplatsOfSizeD']}), the projection of a point $p$ and its vicinity to the tangent plane $T_p\mathcal{M}$ (\ref{['fig:ProjectionToTangentPlane']}), and the Voronoi cells with the farthest point circled (\ref{['fig:VoronoiCells']}), leading to individual splat sizes (\ref{['fig:IndividualSplatSizes']}).
  • Figure 3: Possibilities when creating new vertices and edge connections in the graph $\mathcal{G}$: In \ref{['fig:borderConnectionsCase1']}, the new vertex $v$ and its two edges connect elements of the same border. Here, $v$ is created either in the in- or the outside region of the border. In \ref{['fig:borderConectionsCase2']}, the new vertex $v$ is connecting two borders. After introducing $v$ and its edges, the respective outside regions are still connected, then $v$ and its edges join the borders. However, if the outside regions are split by $v$ and its edges, new borders are created which induce the corresponding regions.
  • Figure 4: An input geometry with a vertex candidate $v_c$ (\ref{['fig:Roof3D']}), the region projection shows an illegal edge crossing (\ref{['fig:RoofProjection']}). Edges to parent vertices are shown dotted. Computation of vertex candidate positions (\ref{['fig:IntersectingSpheres']}).
  • Figure 5: Update steps of the algorithm.
  • ...and 86 more figures

Theorems & Definitions (2)

  • Definition 3.1
  • Lemma 3.1