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Reconstructing Curves from Sparse Samples on Riemannian Manifolds

Diana Marin, Filippo Maggioli, Simone Melzi, Stefan Ohrhallinger, Michael Wimmer

TL;DR

The paper tackles reconstructing closed curves directly on Riemannian manifolds from sparse samples, extending planar curve reconstruction to non-Euclidean domains. It introduces a manifold-generalized proximity graph (SIGDV), a non-uniform sampling framework with injective local feature size ilfs and ireach-based guarantees, and a biased TSP-based reconstruction that recovers the curve order within SIGDV under sampling conditions. The key contributions are the ilfs/ireach-based sampling relaxation, the SIGDV graph construction on manifolds via geodesic Voronoi duals, and the TSP-based ordering with theoretical guarantees, validated on diverse tasks such as motion tracking, cultural heritage contouring, contour matching, and sparse data visualization. Collectively, the approach enables automated, on-surface curve reconstruction from sparse data, with wide potential impact in graphics, archaeology, pattern extraction, and scientific visualization.

Abstract

Reconstructing 2D curves from sample points has long been a critical challenge in computer graphics, finding essential applications in vector graphics. The design and editing of curves on surfaces has only recently begun to receive attention, primarily relying on human assistance, and where not, limited by very strict sampling conditions. In this work, we formally improve on the state-of-the-art requirements and introduce an innovative algorithm capable of reconstructing closed curves directly on surfaces from a given sparse set of sample points. We extend and adapt a state-of-the-art planar curve reconstruction method to the realm of surfaces while dealing with the challenges arising from working on non-Euclidean domains. We demonstrate the robustness of our method by reconstructing multiple curves on various surface meshes. We explore novel potential applications of our approach, allowing for automated reconstruction of curves on Riemannian manifolds.

Reconstructing Curves from Sparse Samples on Riemannian Manifolds

TL;DR

The paper tackles reconstructing closed curves directly on Riemannian manifolds from sparse samples, extending planar curve reconstruction to non-Euclidean domains. It introduces a manifold-generalized proximity graph (SIGDV), a non-uniform sampling framework with injective local feature size ilfs and ireach-based guarantees, and a biased TSP-based reconstruction that recovers the curve order within SIGDV under sampling conditions. The key contributions are the ilfs/ireach-based sampling relaxation, the SIGDV graph construction on manifolds via geodesic Voronoi duals, and the TSP-based ordering with theoretical guarantees, validated on diverse tasks such as motion tracking, cultural heritage contouring, contour matching, and sparse data visualization. Collectively, the approach enables automated, on-surface curve reconstruction from sparse data, with wide potential impact in graphics, archaeology, pattern extraction, and scientific visualization.

Abstract

Reconstructing 2D curves from sample points has long been a critical challenge in computer graphics, finding essential applications in vector graphics. The design and editing of curves on surfaces has only recently begun to receive attention, primarily relying on human assistance, and where not, limited by very strict sampling conditions. In this work, we formally improve on the state-of-the-art requirements and introduce an innovative algorithm capable of reconstructing closed curves directly on surfaces from a given sparse set of sample points. We extend and adapt a state-of-the-art planar curve reconstruction method to the realm of surfaces while dealing with the challenges arising from working on non-Euclidean domains. We demonstrate the robustness of our method by reconstructing multiple curves on various surface meshes. We explore novel potential applications of our approach, allowing for automated reconstruction of curves on Riemannian manifolds.
Paper Structure (15 sections, 21 theorems, 16 equations, 14 figures, 1 algorithm)

This paper contains 15 sections, 21 theorems, 16 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Related work
  2. Background and notations
  3. Curve sampling
  4. Differential geometry
  5. Method
  6. Non-uniform sparse sampling on surfaces
  7. SIGDV graph on manifolds
  8. Curve reconstruction as a Traveling Salesman Problem
  9. Applications
  10. Motion tracking
  11. Virtual cultural heritage
  12. Contour matching
  13. Sparse data visualization
  14. Conclusions
  15. Proofs

Key Result

Theorem 1

Let $\mathcal{M}$ be a $d$-dimensional Riemannian manifold. For every point $p \in \mathcal{M}$, the restriction of the exponential map $\exp_p$ to $U \subset T_p(\mathcal{M})$, such that $\exp_p(U) = \mathcal{B}_{{p}, {r}}$, $\mathcal{B}_{{p}, {r}}$ is injective for all $r \leq i_{\mathcal{M}}(p)$

Figures (14)

  • Figure 1: Left: the local feature size at each point of the curve represents its distance to the medial axis (in black). Right: a $\rho$-sampling of the curve with $\rho = 1.0$ shows that the samples are denser where the medial axis is closer to the curve.
  • Figure 2: Difference between uniform sampling (left) and non-uniform sampling (right) with non-uniformity ratio $u=1.5$.
  • Figure 3: Examples of cut locus (in red) given a point (in blue) on different surfaces. The cut locus can be a single point (left) or an entire curve (right).
  • Figure 4: Two examples of curves on surfaces (solid black) where the local feature size exhibits undesired behaviors. Left: an $r$-ball (shaded in light blue) around $p$ that contains the entire curve $\mathcal{C}_1$, but no point of the medial axis. Right: a curve $\mathcal{C}_2$ on the surface with an empty medial axis (i.e., the local feature size is undefined).
  • Figure 5: A comparison between a dense uniform sampling, with 281 samples, satisfying the conditions described by Shah et al.shah:2013:curve (on the left) and a non-uniform $\rho$-sampling scheme with only 124 samples (on the right).
  • ...and 9 more figures

Theorems & Definitions (41)

  • Definition 1: blum:1967:medialaxis
  • Definition 2: ruppert:1993:lfs
  • Definition 3: federer:1959:curvature
  • Definition 4: ohrhallinger:2016:hnncrust
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9: docarmo:2016:geometry
  • Definition 10
  • ...and 31 more