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A Primal-Dual Level Set Method for Computing Geodesic Distances

Hailiang Liu, Laura Zinnel

TL;DR

The work addresses computing geodesic distances on surfaces without explicit surface discretization by representing the surface as the zero level set of a function $\phi$ and formulating the problem as a constrained variational saddle-point problem. A primal-dual level-set approach with regularization and relaxation is developed, yielding a PDHG-inspired update scheme that alternates between a path $\gamma$ on the surface and a multiplier $\lambda$ enforcing $\phi(\gamma)=0$. The authors provide a continuous PDE analysis showing convergence to equilibria that correspond to geodesics as $\varepsilon \to 0$, and demonstrate ergodic $O(1/k)$ convergence for planar cases along with robust numerical results on spheres, tori, and the Stanford Bunny. The method is simple to implement, mesh-free, and capable of handling nontrivial geometries, with potential extensions to moving surfaces and general Lagrangians. Overall, the work offers a scalable framework for geodesic computation on implicit surfaces with theoretical convergence guarantees and practical verification.

Abstract

The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high-resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.

A Primal-Dual Level Set Method for Computing Geodesic Distances

TL;DR

The work addresses computing geodesic distances on surfaces without explicit surface discretization by representing the surface as the zero level set of a function and formulating the problem as a constrained variational saddle-point problem. A primal-dual level-set approach with regularization and relaxation is developed, yielding a PDHG-inspired update scheme that alternates between a path on the surface and a multiplier enforcing . The authors provide a continuous PDE analysis showing convergence to equilibria that correspond to geodesics as , and demonstrate ergodic convergence for planar cases along with robust numerical results on spheres, tori, and the Stanford Bunny. The method is simple to implement, mesh-free, and capable of handling nontrivial geometries, with potential extensions to moving surfaces and general Lagrangians. Overall, the work offers a scalable framework for geodesic computation on implicit surfaces with theoretical convergence guarantees and practical verification.

Abstract

The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high-resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.
Paper Structure (26 sections, 8 theorems, 113 equations, 10 figures, 2 tables)

This paper contains 26 sections, 8 theorems, 113 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Setup
  4. Contribution
  5. Related work
  6. Organization of the Paper
  7. Method
  8. Optimization Problem
  9. Optimality conditions
  10. Gradient Descent-Ascent
  11. Regularization
  12. Acceleration
  13. Analysis of dynamic behavior
  14. PDE System
  15. Equilibrium curves
  16. ...and 11 more sections

Key Result

Lemma 1

If $\gamma$ is a geodesic on $\Omega$ from $p$ to $q$, then $L(p,q) = d(p,q)$. Moreover, the following statements are equivalent: (i) $\gamma$ is a constant-speed geodesic. (ii) $\gamma \in C^1[0, 1]$ and $|\dot \gamma(t)| = d(\gamma(0), \gamma(1))$ for all $t \in[0, 1]$. (iii) Let $m>1$, $\gamma$ s

Figures (10)

  • Figure 1: Absolute error of the approximated geodesic distance between antipodal points on the sphere, plotted against the number of iterations for the update scheme described above.
  • Figure 2: Left: Resulting curve $\gamma$ (in blue) after 3500 iterations of the algorithm to find the path between antipodal points $p$ and $q$, using a straight line as initial guess. Right: Resulting curve after 3500 iterations for the same antipodal points $p$ and $q$, using random initialization.
  • Figure 3: Plot of absolute error (left) and surface error (right) over iterations of the algorithm for finding a path $\gamma$ between antipodal points on a sphere. Each colored curve represents results for a different value of the regularization coefficient $\varepsilon$.
  • Figure 4: No regularization. Absolute error (left) and surface error (right) over iterations of the algorithm for finding a path $\gamma$ between antipodal points on the sphere. Different colors indicate different values of $\tau_{\lambda}$.
  • Figure 5: With regularization ($\varepsilon = 0.01$). Absolute error (left) and surface error (right) vs. iterations for computing a path $\gamma$ between antipodal points on the sphere. Different colors indicate different values of $\tau_{\lambda}$.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 5 more