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Minimization of Curve Length through Energy Minimization using Finite Difference and Numerical Integration in Real Coordinate Space

Akira Kitaoka

TL;DR

The paper addresses the problem of finding minimal-length curves under a Riemannian metric by recasting geodesic computation as an energy minimization problem and solving it via discretization with finite differences and numerical integration. It proves that, for complete real-coordinate spaces, discretizations based on the trapezoidal and left-endpoint rules yield minimum energy values that converge to the true minimum energy at rate $O(N^{-1/2})$, and that the corresponding interpolated curves converge in length at the same rate. The work also establishes Morrey-type inequalities, provides a priori estimates for minimal geodesics, and analyzes the limitations of discretized length formulations through counterexamples. These results offer a scalable, implementable framework for computing minimal geodesics without resorting to full Christoffel-symbol-based formulations, with direct implications for computer vision, robotics, and related fields. It also clarifies the conditions under which energy-based discretizations faithfully reproduce true geodesic length, and when careful interpretation of discrete energies is required.

Abstract

The problem of determining the minimal length is garnering attention in various fields such as computer vision, robotics, and machine learning. One solution to this problem involves linearly interpolating the solution of a nonlinear optimization problem that approximates the curve's energy minimization problem using finite differences and numerical integration. This method tends to be easier to implement compared to others. However, it was previously unknown whether this approach successfully minimizes the curve's length under the Riemannian metric in real coordinate spaces. In this paper, we prove that the length of a curve obtained by linear interpolation of the solution to an optimization problem, where the energy of the curve is approximated using finite differences and the trapezoidal rule, converges to the minimal curve length at a rate of $1/2$ in terms of the number of points used in the numerical integration. Similarly, we prove that when using the left-point rule, the approximated curve's length likewise converges to the minimal curve length at a rate of $1/2$ in terms of the number of points used in the numerical integration.

Minimization of Curve Length through Energy Minimization using Finite Difference and Numerical Integration in Real Coordinate Space

TL;DR

The paper addresses the problem of finding minimal-length curves under a Riemannian metric by recasting geodesic computation as an energy minimization problem and solving it via discretization with finite differences and numerical integration. It proves that, for complete real-coordinate spaces, discretizations based on the trapezoidal and left-endpoint rules yield minimum energy values that converge to the true minimum energy at rate , and that the corresponding interpolated curves converge in length at the same rate. The work also establishes Morrey-type inequalities, provides a priori estimates for minimal geodesics, and analyzes the limitations of discretized length formulations through counterexamples. These results offer a scalable, implementable framework for computing minimal geodesics without resorting to full Christoffel-symbol-based formulations, with direct implications for computer vision, robotics, and related fields. It also clarifies the conditions under which energy-based discretizations faithfully reproduce true geodesic length, and when careful interpretation of discrete energies is required.

Abstract

The problem of determining the minimal length is garnering attention in various fields such as computer vision, robotics, and machine learning. One solution to this problem involves linearly interpolating the solution of a nonlinear optimization problem that approximates the curve's energy minimization problem using finite differences and numerical integration. This method tends to be easier to implement compared to others. However, it was previously unknown whether this approach successfully minimizes the curve's length under the Riemannian metric in real coordinate spaces. In this paper, we prove that the length of a curve obtained by linear interpolation of the solution to an optimization problem, where the energy of the curve is approximated using finite differences and the trapezoidal rule, converges to the minimal curve length at a rate of in terms of the number of points used in the numerical integration. Similarly, we prove that when using the left-point rule, the approximated curve's length likewise converges to the minimal curve length at a rate of in terms of the number of points used in the numerical integration.

Paper Structure

This paper contains 23 sections, 27 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Differential geometry
  4. Minimal curves and minimal geodesics
  5. Minimal length problem and numerical integration
  6. Length of minimal path
  7. Left-endpoint rule
  8. Trapezoidal Rule
  9. Characterization of minimal geodesics via energy minimization
  10. Case of two points
  11. Case of two sets
  12. Approximation of energy minimization problem via trapezoidal rule
  13. Morrey-type Inequality
  14. Error between curve energy and energy approximated by trapezoidal rule
  15. Points which attain minimum of energy approximated trapezoidal rule, and its linear interpolation
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $g \in C^{\infty}$ be a Riemann metric on $\mathbb{R}^D$ satisying "there esist $c_1 ,c_2>0$ such that for any $u \in \mathbb{R}^D$, $c_1 \| u \|_2^2 \leq g_x (u,u) \leq c_2 \| u \|_2^2$" and "written by $g_x (u,u) = u^{\top} H(x) u$ satisfying $\| H (x) - H ( y) \|_{\mathcal{B}} \leq L_H \| x -

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 3.1: C.f. milnor1963morse*§ 10
  • Proposition 3.2: hopg1931ueber
  • Proposition 3.3
  • proof
  • Proposition 4.1: C.f. konno2013differentiable*Theorem 4.2.1
  • Proposition 4.2
  • ...and 37 more