Fairing of Discrete Planar Curves by Discrete Euler's Elasticae

TL;DR

This paper develops a method to approximate discrete planar curves with discrete Euler's elasticae by minimizing an L2 distance, using gradient optimization and a specialized initial guess to handle non-convexity.

Contribution

It introduces a discrete analogue of Euler's elastica and a practical optimization approach for fairing discrete curves, improving stability and accuracy.

Findings

Successful approximation of discrete curves with elasticae

Effective use of gradient-based optimization with initial guess

Enhanced numerical stability in curve fairing

Abstract

After characterizing the integrable discrete analogue of the Euler's elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Euler's elastica. We carry out the fairing process via a $L^2\!$-distance minimization to avoid the numerical instabilities. The optimization problem is solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess, so that we use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method.