A Continuation Method for Fitting a Bandlimited Curve to Points in the Plane

TL;DR

This paper introduces an efficient algorithm for fitting smooth, bandlimited curves to points in the plane by iteratively filtering Fourier or Chebyshev coefficients, achieving high accuracy with minimal coefficients.

Contribution

The paper presents a novel iterative filtering approach for fitting analytic, bandlimited curves to points, reducing bandwidth and avoiding ringing artifacts.

Findings

Achieves machine-precision accuracy in curve fitting.

Produces smooth, visually appealing curves with minimal coefficients.

Operates with O(N log N) complexity per iteration.

Abstract

In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in $\mathbb{R}^{2}$. The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc length function and the tangential angle of the curve, and applying smooth perturbations, after each filtering step, until the curve is represented by a reasonably small number of coefficients. The algorithm produces a curve passing through the set of points to an accuracy of machine precision, after a limited number of iterations. It costs O($N\log{N}$) operations at each iteration, provided that the number of discretization nodes is $N$. The resulting curves are smooth and visually appealing, and do not exhibit any ringing artifacts. The bandwidths of the constructed curves are much smaller than those of curves constructed by previous methods. We demonstrate the performance of our algorithm with several numerical experiments.