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, affine invariant, 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.
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Data Availability
The implementation of our algorithm is provided in https://doi.org/10.5281/zenodo.7742917.
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Funding
This work was supported in part by the NSERC Discovery Grants RGPIN-2020-06022 and DGECR-2020-00356.
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Communicated by: Zydrunas Gimbutas
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Zhao, M., Serkh, K. A continuation method for fitting a bandlimited curve to points in the plane. Adv Comput Math 50, 74 (2024). https://doi.org/10.1007/s10444-024-10144-5
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DOI: https://doi.org/10.1007/s10444-024-10144-5
Keywords
- Parametrization
- Bandlimited functions
- \(C^{\infty }\) functions
- Approximation theory
- Filtering
- Bézier splines
- Smooth interpolation