Abstract
Abstract Suppose a large and dense point cloud of an object with complex geometry is available that can be approximated by a smooth univariate function. In general, for such point clouds the “best” approximation using the method of least squares is usually hard or sometimes even impossible to compute. In most cases, however, a “near-best” approximation is just as good as the “best”, but usually much easier and faster to calculate. Therefore, a fast approach for the approximation of point clouds using Chebyshev polynomials is described, which is based on an interpolation in the Chebyshev points of the second kind. This allows to calculate the unknown coefficients of the polynomial by means of the Fast Fourier transform (FFT), which can be extremely efficient, especially for high-order polynomials. Thus, the focus of the presented approach is not on sparse point clouds or point clouds which can be approximated by functions with few parameters, but rather on large dense point clouds for whose approximation perhaps even millions of unknown coefficients have to be determined.
Highlights
Modern measuring instruments can record information about the geometry of physical objects and provide the user with discrete points in a Cartesian coordinate system
A fast approach for the approximation of point clouds using Chebyshev polynomials is described, which is based on an interpolation in the Chebyshev points of the second kind
The problems of knot placement and model selection can be considered as separate research topics and have a significant impact on the quality of the approximation using B-Splines. Motivated by these challenges and since these approaches are usually used for small point clouds with only a few unknown parameters to be determined, we present a strategy for the approximation of point clouds using Chebyshev polynomials, which is based on an interpolation in the Chebyshev points of the second kind
Summary
Modern measuring instruments can record information about the geometry of physical objects and provide the user with discrete points in a Cartesian coordinate system. Several criteria have been presented in the literature for choosing the optimal number of control points, which is known as model selection. The problems of knot placement and model selection can be considered as separate research topics and have a significant impact on the quality of the approximation using B-Splines Motivated by these challenges and since these approaches are usually used for small point clouds with only a few unknown parameters to be determined, we present a strategy for the approximation of point clouds using Chebyshev polynomials, which is based on an interpolation in the Chebyshev points of the second kind. An additional model selection criterion is not needed In this contribution we consider at first only point clouds that represent curves without any kinks, jumps or data gaps, which can be approximated by continuous smooth functions. An outlook on current research regarding the presented approach concludes this paper
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