GLS-PIA: n-Dimensional Spherical B-Spline Curve Fitting based on Geodesic Least Square with Adaptive Knot Placement

Abstract

Due to the widespread applications of curves on n-dimensional spheres, fitting curves on n-dimensional spheres has received increasing attention in recent years. However, due to the non-Euclidean nature of spheres, curve fitting methods on n-dimensional spheres often struggle to balance fitting accuracy and curve fairness. In this paper, we propose a new fitting framework, GLSPIA, for parameterized point sets on n-dimensional spheres to address the challenge. Meanwhile, we provide the proof of the method. Firstly, we propose a progressive iterative approximation method based on geodesic least squares which can directly optimize the geodesic least squares loss on the n-sphere, improving the accuracy of the fitting. Additionally, we use an error allocation method based on contribution coefficients to ensure the fairness of the fitting curve. Secondly, we propose an adaptive knot placement method based on geodesic difference to estimate a more reasonable distribution of control points in the parameter domain, placing more control points in areas with greater detail. This enables B-spline curves to capture more details with a limited number of control points. Experimental results demonstrate that our framework achieves outstanding performance, especially in handling imbalanced data points. (In this paper, ''sphere'' refers to n-sphere (n = 2) unless otherwise specified.)