Non-linear least squares fitting of bézier surfaces to unstructured point clouds

Abstract

Algorithms for linear and non-linear least squares fitting of Bézier surfaces to unstructured point clouds are derived from first principles. The presented derivation includes the analytical form of the partial derivatives that are required for minimising the objective functions, these have been computed numerically in previous work concerning Bézier curve fitting, not surface fitting. Results of fitting fourth degree Bézier surfaces to complex simulated and measured surfaces are presented, a quantitative comparison is made between fitting Bézier surfaces and fitting polynomial surfaces. The developed fitting algorithm is used to remove the geometric form of a complex engineered surface such that the surface roughness can be evaluated.