Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile P represented as a set of points in the space R3 and four families of low-degree polynomial curves that respectively depend on the parameters in the space R4, our goal is to identify the piecewise space polynomial curve best fitting the profile P. We use a parametric representation of the space curves and subdivide the profile into smaller portions that can be fitted with regular curves. We provide theoretical guarantees to the existence of such an approximation and an algorithm for the profile approximation. We take advantage of the implicit function theorem to locally project a space curve on at most two planes and to locally recognise it with a low-degree polynomial curve obtained by applying the Hough transform. Finally, we recombine the curve expressions on the two planes backwards in the space R3. The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to approximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.