Piecewise polynomial approximation of spatial curvilinear profiles using the Hough transform

Given a curvilinear profile Prepresented as a set of points in the space R 3 and four fam- ilies of low-degree polynomial curves that respectively depend on the parameters in the space R 4 , 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 pro- file 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 ap- proximation. 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 expres- sions on the two planes backwards in the space R 3 . The outcome of the algorithm is thus a piecewise polynomial curve approximating the profile. We validate our approach to ap- proximate curvilinear profiles extracted from 3D point clouds representing real objects and to simplify and resample point clouds.