Point data reconstruction and smoothing using cubic splines and clusterization

Abstract

An algorithm to smooth a sequence of noisy data in Rd with cubic polynomials is herein presented. The data points are assumed to be sequentially ordered, with the idea that they are sampled in time/space and represent an unknown curve to be reconstructed. An example is the reconstruction of left and right borders of a road from GPS and/or lidars data, that is a common problem encountered in applications for autonomous vehicles or for the generation of high definition digital maps. The result is a sequence of denoised points, that ideally, belong to the original unknown curve, which is the basis for an interpolation process that aims at building the best approximating curve. The problem is solved employing a least squares approach, with quasi-orthogonal projections. The possibility to weight the samples is provided, as well as a Tikhonov regularisation term to penalise the magnitude of the derivative of the cubic curve used to smooth the data. The Tikhonov term is weighted with a parameter that is determined applying the Generalised Cross-Validation (GCV), which, in particular cases, can be written in closed form. We show and validate the algorithm on a real application example reconstructing the road borders of the circuit track of Doha, Qatar.