Maximizing Likelihood Function for Parameter Estimation in Point Clouds via Groebner Basis

Abstract

Nowadays, surface reconstruction from point clouds generated by laser scanning technology has become a fundamental task in many fields, such as robotics, computer vision, digital photogrammetry, computational geometry, digital building modeling, forest planning and operational activities. The point clouds produced by laser scanning, however, are limited due to the occurrence of occlusions, multiple reflectance and noise, and off-surface points (outliers), thus necessitating the need for robust fitting techniques. These techniques require repeated parameter estimation while eliminating outliers. Employing maximum likelihood estimation, the parameters of the model are estimated by maximizing the likelihood function, which maps the parameters to the likelihood of observing the given data. The transformation of this optimization problem into the solution of a multivariate polynomial system via computer algebra can provide two advantages. On the one hand, since all of the solutions can be computed, a single solution that provides global maximum can be selected. On the other hand, once the symbolic result has been computed, it can be used in numerical evaluations in a split second, which reduces the computation time. In our presentation, we applied Groebner basis to solve the maximization of the likelihood function in various robust techniques. A numerical example with data from a real laser scanner experiment illustrates the method. Computations have been carried out in the Mathematica environment.