Revisiting uncertainty analysis for optimum planes extracted from 3D range sensor point-clouds

In this work, we utilize a recently studied more accurate range noise model for 3D sensors to derive from scratch the expressions for the optimum plane which best fits a point-cloud and for the combined covariance matrix of the plane´s parameters. The parameters in question are the plane´s normal and its distance to the origin. The range standard-deviation model used by us is a quadratic function of the true range and is a function of the incidence angle as well. We show that for this model, the maximum-likelihood plane is biased, whereas the least-squares plane is not. The plane-parameters´ covariance matrix for the least-squares plane is shown to possess a number of desirable properties, e.g., the optimal solution forms its null-space and its components are functions of easily understood terms like the planar-patch´s center and scatter. We verify our covariance expression with that obtained by the eigenvector perturbation method. We further compare our method to that of renormalization with respect to the theoretically best covariance matrix in simulation. The application of our approach to real-time range-image registration and plane fusion is shown by an example using a commercially available 3D range sensor. Results show that our method has good accuracy, is fast to compute, and is easy to interpret intuitively.