A Closed-Form Expression of the Positional Uncertainty for 3D Point Clouds

We present a novel closed-form expression of positional uncertainty measured by a near-monostatic and time-of-flight laser range finder with consideration of its measurement uncertainties. An explicit form of the angular variance of the estimated surface normal vector is also derived. This expression is useful for the precise estimation of the surface normal vector and the outlier detection for finding correspondence in order to register multiple three-dimensional point clouds. Two practical algorithms using these expressions are presented: a method for finding optimal local neighbourhood size which minimizes the variance of the estimated normal vector and a resampling method of point clouds.