This paper addresses the weighted orthogonal Procrustes problem of matching stochastically perturbed point clouds, formulated as an optimization problem with a closed-form solution. A novel uncertainty characterization of the solution of this problem is proposed resorting to perturbation theory concepts, which admits arbitrary transformations between point clouds and individual covariance and cross covariance matrices for the points of each cloud. The method is thoroughly validated through extensive Monte Carlo simulations, and particularly interesting cases where nonlinearities may arise are further analyzed.