Optimal computation of 3-D similarity: Gauss–Newton vs. Gauss–Helmert

Because 3-D data are acquired using 3-D sensing such as stereo vision and laser range finders, they have inhomogeneous and anisotropic noise. This paper studies optimal computation of the similarity (rotation, translation, and scale change) of such 3-D data. We first describe two well known methods for this: the Gauss–Newton and the Gauss–Helmert methods, which are often regarded as different techniques. We then point out that they have similar mathematical structures and combine them to define a hybrid, which we call the modified Gauss–Helmert method. Doing stereo vision simulation, we demonstrate that the proposed method is superior to either of the two methods in convergence performance. Finally, we show an application to real GPS geodetic data and point out that the widely used homogeneous and isotropic noise model is insufficient. We also discuss some numerical issues about GPS data.