A proof that fusing measurements using Point-to-hyperplane registration is invariant to relative scale

The objective of this paper is to demonstrate that the metric error between different types of measurements can be jointly minimized without a scaling factor for the estimation processes if a Point-to-hyperplane approach is employed. This article is an extension of previous work based on the Point-tohyperplane approach in 4 dimensions applied to pose estimation, where the proposed method minimized a fused error (3D Euclidean points + Image intensities) and it was experimentally demonstrated that the method is invariant to the choice of scale factor. In this paper, the invariance to the scale factor will be mathematically demonstrated. By doing this, it will be shown how the proposed method can further improve the convergence domain in 4D (or higher dimensions) and speed up the alignment between augmented frames (color + depth) whilst maintaining the robust and accurate properties of hybrid approaches when different types of measurements are available.