On Similarity Transformation Problems: Globally Optimal Results and Applications

Similarity transformation problems are important in robotic instrumentation and computer vision-based measurements since in many cases the information of visually observed scene scale is unknown and must be restored for accurate 3-D reconstruction. In existing solvers, the scale is often considered as a scalar, i.e., isotropic, which may be invalid for anisotropic-scale setups. This article exploits some mathematical coincidences that will lead to efficient solutions to these problems. Possible further applications also include hand-eye calibration and structure-from-motion. We revisit pose estimation problems within the framework of similarity transformation, the one that considers scale-stretching, rotation, and translation simultaneously. Two major problems are taken into account, i.e., the scale-stretching point-cloud registration (PCR) and perspective-n-points (PnPs). It has been found out that these two problems are quite similar. Moreover, we solve the anisotropic-scale registration problem which is important and is a remaining unsolved one in previous literatures. To compute the globally optimal solution of these nonconvex problems, the algebraic solution is obtained to compute all local minima using computationally efficient methods. The designed algorithm is deployed for robotic-arm pose estimation. We also extend the algorithm for solving the problem of robust magnetometer calibration. Visual pose experiments verify the superiority of the proposed method compared with representatives, including P3P, Lambda-Twist P3P, and EPnP, which can be reproduced by the repository in https://github.com/zarathustr/APnP.