Deterministic Point Cloud Registration via Novel Transformation Decomposition

Given a set of putative 3D-3D point correspondences, we aim to remove outliers and estimate rigid transformation with 6 degrees of freedom (DOF). Simultaneously estimating these 6 DOF is time-consuming due to high-dimensional parameter space. To solve this problem, it is common to decompose 6 DOF, i.e. independently compute 3-DOF rotation and 3-DOF translation. However, high non-linearity of 3-DOF rotation still limits the algorithm efficiency, especially when the number of correspondences is large. In contrast, we propose to decompose 6 DOF into <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(2+1)$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(1+2)\ DOF$</tex> . Specifically, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(2+1)DOF$</tex> represent 2-DOF rotation axis and 1-DOF displacement along this rotation axis. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(1+2)\ DOF$</tex> indicate 1-DOF rotation angle and 2-DOF displacement orthogonal to the above rotation axis. To compute these DOF, we design a novel two-stage strategy based on inlier set maximization. By leveraging branch and bound, we first search for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(2+1)\ DOF$</tex> , and then the remaining <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(1+2)\ DOF$</tex> . Thanks to the proposed transformation decomposition and two-stage search strategy, our method is deterministic and leads to low computational complexity. We extensively compare our method with state-of-the-art approaches. Our method is more accurate and robust than the approaches that provide similar efficiency to ours. Our method is more efficient than the approaches whose accuracy and robustness are comparable to ours.