Kabsch algorithm

Abstract: We tackle data-driven 3D point cloud registration. Given point correspondences, the standard Kabsch algorithm provides an optimal rotation estimate. This allows to train registration models in an end-to-end manner by differentiating the SVD operation. However, given the initial rotation estimate supplied by Kabsch, we show we can improve point correspondence learning during model training by extending the original optimization problem. In particular, we linearize the governing constraints of the rotation matrix and solve the resulting linear system of equations. We then iteratively produce new solutions by updating the initial estimate. Our experiments show that, by plugging our differentiable layer to existing learning-based registration methods, we improve the correspondence matching quality. This yields up to a 7% decrease in rotation error for correspondence-based data-driven registration methods.

Abstract: This work deals with the analysis of Kabsch and quaternion algorithms, which may be used for 3D superimposition of molecules by rigid roto-translation in computational chemistry and biology. Both algorithms, which are very important for in silico drug design, were studied from the point of view of their non-trivial mathematical structure. Their computational complexity was investigated by a superimposition of various random pseudo-molecules with 2 – 100,000 atoms in Matlab. It was found that both proposed algorithm implementations exhibit the same asymptotic time computational complexity of O(n), with the quaternion algorithm involving a higher number of floating-point operations (FLOPs) and showing lower computational performance in terms of serial CPU time.

Abstract: Protein complex formation is a central problem in biology, being involved in most of the cell's processes, and essential for applications, e.g. drug design or protein engineering. We tackle rigid body protein-protein docking, i.e., computationally predicting the 3D structure of a protein-protein complex from the individual unbound structures, assuming no conformational change within the proteins happens during binding. We design a novel pairwise-independent SE(3)-equivariant graph matching network to predict the rotation and translation to place one of the proteins at the right docked position relative to the second protein. We mathematically guarantee a basic principle: the predicted complex is always identical regardless of the initial locations and orientations of the two structures. Our model, named EquiDock, approximates the binding pockets and predicts the docking poses using keypoint matching and alignment, achieved through optimal transport and a differentiable Kabsch algorithm. Empirically, we achieve significant running time improvements and often outperform existing docking software despite not relying on heavy candidate sampling, structure refinement, or templates.

Abstract: The present invention provides a variable precision optimization coupling method of a NURBS curve based on a Kabsch algorithm. The method employs a variable precision coupling method with rough first and fine second and employs the NURBS to describe any curve. The splitting and the combination of the curve have a uniform mathematical version, firstly, a similar threshold is used for balancing the similar degree of two tracks, the motif similarity is reduced and the search step is increased to search tracks having more similar features with an expect track in lots of tracks; secondly, the corresponding coupling precision index is improved, and in the condition of satisfying the similarity, the track motif which allow the expect reference curve segment to have the fewest times in the searched curves; and finally, all the coupling motifs are combined in order, wherein the combination curve and the original expect curve are similar.