Preconditioned Single‐step Transforms for Non‐rigid ICP

Abstract Non‐rigid iterative closest point (ICP) is a popular framework for shape alignment, typically formulated as alternating iteration of correspondence search and shape transformation. A common approach in the shape transformation stage is to solve a linear least squares problem to find a smoothness‐regularized transform that fits the target shape. However, completely solving the linear least squares problem to obtain a transform is wasteful because the correspondences used for constructing the problem are imperfect, especially at early iterations. In this work, we design a novel framework to compute a transform in single step without the exact linear solve. Our key idea is to use only a single step of an iterative linear system solver, conjugate gradient, at each shape transformation stage. For this single‐step scheme to be effective, appropriate preconditioning of the linear system is required. We design a novel adaptive Sobolev‐Jacobi preconditioning method for our single‐step transform to produce a large and regularized shape update suitable for correspondence search in the next iteration. We demonstrate that our preconditioned single‐step transform stably accelerates challenging 3D surface registration tasks.