Structured Analytic Coherent Point Drift for Non-Rigid Point Set Registration

2026-04-30

AI Summary

This work addresses the limitations of traditional Coherent Point Drift (CPD) methods in non-rigid point set registration—namely, parameter redundancy, slow convergence, and insufficient stability under large deformations—by introducing a novel approach based on a structured analytical deformation model. Instead of representing displacements via Gaussian kernel-based fields, the method employs a finite-dimensional analytical mapping constructed from a truncated multivariate Taylor expansion. Registration is reformulated as a weighted analytical fitting problem, where soft target points are generated using posterior probabilities from a Gaussian Mixture Model. The number of parameters depends only on spatial dimensionality and expansion order, substantially reducing model complexity. An incremental order-increasing strategy is further incorporated to enhance robustness under large deformations. Experiments demonstrate that the proposed method achieves lower registration errors and faster convergence than standard CPD in both 2D analytical and 3D smooth non-analytical deformation scenarios.

Abstract

We introduce Analytic-CPD, a structured analytic variant of coherent point drift for non-rigid point set registration. The method retains the CPD posterior correspondence layer, but replaces the point-indexed Gaussian-kernel displacement-field M-step with a finite-dimensional structured analytic mapping estimator. Posterior probabilities from the Gaussian mixture model are condensed through a barycentric identity into weighted soft target points, converting the CPD pairwise soft-correspondence objective into a weighted analytic fitting problem. The deformation is represented by a truncated multivariate Taylor mapping of a vector-valued function, so the number of deformation parameters is controlled by the ambient dimension and the analytic order rather than by an M-by-M kernel system over the moving points. A degree-continuation strategy is further introduced to stabilize large-deformation registration by progressively activating higher-order analytic modes. Experiments on two-dimensional analytic deformations and three-dimensional smooth non-analytic deformations show that Analytic-CPD achieves lower final errors and faster convergence than standard CPD in representative large-deformation settings. The results suggest that CPD-style probabilistic correspondences and structured analytic mappings provide a compact and interpretable alternative to kernel-based non-rigid registration. Code is available at https://github.com/monge-ampere/Analytic-CPD.

Problem

• non-rigid point set registration
• coherent point drift
• large-deformation
• analytic mapping
• point correspondence

Innovation

• Analytic-CPD
• non-rigid registration
• structured analytic mapping
• coherent point drift
• Taylor expansion