A variational Bayesian algorithm for probabilistic affine and non-rigid point cloud registration

Point cloud registration (PCR) is a fundamental problem in computer vision and pattern recognition. However, its performance in real-world scenarios is hindered by low overlap, non-rigid deformation, noise, and outliers. We present VBReg, a variational Bayesian framework that jointly models affine transformation, estimates probabilistic point distributions, and performs non-rigid refinement under a unified inference paradigm. Our VBReg introduces three key innovations. First, an affine transformation is estimated using a Student's t mixture model (SMM), which leverages heavy-tailed distributions to suppress outliers while preserving geometric topology, thereby providing robust initialization for subsequent non-rigid alignment. Second, we extend the SMM to a heterogeneous Student's t mixture model (HSMM) by incorporating Gamma-distributed components for explicit separation of noise and outliers, along with Dirichlet priors for adaptive mixture weighting. Third, variational Bayesian inference is employed to optimize correspondences under non-rigid transformations, enabling principled and probabilistic registration. Extensive experiments on six 2D/3D benchmarks demonstrate that our VBReg consistently outperforms state-of-the-art iterative and learning-based methods in both robustness and accuracy. Notably, VBReg does not require training data and completes inference on 5k-point clouds in less than two seconds on a CPU for both multi-centric lesions and low overlap tasks. Source code is released at https://github.com/XinkeMa/VBReg.