Scalable and Differentiable Point-Cloud Registration Using Maximum Mean Discrepancy
Authors: Rixon Crane, Fahira Afzal Maken, Nicholas Lawrance, Stanislav Funiak, Kasra Khosoussi, Ming Xu, Russell Tsuchida
Organizations: Data61, CSIRO, Brisbane, Australia · Data61, CSIRO, Sydney, Australia · Work done while at Data61, CSIRO, Brisbane, Australia · School of Electrical Engineering and Computer Science, The University of Queensland, Brisbane, Australia · CVLab, EPFL, Lausanne, Switzerland · Department of Data Science & AI, Monash University, Melbourne, Australia
We present MMD-Reg, a novel correspondence-free approach to point-cloud registration that is differentiable and has linear computational complexity in the number of points. We model registration as a nonlinear least-squares problem based on the Maximum Mean Discrepancy, approximated using random Fourier features. The resulting objective can be solved efficiently with standard methods such as Levenberg-Marquardt, and the solution is differentiable via the implicit function theorem. This allows MMD-Reg to be used as a differentiable optimization layer within end-to-end trainable models, supporting registration under challenging conditions such as poor initial alignment and partial overlap. We demonstrate this Neural MMD-Reg formulation by integrating the layer with a set transformer, training the resulting model in supervised and unsupervised settings, and comparing its performance against recent learning-based methods. We also evaluate standalone MMD-Reg, comparing its accuracy and scalability against widely used non-learning-based registration methods.
Coherent Point Drift (CPD) is widely used for rigid point cloud registration because of its soft correspondences and closed-form parameter updates. However, CPD's target-side marginal constraint forces every observation, including outliers, to receive exactly unit probability mass. This assumption degrades registration accuracy under heavy outliers and partial overlap. Optimal transport (OT) methods can handle missing mass through unbalanced formulations, but require hand-tuned annealing schedules. In this paper, we propose Sinkhorn-CPD, which replaces CPD's target-side marginal constraint with dual Kullback-Leibler penalties, allowing the algorithm to discard outliers on both sides. The resulting formulation is a fully unbalanced entropic optimal transport problem, which can be efficiently solved by generalized Sinkhorn iterations. Moreover, Sinkhorn-CPD preserves the closed-form Procrustes and variance updates of CPD. In our method, the variance sigma^2 plays the role of the entropic regularization parameter, which induces an automatic annealing schedule from diffuse to sharp correspondences without manual temperature tuning. Experiments on synthetic, cross-category, and scan-to-CAD benchmarks show that Sinkhorn-CPD achieves state-of-the-art accuracy, with strong robustness to outliers and partial overlap.
3D human registration has historically been treated as a regression task, assuming a unique ground-truth alignment exists between the template and an input point cloud. In reality, acquisition noise, occlusions, and unknown soft tissue dynamics introduce inherent ambiguity into human scans. Regression-based methods consequently converge to an average prediction, often failing to represent a plausible geometry. In our work, we embrace such uncertainty by modeling the registration as a distribution of alignments. We propose ODin, which formulates registration as a 3D diffusion process that generates a point cloud aligned with the target geometry while preserving template semantics through consistent point ordering. To achieve this, ODin relies on global, local, and positional conditioning, guiding each point to its correct location. Our experiments demonstrate that such a generative formulation not only outperforms its regression-based baseline, but also establishes a new state of the art, surpassing highly engineered methods while reducing the registration time by two-thirds. Pre-trained models and code are available at https://riccardomarin.github.io/odin/.
Non-rigid point cloud registration is central to soft-tissue shape analysis, but large deformations, noise, and outliers make correspondence estimation challenging. Most learning-based methods rely on local objectives such as Chamfer distance, which encourage point-wise proximity but do not constrain the global plausibility of the predicted deformation field. We address this limitation with DINE, a maximum a posteriori framework that augments distance-based registration with a learned statistical prior over displacement vector fields. DINE is applied to two registration backbones, Robust-DefReg and DefTransNet, using a two-stage strategy: a first-stage model is trained with Chamfer distance, its predicted deformation fields are used to estimate a prior, and the model is then refined with a combined distance and negative log-prior objective. We compare a full-field PCA Gaussian prior with a per-vector normalizing-flow prior. Experiments on DeformedTissue and SynBench show lower mean Chamfer distance under deformation and corruption. On DeformedTissue, DINE-PCA reduces Chamfer distance by approximately 27--69% relative to the corresponding Stage-1 backbone across deformation levels, and improves robustness by up to 66% for outliers and 83% for Gaussian noise. On SynBench, improvements are modest at the smallest deformation levels and reach approximately 59--79% from moderate to severe deformation. These results suggest that global deformation plausibility is an important constraint for reliable soft-tissue point cloud registration. (The code will be published soon.)