Abstract
Nonrigid registration is conventionally divided into point set registration, which aligns sparse geometries, and image registration, which aligns continuous intensity fields on regular grids. This dichotomy is limiting for emerging scientific data such as spatial transcriptomics, where high-dimensional vector-valued functions, e.g., gene expression, are defined on irregular sparse manifolds. Researchers must therefore either sacrifice single-cell resolution through voxelization or ignore functional signals in favor of geometric alignment. We propose Domain Elastic Transform (DET), a grid-free probabilistic framework that jointly aligns geometry and function. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. Within a generalized Bayesian formulation, domain deformation is modeled as elastic motion guided by a joint spatial-functional likelihood. DET is fully unsupervised and scalable through registration on sampled points followed by displacement interpolation. We evaluate DET on MERFISH mouse-brain slices and Stereo-seq mouse-embryo atlases. On a 90-case MERFISH benchmark with severe perturbations and no prior initialization, DET achieved the strongest spatial overlap and topology among the evaluated pipelines, while an accelerated PASTE2 variant achieved the highest label-transfer ARI. In an atlas-scale MOSTA feasibility study without cross-stage ground truth, nonrigid refinement improved several within-pipeline anatomical-domain and boundary-consistency measures. These results suggest that grid-free function registration complements point-set, image-based, and optimal-transport approaches for high-dimensional scientific data. The DET implementation is available at https://github.com/ohirose/bcpd (since Mar, 2025).
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May 1, 2026cs.LG
Coherent Point Drift (CPD) is a representative probabilistic framework for unsupervised non-rigid point set registration. Its standard non-rigid M-step, however, relies on a point-indexed Gaussian-kernel system whose size grows with the number of moving points, making deformation estimation computationally heavy for large point sets and difficult to control in complexity during registration. To address these limitations, we propose Analytic-CPD, a new unsupervised non-rigid registration framework that gives CPD a structured analytic reformulation. Analytic-CPD preserves the CPD posterior correspondence layer, but lifts the M-step from point-indexed kernel displacement estimation to structured analytic mapping estimation. By coupling the Gaussian-mixture posterior mechanism of CPD with Structured Analytic Mappings (SAM), the method obtains a deformation model whose coefficient dimension is governed by the ambient dimension and analytic order rather than by the number of moving points. More importantly, deformation estimation is organized over an interpretable hierarchy of analytic function spaces, so the analytic order can be increased progressively as posterior correspondences become more reliable. We implement this idea through an increasing-degree continuation strategy with decreasing stage lengths: low-order analytic maps first stabilize the posterior correspondence structure, while higher-order modes later refine nonlinear residual deformation. Experiments on controlled model-matched, smooth model-mismatch, and registered human-shape data demonstrate the effectiveness and favorable accuracy--efficiency performance of Analytic-CPD.
Wei Feng, Haiyong Zheng
Aug 6, 2026cs.CV
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/.
Mattia Masiero, Ilya A. Petrov, Daniel Cremers +2
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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.
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