Domain Elastic Transform: Bayesian Function Registration for High-Dimensional Scientific Data

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).