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. However, this dichotomy creates a bottleneck for emerging scientific data, such as spatial transcriptomics, where high-dimensional vector-valued functions, e.g., gene expression, are defined on irregular, sparse manifolds. Consequently, researchers currently face a forced choice: either sacrifice single-cell resolution via voxelization to utilize image-based tools, or ignore the functional signal to utilize geometric tools. To resolve this dilemma, we propose Domain Elastic Transform (DET), a grid-free probabilistic framework that combines geometric and functional alignment. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. We for mulate the problem within a generalized Bayesian framework, modeling domain deformation as an elastic motion guided by a joint spatial functional likelihood. The method is fully unsupervised and scalable through registration on sampled points followed by displacement interpolation. We evaluate DET on spatial-transcriptomics registration tasks using MERFISH mouse-brain slices and Stereo-seq mouse-embryo at lases. On a 90-case MERFISH benchmark under severe perturbations without 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, non rigid refinement improved several within-pipeline anatomical-domain and boundary-consistency measures. The results suggest that grid free function registration is a useful complement to existing point set-based, image-based, and optimal-transport approaches for high dimensional scientific data. The implementation of DET is available at https://github.com/ohirose/bcpd.