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

TL;DR

The paper introduces Domain Elastic Transform (DET), a Bayesian, grid-free framework for high-dimensional, irregular domain data registration, unifying geometric and functional alignment without sacrificing resolution or requiring binning.

Contribution

DET is a novel probabilistic method that directly registers high-dimensional functions on irregular domains, overcoming limitations of existing point set and image registration techniques.

Findings

Achieves 92% topological preservation on MERFISH data

Outperforms state-of-the-art optimal transport methods

Successfully registers large-scale, complex developmental atlases

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. However, this dichotomy creates a critical 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 critical functional signal to utilize geometric tools. To resolve this dilemma, we propose Domain Elastic Transform (DET), a grid-free probabilistic framework that unifies geometric and functional alignment. By treating data as functions on irregular domains, DET registers high-dimensional signals directly without binning. We formulate the problem within a rigorous Bayesian framework, modeling domain deformation as an elastic motion guided by a joint spatial-functional likelihood. The method is fully unsupervised and scalable, utilizing feature-sensitive downsampling to handle massive atlases. We demonstrate that DET achieves 92\% topological preservation on MERFISH data where state-of-the-art optimal transport methods struggle ($<$5\%), and successfully registers whole-embryo Stereo-seq atlases across developmental stages -- a task involving massive scale and complex nonrigid growth. The implementation of DET is available on {https://github.com/ohirose/bcpd} (since Mar, 2025).