The tissue warps. The genes follow. Align them together.
Spatial transcriptomics maps gene expression to physical locations in tissue. Different samples — same tissue type, different individuals or developmental stages — have different shapes but express the same genes. Registering them (aligning the spatial domains so gene expression patterns overlap) is necessary for comparison. Standard approaches force a choice: align the points (ignoring gene expression) or bin the data into grids and align as images (losing spatial resolution).
Domain Elastic Transform (arXiv:2603.21235): register high-dimensional signals directly on irregular domains without binning. The deformation is Bayesian — modeled as a smooth spatial warping with uncertainty. Spatial alignment (make the shapes match) and functional alignment (make the gene expression patterns match) guide the deformation jointly.
92% topological preservation on MERFISH data versus under 5% for competing optimal transport methods. Successfully registers whole-embryo developmental atlases across different stages — aligning not just shape but the gene expression gradients that define developmental axes.
The structural insight: point registration and image registration fail for spatial transcriptomics because the data is neither points (it has high-dimensional features at each location) nor images (it lives on irregular domains, not grids). The data is a signal on a domain, and the domain deforms. You need to register the domain and the signal simultaneously, because neither alone constrains the alignment well enough. The shape without the genes is ambiguous (many tissues have similar shapes). The genes without the shape are ambiguous (the same genes express in different locations). Together they constrain.