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).

In these frameworks, templates are constructed using an iterative process of template estimation and alignment, which is often computationally very expensive. Due in part to this shortcoming, most methods compute asingle template for the entire population of images, or a few templates for specific sub-groups of the data.

While optimization-based methods boast gen-eralizability across modalities and robust performance, learning-based methods promise peak performance, incorporating weak supervision and amortized optimization. However, the exact conditions for either paradigm to perform well over the other are shrouded and not explicitly outlined in the existing literature.

Neural Information Processing Systems http://nips.cc/

Recent methods introduce deep neural networks to predict the controlling parameters of hand-crafted geometric transformation models (e.g.

We demonstrate the effectiveness of our method on both simulated 2D images andreal3Dbrain magnetic resonance (MR) images.

We propose contrastive coding to learn shared, dense image representations, referredtoasCoMIRs (ContrastiveMultimodal ImageRepresentations).