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

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).

• The paper introduces DET, a grid-free Bayesian framework that aligns high-dimensional function data on irregular domains while preserving single-cell resolution.
• It employs variational inference with a Gaussian process prior and ARD to balance spatial proximity and functional similarity for robust registration.
• Empirical results in spatial transcriptomics demonstrate DET’s superior performance with a 92% topology score and high geometric accuracy.

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

Introduction and Motivation

The traditional dichotomy in non-rigid registration—point set registration for sparse geometries and image registration for continuous fields—has become inadequate for modern high-dimensional scientific data, particularly in spatial transcriptomics. Such data are characterized by high-dimensional vector-valued functions (e.g., gene expression) defined on irregular, sparse manifolds, where neither geometric nor grid-based approaches suffice. The limitations of voxelization for image registration and the lack of functional signal in geometry-only alignment lead to either loss of resolution or ambiguity in correspondence.

To address these constraints, the paper introduces Domain Elastic Transform (DET), a grid-free, Bayesian, and training-free framework for direct function registration of high-dimensional signals on irregular domains (2603.21235). DET unifies geometric and functional alignment using a probabilistic approach, allowing the preservation of single-cell resolution and high-frequency signal structure in large-scale, irregular scientific datasets.

Bayesian Framework and Generative Model

DET formulates function registration as the Bayesian inference of a nonlinear domain transformation $\mathcal{T}$, which elastically warps the source domain to align with the target, guided by both spatial and functional similarity. The framework leverages a generative model where the target function is constructed by transforming the source via smooth deformations, modeled by a Gaussian process-based motion coherence prior.

Figure 2: Illustration of the DET generative model for function registration, highlighting domain deformation under the motion coherence prior.

Correspondences are established probabilistically between discretized source and target points. Outliers are explicitly modeled through a variable $c_n$ with a Bernoulli prior, and matching probabilities are maintained for all candidate associations. The transformation $\mathcal{T}$ combines a global similarity component (scale, rotation, translation) and a spatially smooth, nonrigid displacement field.

DET's likelihood incorporates both spatial proximity and functional similarity, with a balancing parameter ensuring that high-dimensional functional information neither overwhelms nor is ignored relative to geometry.

Figure 3: Illustration of function registration with spatial and functional constraints on irregular domains.

Figure 1: Notation for point correspondences and outlier modeling in DET.

Variational Inference and Algorithmic Design

Exact Bayesian inference in this setting is computationally infeasible due to the combinatorial number of correspondence assignments. DET employs variational inference with factorized posteriors over transformation, correspondence, and noise parameters. Updates for transformation parameters, matching probabilities, and functional-noise covariance are derived in closed form for efficient optimization.

The DET algorithm alternates between updating the matching matrix (P), estimating local and global transformations, and re-estimating noise scales. Functional similarity is automatically down-weighted in high dimensions using automatic relevance determination (ARD), implemented by constraining the functional noise covariance to a diagonal structure and updating it accordingly.

Figure 4: Progression of function registration with DET, demonstrating domain deformation and correspondence refinement.

Implementation Strategies for Scalability and Robustness

DET incorporates several methodological advances to enable practical registration for large-scale, high-dimensional datasets:

• Nyström and k-D tree acceleration scale the core computations (matching matrix and coherence kernel) to millions of points, maintaining sublinear complexity with regard to dataset size.
• Feature-sensitive sampling (VGIS) prioritizes landmark selection in regions of high functional and geometric variability, preserving critical anatomical interfaces and boundaries.
• Surface-aware coherence matrices combine geodesic and Euclidean distances in the prior, allowing complex, boundary-aware deformations across tissue interfaces and irregular domains.
• Robust adaptive outlier modeling and functional-geometry likelihood balancing ensure that neither dimensionality nor data noise skews correspondences.

  Figure 6: Scalability of DET runtime as a function of the total number of cells and subsampling size.

Empirical Evaluation

DET is evaluated on challenging benchmarks in spatial transcriptomics:

Spatiotemporal Registration of Embryonic Atlases

DET aligns mouse embryo Stereo-seq data across developmental stages (E14.5 → E15.5), a task requiring highly elastic and biologically plausible transformations due to large growth-induced deformations. The result demonstrates accurate alignment of more than 100,000 points, preserving tissue boundaries and topological structure.

Figure 7: Spatiotemporal registration of Mouse Embryo Stereo-seq data (E14.5 $\rightarrow$ E15.5), showing effective alignment of complex anatomical domains.

High-Dimensional Slice Registration with MERFISH

Comparison with SOTA methods on MERFISH brain slice data demonstrates several critical strengths of DET:

• Topology preservation: DET achieves a 92% topology score, far exceeding optimal transport methods that fragment tissue integrity ($<$5%).
• Alignment robustness: DET maintains geometric accuracy (Jaccard 0.88 $\pm$ 0.04) even in the presence of large initial misalignment, significantly outperforming BCPD (0.69 $\pm$ 0.36) and PASTE (0.64 $\pm$ 0.01).
• Functional correspondence: Though PASTE achieves the highest functional PCC (0.85), it does so at the cost of tissue topology. DET achieves a strong compromise (functional PCC 0.77) while retaining high geometric and structural integrity.

  Figure 8: Comparative results of DET and SOTA methods for MERFISH data, highlighting DET's balance between functional and geometric alignment.

Practical and Theoretical Implications

DET provides a systematic solution to the "grid vs. geometry" bottleneck in high-dimensional data registration by enabling resolution-preserving, training-free correspondence across irregular, non-Euclidean domains. Its grid-free, function-based Bayesian model opens the possibility for robust multimodal alignment in settings where deep learning either cannot generalize or lacks training data.

Theoretically, the probabilistic modeling of function registration via smooth, regularized deformation and ARD-guided likelihood balancing provides a principled framework applicable to new data modalities beyond spatial transcriptomics, such as single-cell multi-omics, shapes with heterogeneous features, or multimodal medical imaging.

Future Directions

Several avenues emerge for further research:

• Generalization to manifold-valued data: Extending DET's principles to more abstract geometric domains, such as meshes and hypergraphs, could broaden its impact.
• Integration with uncertainty quantification: Since the Bayesian machinery yields posteriors over transformations and correspondences, downstream inference pipelines could benefit from explicit uncertainty estimates.
• End-to-end learning for parameter tuning: While DET is training-free, meta-optimization approaches could automate or adapt DET's priors and balancing parameters for particular domains.
• Function registration in implicit or continuous settings: Coupling DET with neural implicit representations holds promise for ultra-high resolution or time-varying data.

Conclusion

DET introduces a grid-free, scalable, and fully unsupervised Bayesian framework for function registration that robustly aligns high-dimensional scientific data on irregular domains. Empirically, it achieves high topological fidelity and functional accuracy where existing geometry- and image-based methods fail to generalize or scale. DET is poised to serve as a foundational algorithmic tool for analysis of next-generation multimodal scientific datasets, particularly where annotation is scarce and resolution must be maintained (2603.21235).