Coherent Point Drift (CPD) is a representative probabilistic framework for unsupervised non-rigid point set registration. Its standard non-rigid M-step, however, relies on a point-indexed Gaussian-kernel system whose size grows with the number of moving points, making deformation estimation computationally heavy for large point sets and difficult to control in complexity during registration. To address these limitations, we propose Analytic-CPD, a new unsupervised non-rigid registration framework that gives CPD a structured analytic reformulation. Analytic-CPD preserves the CPD posterior correspondence layer, but lifts the M-step from point-indexed kernel displacement estimation to structured analytic mapping estimation. By coupling the Gaussian-mixture posterior mechanism of CPD with Structured Analytic Mappings (SAM), the method obtains a deformation model whose coefficient dimension is governed by the ambient dimension and analytic order rather than by the number of moving points. More importantly, deformation estimation is organized over an interpretable hierarchy of analytic function spaces, so the analytic order can be increased progressively as posterior correspondences become more reliable. We implement this idea through an increasing-degree continuation strategy with decreasing stage lengths: low-order analytic maps first stabilize the posterior correspondence structure, while higher-order modes later refine nonlinear residual deformation. Experiments on controlled model-matched, smooth model-mismatch, and registered human-shape data demonstrate the effectiveness and favorable accuracy--efficiency performance of Analytic-CPD.

In this section, we describe in detail the proposed architecture and its implementation. Our architecture is composed by an encoder and a decoder. The encoder receives as input a predefined number of learnable latent probes LP, together with the point coordinates of the target point cloud XT. Each layer of the encoder performs an operation of cross-attention between LP and XT followed by a self-attention on LP. Each attention is followed by a feed-forward layer.

In this paper, we propose a transformer-based procedure for the efficient registration of non-rigid 3D point clouds. The proposed approach is data-driven and adopts for the first time the transformer architecture in the registration task. Our method is general and applies to different settings. Given a fixed template with some desired properties (e.g.

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).

Classical optimization and learning-based methods are the two reigning paradigms in deformable image registration. While optimization-based methods boast generalizability 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. In this paper, we make an explicit correspondence between the mutual information of the distribution of per-pixel intensity and labels, and the performance of classical registration methods. This strong correlation hints to the fact that architectural designs in learning-based methods is unlikely to affect this correlation, and therefore, the performance of learning-based methods. This hypothesis is thoroughly validated with state-of-the-art classical and learning-based methods. However, learning-based methods with weak supervision can perform high-fidelity intensity and label registration, which is not possible with classical methods. Next, we show that this high-fidelity feature learning does not translate to invariance to domain shift, and learning-based methods are sensitive to such changes in the data distribution. We reassess and recalibrate performance expectations from classical and DLIR methods under access to label supervision, training time, and its generalization capabilities under minor domain shifts.

Poodle crossbreeds have grown overwhelmingly popular, sparking controversy in dog parks and kennel clubs alike. The features of doodles such as Peaches (above), a goldendoodle, have become the canine equivalent of Instagram face. Meet the Breeds, the American Kennel Club's annual showcase of purebred dogs, took place over two eye-wateringly cold days in early February at the Javits Center, in Manhattan. About a hundred and fifty of the two hundred and five varieties recognized as official breeds by the A.K.C., the long-standing authority in the U.S. dog world, were in attendance for the public to ogle, fondle, and coo "So cute!" to, including the basset fauve de Bretagne, a hunting hound from France that's one of three newly recognized breeds recently allowed into the purebred pantheon. Some of the dogs had competed in the Westminster Kennel Club Dog Show earlier in the week, and past champions had their ribbons on display. In spite of the frigid weather, pavilions hosting the more popular breeds--the pug, the Doberman pinscher, the Great Dane, the St. Bernard--were packed. Lesser-known varieties, such as the saluki, the Löwchen, and the Lapponian herder, drew sparser crowds. There were exhibition spaces for each breed, and on the back walls were three adjectives supposedly describing that particular type of dog's temperament. There is, in fact, no evidence that temperament is consistent within a breed, but the idea is deeply rooted in dogdom. I stopped to caress the velvety ear leather of a pharaoh hound ("Friendly, Smart, Noble"), a sprinting breed once used to hunt rabbits in Malta; accept kisses from a Portuguese water dog, bred to assist with retrieving tackle ("Affectionate, Adventurous, Athletic"); and have my photograph taken with a Leonberger, a German breed from the town of Leonberg, in southwest Germany ("Friendly, Gentle, Playful"). No one was supposed to be openly selling dogs, but, if you asked, the breeders would share their information. Excluding what are known as companion dogs, like the Leonberger, most of the animals at the show were designed for a purpose that is no longer required of them. In Great Britain, foxhounds are legally barred from chasing foxes. Consider the fate of the otterhound, an ancient variety with a noble heritage which was once used in the U.K. to hunt river otters, which were prized for their thick fur and disliked by wealthy landowners because they ate fish in their stocked ponds.

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.