# X-Metric: An N-Dimensional Information-Theoretic Framework for Groupwise Registration and Deep Combined Computing

arXiv: https://arxiv.org/abs/2211.01631 · HTML: https://mlsanity.com/paper/2211.01631/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 62 references that are in the map.
- First posted on arXiv 3 Nov 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Domain-Specific Batch Normalization for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1906.03950/) (2019)
- [Multivariate mixture model for myocardium segmentation combining multi-source images](https://mlsanity.com/paper/1612.08820/) (2016)
- [Learn2Reg: comprehensive multi-task medical image registration challenge, dataset and evaluation in the era of deep learning](https://mlsanity.com/paper/2112.04489/) (2021)
- [HeMIS: Hetero-Modal Image Segmentation](https://mlsanity.com/paper/1607.05194/) (2016)
- [Weakly-Supervised Convolutional Neural Networks for Multimodal Image Registration](https://mlsanity.com/paper/1807.03361/) (2018)
- [Multi-Atlas Segmentation of Biomedical Images: A Survey](https://mlsanity.com/paper/1412.3421/) (2014)
- [Cardiac Segmentation on Late Gadolinium Enhancement MRI: A Benchmark Study from Multi-Sequence Cardiac MR Segmentation Challenge](https://mlsanity.com/paper/2006.12434/) (2020)
- [A Deep Metric for Multimodal Registration](https://mlsanity.com/paper/1609.05396/) (2016)
- [Joint Motion Correction and Super Resolution for Cardiac Segmentation via Latent Optimisation](https://mlsanity.com/paper/2107.03887/) (2021)
- [Deformation corrected compressed sensing (DC-CS): a novel framework for accelerated dynamic MRI](https://mlsanity.com/paper/1405.7718/) (2014)
- [Variational Multi-Task MRI Reconstruction: Joint Reconstruction, Registration and Super-Resolution](https://mlsanity.com/paper/1908.05911/) (2019)

## Built on by (most cited first, 10 of 10)

- [SA-Med2D-20M Dataset: Segment Anything in 2D Medical Imaging with 20 Million masks](https://mlsanity.com/paper/2311.11969/) (2023)
- [Interactive Medical Image Segmentation: A Benchmark Dataset and Baseline](https://mlsanity.com/paper/2411.12814/) (2024)
- [MAPSeg: Unified Unsupervised Domain Adaptation for Heterogeneous Medical Image Segmentation Based on 3D Masked Autoencoding and Pseudo-Labeling](https://mlsanity.com/paper/2303.09373/) (2023)
- [FPL+: Filtered Pseudo Label-based Unsupervised Cross-Modality Adaptation for 3D Medical Image Segmentation](https://mlsanity.com/paper/2404.04971/) (2024)
- [MedSG-Bench: A Benchmark for Medical Image Sequences Grounding](https://mlsanity.com/paper/2505.11852/) (2025)
- [Volumetric medical image segmentation through dual self-distillation in U-shaped networks](https://mlsanity.com/paper/2306.03271/) (2023)
- [CAR: Contrast-Agnostic Deformable Medical Image Registration with Contrast-Invariant Latent Regularization](https://mlsanity.com/paper/2408.05341/) (2024)
- [Bayesian Unsupervised Disentanglement of Anatomy and Geometry for Deep Groupwise Image Registration](https://mlsanity.com/paper/2401.02141/) (2024)
- [MedCL: Learning Consistent Anatomy Distribution for Scribble-supervised Medical Image Segmentation](https://mlsanity.com/paper/2503.22890/) (2025)
- [Adaptive Conditional Contrast-Agnostic Deformable Image Registration with Uncertainty Estimation](https://mlsanity.com/paper/2601.05981/) (2026)

## Abstract (arXiv)

This paper presents a generic probabilistic framework for estimating the statistical dependency and finding the anatomical correspondences among an arbitrary number of medical images. The method builds on a novel formulation of the N-dimensional joint intensity distribution by representing the common anatomy as latent variables and estimating the appearance model with nonparametric estimators. Through connection to maximum likelihood and the expectation-maximization algorithm, an informationhyptheoretic metric called X-metric and a co-registration algorithm named X-CoReg are induced, allowing groupwise registration of the N observed images with computational complexity of O(N). Moreover, the method naturally extends for a weakly-supervised scenario where anatomical labels of certain images are provided. This leads to a combinedhypcomputing framework implemented with deep learning, which performs registration and segmentation simultaneously and collaboratively in an end-to-end fashion. Extensive experiments were conducted to demonstrate the versatility and applicability of our model, including multimodal groupwise registration, motion correction for dynamic contrast enhanced magnetic resonance images, and deep combined computing for multimodal medical images. Results show the superiority of our method in various applications in terms of both accuracy and efficiency, highlighting the advantage of the proposed representation of the imaging process.