# ResNet-LDDMM: Advancing the LDDMM Framework using Deep Residual Networks arXiv: https://arxiv.org/abs/2102.07951 ยท HTML: https://mlsanity.com/paper/2102.07951/ ## Key facts - Cited by 13 papers in the map (data through 14 Oct 2026). - Builds on 28 of its 77 references that are in the map. - First posted on arXiv 16 Feb 2021. ## Builds on (most cited first, 28 of 28) - [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016) - [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015) - [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013) - [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017) - [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018) - [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019) - [3D-CODED : 3D Correspondences by Deep Deformation](https://mlsanity.com/paper/1806.05228/) (2018) - [Deep Functional Maps: Structured Prediction for Dense Shape Correspondence](https://mlsanity.com/paper/1704.08686/) (2017) - [Deep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence](https://mlsanity.com/paper/2003.14286/) (2020) - [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018) - [Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks](https://mlsanity.com/paper/2003.09514/) (2020) - [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020) - [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019) - [Learning a Probabilistic Model for Diffeomorphic Registration](https://mlsanity.com/paper/1812.07460/) (2018) - [Partial Functional Correspondence](https://mlsanity.com/paper/1506.05274/) (2015) - [Networks for Joint Affine and Non-parametric Image Registration](https://mlsanity.com/paper/1903.08811/) (2019) - [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017) - [Metric Learning for Image Registration](https://mlsanity.com/paper/1904.09524/) (2019) - [Deep Learning Theory Review: An Optimal Control and Dynamical Systems Perspective](https://mlsanity.com/paper/1908.10920/) (2019) - [Efficient Deformable Shape Correspondence via Kernel Matching](https://mlsanity.com/paper/1707.08991/) (2017) - [The varifold representation of non-oriented shapes for diffeomorphic registration](https://mlsanity.com/paper/1304.6108/) (2013) - [Region-specific Diffeomorphic Metric Mapping](https://mlsanity.com/paper/1906.00139/) (2019) - [Quasi-Newton Solver for Robust Non-Rigid Registration](https://mlsanity.com/paper/2004.04322/) (2020) - [Spectral Generalized Multi-Dimensional Scaling](https://mlsanity.com/paper/1311.2187/) (2013) - [A Lagrangian Gauss-Newton-Krylov Solver for Mass- and Intensity-Preserving Diffeomorphic Image Registration](https://mlsanity.com/paper/1703.04446/) (2017) - [Diffeomorphic Learning](https://mlsanity.com/paper/1806.01240/) (2018) ## Built on by (most cited first, 13 of 13) - [Unsupervised Diffeomorphic Surface Registration and Non-Linear Modelling](https://mlsanity.com/paper/2109.13630/) (2021) - [RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations](https://mlsanity.com/paper/2305.12854/) (2023) - [A deep residual learning implementation of Metamorphosis](https://mlsanity.com/paper/2202.00676/) (2022) - [DFORM: Diffeomorphic vector field alignment for assessing dynamics across learned models](https://mlsanity.com/paper/2402.09735/) (2024) - [Self Supervised Networks for Learning Latent Space Representations of Human Body Scans and Motions](https://mlsanity.com/paper/2411.03475/) (2024) - [Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEs](https://mlsanity.com/paper/2506.00947/) (2025) - [AI-Powered Automated Model Construction for Patient-Specific CFD Simulations of Aortic Flows](https://mlsanity.com/paper/2503.12515/) (2025) - [Stationary Velocity Fields on Matrix Groups for Deformable Image Registration](https://mlsanity.com/paper/2410.10997/) (2024) - [FalconBC: Flow matching for Amortized inference of Latent-CONditioned physiologic Boundary Conditions](https://mlsanity.com/paper/2603.19331/) (2026) - [Automatic classification of deformable shapes](https://mlsanity.com/paper/2211.02530/) (2022) - [FineMorphs: Affine-diffeomorphic sequences for regression](https://mlsanity.com/paper/2305.17255/) (2023) - [Shape Modeling of Longitudinal Medical Images: From Diffeomorphic Metric Mapping to Deep Learning](https://mlsanity.com/paper/2503.21489/) (2025) - [LDDMM stochastic interpolants: an application to domain uncertainty quantification in hemodynamics](https://mlsanity.com/paper/2603.28324/) (2026) ## Abstract (arXiv) In deformable registration, the geometric framework - large deformation diffeomorphic metric mapping or LDDMM, in short - has inspired numerous techniques for comparing, deforming, averaging and analyzing shapes or images. Grounded in flows, which are akin to the equations of motion used in fluid dynamics, LDDMM algorithms solve the flow equation in the space of plausible deformations, i.e. diffeomorphisms. In this work, we make use of deep residual neural networks to solve the non-stationary ODE (flow equation) based on a Euler's discretization scheme. The central idea is to represent time-dependent velocity fields as fully connected ReLU neural networks (building blocks) and derive optimal weights by minimizing a regularized loss function. Computing minimizing paths between deformations, thus between shapes, turns to find optimal network parameters by back-propagating over the intermediate building blocks. Geometrically, at each time step, ResNet-LDDMM searches for an optimal partition of the space into multiple polytopes, and then computes optimal velocity vectors as affine transformations on each of these polytopes. As a result, different parts of the shape, even if they are close (such as two fingers of a hand), can be made to belong to different polytopes, and therefore be moved in different directions without costing too much energy. Importantly, we show how diffeomorphic transformations, or more precisely bilipshitz transformations, are predicted by our algorithm. We illustrate these ideas on diverse registration problems of 3D shapes under complex topology-preserving transformations. We thus provide essential foundations for more advanced shape variability analysis under a novel joint geometric-neural networks Riemannian-like framework, i.e. ResNet-LDDMM.