# Learning Geodesics of Geometric Shape Deformations From Images

arXiv: https://arxiv.org/abs/2410.18797 · HTML: https://mlsanity.com/paper/2410.18797/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 24 Oct 2024.

## Builds on (most cited first, 17 of 17)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Unsupervised Learning of Probabilistic Diffeomorphic Registration for Images and Surfaces](https://mlsanity.com/paper/1903.03545/) (2019)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [DiffuseMorph: Unsupervised Deformable Image Registration Using Diffusion Model](https://mlsanity.com/paper/2112.05149/) (2021)
- [Topology-Preserving Shape Reconstruction and Registration via Neural Diffeomorphic Flow](https://mlsanity.com/paper/2203.08652/) (2022)
- [DeepFLASH: An Efficient Network for Learning-based Medical Image Registration](https://mlsanity.com/paper/2004.02097/) (2020)
- [Metric Learning for Image Registration](https://mlsanity.com/paper/1904.09524/) (2019)
- [NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration](https://mlsanity.com/paper/2108.03443/) (2021)
- [Medical Image Registration via Neural Fields](https://mlsanity.com/paper/2206.03111/) (2022)
- [Geo-SIC: Learning Deformable Geometric Shapes in Deep Image Classifiers](https://mlsanity.com/paper/2210.13704/) (2022)
- [NeurEPDiff: Neural Operators to Predict Geodesics in Deformation Spaces](https://mlsanity.com/paper/2303.07115/) (2023)

## Built on by (most cited first, 2 of 2)

- [Finding geodesics with the Deep Ritz method](https://mlsanity.com/paper/2510.15177/) (2025)
- [Geodesic-informed Generative Diffusion Model For Topology-preserved Image Video Generation](https://mlsanity.com/paper/2609.08153/) (2026)

## Abstract (arXiv)

This paper presents a novel method, named geodesic deformable networks (GDN), that for the first time enables the learning of geodesic flows of deformation fields derived from images. In particular, the capability of our proposed GDN being able to predict geodesics is important for quantifying and comparing deformable shape presented in images. The geodesic deformations, also known as optimal transformations that align pairwise images, are often parameterized by a time sequence of smooth vector fields governed by nonlinear differential equations. A bountiful literature has been focusing on learning the initial conditions (e.g., initial velocity fields) based on registration networks. However, the definition of geodesics central to deformation-based shape analysis is blind to the networks. To address this problem, we carefully develop an efficient neural operator to treat the geodesics as unknown mapping functions learned from the latent deformation spaces. A composition of integral operators and smooth activation functions is then formulated to effectively approximate such mappings. In contrast to previous works, our GDN jointly optimizes a newly defined geodesic loss, which adds additional benefits to promote the network regularizability and generalizability. We demonstrate the effectiveness of GDN on both 2D synthetic data and 3D real brain magnetic resonance imaging (MRI).