# ICON: Learning Regular Maps Through Inverse Consistency

arXiv: https://arxiv.org/abs/2105.04459 · HTML: https://mlsanity.com/paper/2105.04459/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 47 references that are in the map.
- First posted on arXiv 10 May 2021.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Exploring Simple Siamese Representation Learning](https://mlsanity.com/paper/2011.10566/) (2020)
- [FlowNet: Learning Optical Flow with Convolutional Networks](https://mlsanity.com/paper/1504.06852/) (2015)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Optical Flow Estimation using a Spatial Pyramid Network](https://mlsanity.com/paper/1611.00850/) (2016)
- [The Pitfalls of Simplicity Bias in Neural Networks](https://mlsanity.com/paper/2006.07710/) (2020)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Iterative Residual Refinement for Joint Optical Flow and Occlusion Estimation](https://mlsanity.com/paper/1904.05290/) (2019)
- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [Cross-domain Correspondence Learning for Exemplar-based Image Translation](https://mlsanity.com/paper/2004.05571/) (2020)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [A Survey on Deep Learning Techniques for Stereo-based Depth Estimation](https://mlsanity.com/paper/2006.02535/) (2020)
- [CycleMorph: Cycle Consistent Unsupervised Deformable Image Registration](https://mlsanity.com/paper/2008.05772/) (2020)
- [Deforming Autoencoders: Unsupervised Disentangling of Shape and Appearance](https://mlsanity.com/paper/1806.06503/) (2018)
- [Networks for Joint Affine and Non-parametric Image Registration](https://mlsanity.com/paper/1903.08811/) (2019)
- [Neural reparameterization improves structural optimization](https://mlsanity.com/paper/1909.04240/) (2019)
- [Inverse-Consistent Deep Networks for Unsupervised Deformable Image Registration](https://mlsanity.com/paper/1809.03443/) (2018)
- [Metric Learning for Image Registration](https://mlsanity.com/paper/1904.09524/) (2019)
- [SoDeep: a Sorting Deep net to learn ranking loss surrogates](https://mlsanity.com/paper/1904.04272/) (2019)
- [Region-specific Diffeomorphic Metric Mapping](https://mlsanity.com/paper/1906.00139/) (2019)
- [Unsupervised Deformable Image Registration Using Cycle-Consistent CNN](https://mlsanity.com/paper/1907.01319/) (2019)
- [Fast Predictive Image Registration](https://mlsanity.com/paper/1607.02504/) (2016)
- [A Cooperative Autoencoder for Population-Based Regularization of CNN Image Registration](https://mlsanity.com/paper/1908.05825/) (2019)
- [The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile](https://mlsanity.com/paper/1407.7022/) (2014)

## Built on by (most cited first, 26 of 26)

- [uniGradICON: A Foundation Model for Medical Image Registration](https://mlsanity.com/paper/2403.05780/) (2024)
- [A survey on deep learning in medical image registration: new technologies, uncertainty, evaluation metrics, and beyond](https://mlsanity.com/paper/2307.15615/) (2023)
- [textttGradICON: Approximate Diffeomorphisms via Gradient Inverse Consistency](https://mlsanity.com/paper/2206.05897/) (2022)
- [NODEO: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration](https://mlsanity.com/paper/2108.03443/) (2021)
- [multiGradICON: A Foundation Model for Multimodal Medical Image Registration](https://mlsanity.com/paper/2408.00221/) (2024)
- [Inverse Consistency by Construction for Multistep Deep Registration](https://mlsanity.com/paper/2305.00087/) (2023)
- [SAME++: A Self-supervised Anatomical eMbeddings Enhanced medical image registration framework using stable sampling and regularized transformation](https://mlsanity.com/paper/2311.14986/) (2023)
- [textttNePhi: Neural Deformation Fields for Approximately Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2309.07322/) (2023)
- [NAISR: A 3D Neural Additive Model for Interpretable Shape Representation](https://mlsanity.com/paper/2303.09234/) (2023)
- [Improving Generalization of Medical Image Registration Foundation Model](https://mlsanity.com/paper/2505.06527/) (2025)
- [Guiding Registration with Emergent Similarity from Pre-Trained Diffusion Models](https://mlsanity.com/paper/2506.02419/) (2025)
- [Exploring Cycle Consistency Learning in Interactive Volume Segmentation](https://mlsanity.com/paper/2303.06493/) (2023)
- [Disentangling Progress in Medical Image Registration: Beyond Trend-Driven Architectures towards Domain-Specific Strategies](https://mlsanity.com/paper/2512.01913/) (2025)
- [Towards Saner Deep Image Registration](https://mlsanity.com/paper/2307.09696/) (2023)
- [Gaussian Primitives for Deformable Image Registration](https://mlsanity.com/paper/2406.03394/) (2024)
- [Geometric regularization of autoencoders via observed stochastic dynamics](https://mlsanity.com/paper/2604.16282/) (2026)
- [CARL: A Framework for Equivariant Image Registration](https://mlsanity.com/paper/2405.16738/) (2024)
- [LEDA: Log-Euclidean Diffeomorphism Autoencoder for Efficient Statistical Analysis of Diffeomorphisms](https://mlsanity.com/paper/2412.16129/) (2024)
- [Mono-Modalizing Extremely Heterogeneous Multi-Modal Medical Image Registration](https://mlsanity.com/paper/2506.15596/) (2025)
- [Uncertainty Estimation for Pretrained Medical Image Registration Models via Transformation Equivariance](https://mlsanity.com/paper/2509.23355/) (2025)
- [Characterizing and Optimizing the Spatial Kernel of Multi Resolution Hash Encodings](https://mlsanity.com/paper/2602.10495/) (2026)
- [Biomechanical Constraints Assimilation in Deep-Learning Image Registration: Application to sliding and locally rigid deformations](https://mlsanity.com/paper/2504.05444/) (2025)
- [Deformable Medical Image Registration with KAN-based Implicit Neural Representations](https://mlsanity.com/paper/2509.22874/) (2025)
- [Multimodal Diffeomorphic Registration with Neural ODEs and Structural Descriptors](https://mlsanity.com/paper/2512.22689/) (2025)
- [PRISM: A 3D Probabilistic Neural Representation for Interpretable Shape Modeling](https://mlsanity.com/paper/2602.11467/) (2026)
- [Robotic Ultrasound Makes CBCT Alive](https://mlsanity.com/paper/2603.10220/) (2026)

## Abstract (arXiv)

Learning maps between data samples is fundamental. Applications range from representation learning, image translation and generative modeling, to the estimation of spatial deformations. Such maps relate feature vectors, or map between feature spaces. Well-behaved maps should be regular, which can be imposed explicitly or may emanate from the data itself. We explore what induces regularity for spatial transformations, e.g., when computing image registrations. Classical optimization-based models compute maps between pairs of samples and rely on an appropriate regularizer for well-posedness. Recent deep learning approaches have attempted to avoid using such regularizers altogether by relying on the sample population instead. We explore if it is possible to obtain spatial regularity using an inverse consistency loss only and elucidate what explains map regularity in such a context. We find that deep networks combined with an inverse consistency loss and randomized off-grid interpolation yield well behaved, approximately diffeomorphic, spatial transformations. Despite the simplicity of this approach, our experiments present compelling evidence, on both synthetic and real data, that regular maps can be obtained without carefully tuned explicit regularizers, while achieving competitive registration performance.