# A Deep Learning Framework for Diffeomorphic Mapping Problems via Quasi-conformal Geometry applied to Imaging

arXiv: https://arxiv.org/abs/2110.10580 · HTML: https://mlsanity.com/paper/2110.10580/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 62 references that are in the map.
- First posted on arXiv 20 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Object Detection with Deep Learning: A Review](https://mlsanity.com/paper/1807.05511/) (2018)
- [A Deep Learning Framework for Unsupervised Affine and Deformable Image Registration](https://mlsanity.com/paper/1809.06130/) (2018)
- [Subsampled Turbulence Removal Network](https://mlsanity.com/paper/1807.04418/) (2018)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [TEMPO: Feature-Endowed Teichm"uller Extremal Mappings of Point Clouds](https://mlsanity.com/paper/1511.06624/) (2015)
- [Topology-Preserving 3D Image Segmentation Based On Hyperelastic Regularization](https://mlsanity.com/paper/2103.16768/) (2021)
- [Shape analysis via inconsistent surface registration](https://mlsanity.com/paper/2003.01357/) (2020)
- [Tooth morphometry using quasi-conformal theory](https://mlsanity.com/paper/1901.01651/) (2019)
- [Computing quasiconformal folds](https://mlsanity.com/paper/1804.03936/) (2018)
- [Inconsistent Surface Registration via Optimization of Mapping Distortions](https://mlsanity.com/paper/1908.09098/) (2019)
- [Efficient Feature-based Image Registration by Mapping Sparsified Surfaces](https://mlsanity.com/paper/1605.06215/) (2016)
- [Quasi-conformal Geometry based Local Deformation Analysis of Lateral Cephalogram for Childhood OSA Classification](https://mlsanity.com/paper/2006.11408/) (2020)
- [Quasiconformal model with CNN features for large deformation image registration](https://mlsanity.com/paper/2011.00731/) (2020)
- [Geometric Registration of High-genus Surfaces](https://mlsanity.com/paper/1305.2283/) (2013)

## Built on by (most cited first, 7 of 7)

- [A Learning-based Framework for Topology-Preserving Segmentation using Quasiconformal Mappings](https://mlsanity.com/paper/2210.03299/) (2022)
- [Deformation-Invariant Neural Network and Its Applications in Distorted Image Restoration and Analysis](https://mlsanity.com/paper/2310.02641/) (2023)
- [Deep Learning-Based Quasi-Conformal Surface Registration for Partial 3D Faces Applied to Facial Recognition](https://mlsanity.com/paper/2405.09880/) (2024)
- [A Symmetric Dynamic Learning Framework for Diffeomorphic Medical Image Registration](https://mlsanity.com/paper/2411.02888/) (2024)
- [Generalization Bounds for Equivariant Networks on Markov Data](https://mlsanity.com/paper/2503.00292/) (2025)
- [Harmonic Beltrami Signature Network: a Shape Prior Module in Deep Learning Framework](https://mlsanity.com/paper/2603.02907/) (2026)
- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)

## Abstract (arXiv)

Many imaging problems can be formulated as mapping problems. A general mapping problem aims to obtain an optimal mapping that minimizes an energy functional subject to the given constraints. Existing methods to solve the mapping problems are often inefficient and can sometimes get trapped in local minima. An extra challenge arises when the optimal mapping is required to be diffeomorphic. In this work, we address the problem by proposing a deep-learning framework based on the Quasiconformal (QC) Teichmuller theories. The main strategy is to learn the Beltrami coefficient (BC) that represents a mapping as the latent feature vector in the deep neural network. The BC measures the local geometric distortion under the mapping, with which the interpretability of the deep neural network can be enhanced. Under this framework, the diffeomorphic property of the mapping can be controlled via a simple activation function within the network. The optimal mapping can also be easily regularized by integrating the BC into the loss function. A crucial advantage of the proposed framework is that once the network is successfully trained, the optimized mapping corresponding to each input data information can be obtained in real time. To examine the efficacy of the proposed framework, we apply the method to the diffeomorphic image registration problem. Experimental results outperform other state-of-the-art registration algorithms in both efficiency and accuracy, which demonstrate the effectiveness of our proposed framework to solve the mapping problem.