# Free-Boundary Quasiconformal Maps via a Least-squares Operator in Diffeomorphism Optimization

arXiv: https://arxiv.org/abs/2511.11679 · HTML: https://mlsanity.com/paper/2511.11679/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 47 references that are in the map.
- First posted on arXiv 12 Nov 2025.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [GraphNorm: A Principled Approach to Accelerating Graph Neural Network Training](https://mlsanity.com/paper/2009.03294/) (2020)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks](https://mlsanity.com/paper/2003.09514/) (2020)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Instant Multi-View Head Capture through Learnable Registration](https://mlsanity.com/paper/2306.07437/) (2023)
- [Learning a Model-Driven Variational Network for Deformable Image Registration](https://mlsanity.com/paper/2105.12227/) (2021)
- [On Finite Difference Jacobian Computation in Deformable Image Registration](https://mlsanity.com/paper/2212.06060/) (2022)
- [Computational Pattern Making from 3D Garment Models](https://mlsanity.com/paper/2202.10272/) (2022)
- [Diffeomorphic Image Registration with Neural Velocity Field](https://mlsanity.com/paper/2202.12498/) (2022)
- [Landmark and Intensity Based Registration with Large Deformations via Quasi-conformal Maps](https://mlsanity.com/paper/1310.1710/) (2013)
- [Fast Disk Conformal Parameterization of Simply-connected Open Surfaces](https://mlsanity.com/paper/1408.6974/) (2014)
- [Spherical Density-Equalizing Map for Genus-0 Closed Surfaces](https://mlsanity.com/paper/2401.11795/) (2024)
- [Shape analysis via inconsistent surface registration](https://mlsanity.com/paper/2003.01357/) (2020)
- [Density-equalizing maps for simply-connected open surfaces](https://mlsanity.com/paper/1704.02525/) (2017)
- [Bijective Density-Equalizing Quasiconformal Map for Multiply-Connected Open Surfaces](https://mlsanity.com/paper/2308.05579/) (2023)
- [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)

## Built on by (most cited first, 1 of 1)

- [Two-chart Beltrami Optimization for Distortion-Controlled Spherical Bijection with Application to Brain Surface Registration](https://mlsanity.com/paper/2602.01589/) (2026)

## Abstract (arXiv)

Free-boundary diffeomorphism optimization, an important and widely occurring task in geometric modeling, computer graphics, and biological imaging, requires simultaneously determining a planar target domain and a locally bijective map with well-controlled distortion. We formulate this task through the least-squares quasiconformal (LSQC) operator and establish key structural properties of the LSQC minimizer, including well-posedness under mild conditions, invariance under similarity transformations, and resolution-independent behavior with stability under mesh refinement. We further analyze the sensitivity of the LSQC solution with respect to the Beltrami coefficient, establishing stability and differentiability properties that enable gradient-based optimization over the space of Beltrami coefficients. To make this differentiable formulation practical at scale and to facilitate the optimization process, we introduce the Spectral Beltrami Network (SBN), a multiscale mesh-spectral surrogate that approximates the LSQC solution operator in a single differentiable forward pass. This yields SBN-Opt, an optimization framework that searches over admissible Beltrami coefficients and pinning conditions to solve free-boundary diffeomorphism objectives with explicit distortion control. Extensive experiments on equiareal parameterization and inconsistent surface registration demonstrate consistent improvements over traditional numerical algorithms.