# Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory

arXiv: https://arxiv.org/abs/2009.08279 · HTML: https://mlsanity.com/paper/2009.08279/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2020.

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

- [Automatic Landmark Detection and Registration of Brain Cortical Surfaces via Quasi-Conformal Geometry and Convolutional Neural Networks](https://mlsanity.com/paper/2208.07010/) (2022)
- [3D Face Parsing via Surface Parameterization and 2D Semantic Segmentation Network](https://mlsanity.com/paper/2206.09221/) (2022)

## Abstract (arXiv)

Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering. However, most of the existing conformal parameterization algorithms only focus on simply-connected surfaces and cannot be directly applied to surfaces with holes. In this work, we propose two novel algorithms for computing the conformal parameterization of multiply-connected surfaces. We first develop an efficient method for conformally parameterizing an open surface with one hole to an annulus on the plane. Based on this method, we then develop an efficient method for conformally parameterizing an open surface with k holes onto a unit disk with k circular holes. The conformality and bijectivity of the mappings are ensured by quasi-conformal theory. Numerical experiments and applications are presented to demonstrate the effectiveness of the proposed methods.