# Parallelizable global conformal parameterization of simply-connected surfaces via partial welding

## TL;DR

This paper introduces a parallelizable algorithm for global conformal surface parameterization that efficiently handles dense meshes by partitioning, local computation, partial welding, and solving Laplace equations, improving speed and accuracy.

## Contribution

The novel partial welding technique and parallelized approach enable efficient, accurate conformal parameterization of large, dense meshes, extending to open and closed surfaces with shape constraints.

## Key findings

- Significant reduction in computational time compared to existing methods.
- High accuracy in conformal parameterization of dense meshes.
- Successful extension to disk and spherical conformal mappings.

## Abstract

Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution better resemble smooth surfaces, they pose computational difficulties for the existing parameterization algorithms. In this work, we propose a novel parallelizable algorithm for computing the global conformal parameterization of simply-connected surfaces via partial welding maps. A given simply-connected surface is first partitioned into smaller subdomains. The local conformal parameterizations of all subdomains are then computed in parallel. The boundaries of the parameterized subdomains are subsequently integrated consistently using a novel technique called partial welding, which is developed based on conformal welding theory. Finally, by solving the Laplace equation for each subdomain using the updated boundary conditions, we obtain a global conformal parameterization of the given surface, with bijectivity guaranteed by quasi-conformal theory. By including additional shape constraints, our method can be easily extended to achieve disk conformal parameterization for simply-connected open surfaces and spherical conformal parameterization for genus-0 closed surfaces. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. When compared to the state-of-the-art conformal parameterization methods, our method achieves a significant improvement in both computational time and accuracy.