SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1049-1083, January 2020.
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.

中文翻译：

---

通过部分焊接的简单连接表面的可并行全局保形参数化

SIAM影像科学杂志，第13卷，第3期，第1049-1083页，2020年1月。
保形表面参数化在图形，成像和可视化中非常有用，可应用于纹理贴图，图集构造，配准，重新定型等。随着扫描和存储数据能力的增强，如今，密集的3D表面网格非常普遍。尽管具有较高分辨率的网格更好地类似于光滑表面，但是它们为现有的参数化算法带来了计算困难。在这项工作中，我们提出了一种新颖的可并行化算法，用于通过局部焊接图来计算简单连接表面的全局共形参数化。给定的简单连接表面首先被划分为较小的子域。然后并行计算所有子域的局部共形参数化。随后，使用基于局部焊接理论开发的一种称为局部焊接的新技术，始终如一地整合参数化子域的边界。最后，通过使用更新的边界条件求解每个子域的拉普拉斯方程，我们获得了给定表面的全局共形参数化，并通过准保形理论保证了双射性。通过包括其他形状约束，我们的方法可以轻松扩展，以实现简单连接的开放曲面的圆盘保形参数化和类0封闭面的球形保形参数化。实验结果表明该算法的有效性。与最新的保形参数化方法相比，