# Free-boundary conformal parameterization of point clouds

arXiv: https://arxiv.org/abs/2010.15399 · HTML: https://mlsanity.com/paper/2010.15399/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2020.

## Built on by (most cited first, 8 of 8)

- [Flatten Anything: Unsupervised Neural Surface Parameterization](https://mlsanity.com/paper/2405.14633/) (2024)
- [Deformation-Invariant Neural Network and Its Applications in Distorted Image Restoration and Analysis](https://mlsanity.com/paper/2310.02641/) (2023)
- [QIS : Interactive Segmentation via Quasi-Conformal Mappings](https://mlsanity.com/paper/2402.14695/) (2024)
- [FlexPara: Flexible Neural Surface Parameterization](https://mlsanity.com/paper/2504.19210/) (2025)
- [A shallow physics-informed neural network for solving partial differential equations on surfaces](https://mlsanity.com/paper/2203.01581/) (2022)
- [From a few Accurate 2D Correspondences to 3D Point Clouds](https://mlsanity.com/paper/2206.08749/) (2022)
- [Circular Quasiconformal Deturbulence: Geometry-Based Restoration from Multiple Turbulent Frames](https://mlsanity.com/paper/2504.13432/) (2025)
- [SeamFlow: Structure-Aware Flow Matching on Edge Probabilities for Artist-Like UV Unwrapping](https://mlsanity.com/paper/2609.04751/) (2026)

## Abstract (arXiv)

With the advancement in 3D scanning technology, there has been a surge of interest in the use of point clouds in science and engineering. To facilitate the computations and analyses of point clouds, prior works have considered parameterizing them onto some simple planar domains with a fixed boundary shape such as a unit circle or a rectangle. However, the geometry of the fixed shape may lead to some undesirable distortion in the parameterization. It is therefore more natural to consider free-boundary conformal parameterizations of point clouds, which minimize the local geometric distortion of the mapping without constraining the overall shape. In this work, we develop a free-boundary conformal parameterization method for disk-type point clouds, which involves a novel approximation scheme of the point cloud Laplacian with accumulated cotangent weights together with a special treatment at the boundary points. With the aid of the free-boundary conformal parameterization, high-quality point cloud meshing can be easily achieved. Furthermore, we show that using the idea of conformal welding in complex analysis, the point cloud conformal parameterization can be computed in a divide-and-conquer manner. Experimental results are presented to demonstrate the effectiveness of the proposed method.