Oriented normals are common pre-requisites for many geometric algorithms based on point clouds, such as Poisson surface reconstruction. However, it is not trivial to obtain a consistent orientation. In this work, we bridge orientation and reconstruction in the implicit space and propose a novel approach to orient point cloud normals by incorporating isovalue constraints to the Poisson equation. In implicit surface reconstruction, the reconstructed shape is represented as an isosurface of an implicit function defined in the ambient space. Therefore, when such a surface is reconstructed from a set of sample points, the implicit function values at the points should be close to the isovalue corresponding to the surface. Based on this observation and the Poisson equation, we propose an optimization formulation that combines isovalue constraints with local consistency requirements for normals. We optimize normals and implicit functions simultaneously and solve for a globally consistent orientation. Thanks to the sparsity of the linear system, our method can work on an average laptop with reasonable computational time. Experiments show that our method can achieve high performance in non-uniform and noisy data and manage varying sampling densities, artifacts, multiple connected components, and nested surfaces. The source code is available at \url{https://github.com/Submanifold/IsoConstraints}.

翻译：定向法向是基于点云的几何算法（如泊松表面重建）的常见前提条件。然而，获得一致的法向定向并非易事。本文在隐式空间中构建了法向定向与表面重建之间的桥梁，并提出了一种新颖的方法，通过将等值约束引入泊松方程来对点云法向进行定向。在隐式表面重建中，重建形状被表示为定义在空间中的隐函数等值面。因此，当从一组采样点重建此类表面时，这些点处的隐函数值应接近表面对应的等值值。基于这一观察和泊松方程，我们提出了一种将等值约束与法向局部一致性要求相结合的优化模型。我们同时优化法向和隐函数，求解全局一致的法向定向。得益于线性系统的稀疏性，我们的方法可在普通笔记本电脑上以合理的计算时间运行。实验表明，我们的方法在非均匀和含噪数据中能实现高性能，并可处理不同采样密度、伪影、多连通分量及嵌套表面。源代码发布于\url{https://github.com/Submanifold/IsoConstraints}。