Globally Consistent Normal Orientation for Point Clouds by Regularizing the Winding-Number Field (2304.11605v1)

Abstract: Estimating normals with globally consistent orientations for a raw point cloud has many downstream geometry processing applications. Despite tremendous efforts in the past decades, it remains challenging to deal with an unoriented point cloud with various imperfections, particularly in the presence of data sparsity coupled with nearby gaps or thin-walled structures. In this paper, we propose a smooth objective function to characterize the requirements of an acceptable winding-number field, which allows one to find the globally consistent normal orientations starting from a set of completely random normals. By taking the vertices of the Voronoi diagram of the point cloud as examination points, we consider the following three requirements: (1) the winding number is either 0 or 1, (2) the occurrences of 1 and the occurrences of 0 are balanced around the point cloud, and (3) the normals align with the outside Voronoi poles as much as possible. Extensive experimental results show that our method outperforms the existing approaches, especially in handling sparse and noisy point clouds, as well as shapes with complex geometry/topology.

• The paper introduces a novel strategy for achieving globally consistent normal orientation in point clouds by regularizing the winding-number field.
• The method uses an optimization procedure based on the Voronoi diagram, leveraging components like binary winding numbers and normal alignment for robustness.
• Experiments show superior normal orientation accuracy and robustness for practical data with noise, gaps, and irregular density.

Overview of "Globally Consistent Normal Orientation for Point Clouds by Regularizing the Winding-Number Field"

The paper investigates the problem of estimating normals with globally consistent orientations for raw point clouds, a foundational step for various geometry processing applications such as surface reconstruction, shape registration, and shape analysis. Despite extensive research efforts, existing methods struggle with unoriented point clouds containing data sparsity, thin-walled structures, noise, and nearby gaps. This work introduces a novel strategy using the winding-number field, leveraging a smooth objective function to ensure that normal orientations are consistent globally.

Key Components

The proposed method hinges on an intuitive observation of the winding number in geometry processing. Traditionally, the winding number is utilized for inside-outside tests in 3D space, valued as either 0 outside or 1 inside a closed, orientable surface. The authors formalize an optimization procedure that starts from random normals and adjusts them to ensure the winding-number field is binary, maintaining global consistency across the point cloud.

The functional components of the proposed optimization include:

1. Binary Requirement of Winding Number ($f_{01}$): Using a double well potential with additional modifications encourages the winding numbers to stabilize at 0 or 1, representing exterior and interior locations, respectively.
2. Balance in Winding Number Distribution ($f_B$): Enforces a balance of winding number occurrences (0s and 1s) around each point’s Voronoi cell, ensuring spatial coherence and robust handling against spatial inconsistencies.
3. Normal Alignment with Voronoi Poles ($f_A$): Aligns normals with predictable directions from Voronoi poles, enhancing the regularity and alignment accuracy of the computed normals.

Methodology

The paper's method involves constructing the Voronoi diagram of the input point cloud and utilizing its vertices for winding number evaluation. The approach considers these vertices' spatial distribution to minimize the functional using an unconstrained optimization technique. L-BFGS is used for optimization, requiring about 30-50 iterations to converge, ensuring scalability for large datasets.

Results and Comparisons

Extensive experimental results showcase the method's robustness and efficacy. The authors compare their approach against various state-of-the-art methods under different point clouds exhibiting challenges such as noise, gaps, irregular density, and complexity. Significantly, their method achieved superior normal orientation accuracy and enhanced surface reconstruction fidelity even with sparse and noisy data.

Implications and Future Prospects

The authors argue their method’s utility in practical scenarios such as real-world scanned data, asserting its resilience in generating reliable normal estimations even under adverse conditions. The introduction of winding-number field regularization presents a paradigm shift in normal estimation, potentially opening pathways for advances in geometric modeling and shape analysis.

Future development might focus on computational efficiency, particularly in high-resolution contexts, and address cases with significant internal noise or overlapping data, where the winding-number approach presents inherent challenges.

In summary, this work promises considerable improvements in point cloud processing reliability and accuracy, marking an important contribution by integrating geometric insights with computational procedures to handle complex 3D data seamlessly.