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.

Despite recent advances in reconstructing an organic model with the neural signed distance function (SDF), the high-fidelity reconstruction of a CAD model directly from low-quality unoriented point clouds remains a significant challenge. In this paper, we address this challenge based on the prior observation that the surface of a CAD model is generally composed of piecewise surface patches, each approximately developable even around the feature line. Our approach, named NeurCADRecon , is self-supervised, and its loss includes a developability term to encourage the Gaussian curvature toward 0 while ensuring fidelity to the input points (see the teaser figure). Noticing that the Gaussian curvature is non-zero at tip points, we introduce a double-trough curve to tolerate the existence of these tip points. Furthermore, we develop a dynamic sampling strategy to deal with situations where the given points are incomplete or too sparse. Since our resulting neural SDFs can clearly manifest sharp feature points/lines, one can easily extract the feature-aligned triangle mesh from the SDF and then decompose it into smooth surface patches, greatly reducing the difficulty of recovering the parametric CAD design. A comprehensive comparison with existing state-of-the-art methods shows the significant advantage of our approach in reconstructing faithful CAD shapes.

Surface reconstruction is a challenging task when input point clouds, especially real scans, are noisy and lack normals. Observing that the Multilayer Perceptron (MLP) and the implicit moving least-square function (IMLS) provide a dual representation of the underlying surface, we introduce Neural-IMLS , a novel approach that directly learns a noise-resistant signed distance function (SDF) from unoriented raw point clouds in a self-supervised manner. In particular, IMLS regularizes MLP by providing estimated SDFs near the surface and helps enhance its ability to represent geometric details and sharp features, while MLP regularizes IMLS by providing estimated normals. We prove that at convergence, our neural network produces a faithful SDF whose zero-level set approximates the underlying surface due to the mutual learning mechanism between the MLP and the IMLS. Extensive experiments on various benchmarks, including synthetic and real scans, show that Neural-IMLS can reconstruct faithful shapes even with noise and missing parts. The source code can be found at https://github.com/bearprin/Neural-IMLS .

In mesh simplification, common requirements like accuracy, triangle quality, and feature alignment are often considered as a trade-off. Existing algorithms concentrate on just one or a few specific aspects of these requirements. For example, the well-known Quadric Error Metrics (QEM) approach [Garland and Heckbert 1997] prioritizes accuracy and can preserve strong feature lines/points as well, but falls short in ensuring high triangle quality and may degrade weak features that are not as distinctive as strong ones. In this paper, we propose a smooth functional that simultaneously considers all of these requirements. The functional comprises a normal anisotropy term and a Centroidal Voronoi Tessellation (CVT) [Du et al. 1999] energy term, with the variables being a set of movable points lying on the surface. The former inherits the spirit of QEM but operates in a continuous setting, while the latter encourages even point distribution, allowing various surface metrics. We further introduce a decaying weight to automatically balance the two terms. We selected 100 CAD models from the ABC dataset [Koch et al. 2019], along with 21 organic models, to compare the existing mesh simplification algorithms with ours. Experimental results reveal an important observation: the introduction of a decaying weight effectively reduces the conflict between the two terms and enables the alignment of weak features. This distinctive feature sets our approach apart from most existing mesh simplification methods and demonstrates significant potential in shape understanding. Please refer to the teaser figure for illustration.

Motivated by the fact that the medial axis transform is able to encode the shape completely, we propose to use as few medial balls as possible to approximate the original enclosed volume by the boundary surface. We progressively select new medial balls, in a top-down style, to enlarge the region spanned by the existing medial balls. The key spirit of the selection strategy is to encourage large medial balls while imposing given geometric constraints. We further propose a speedup technique based on a provable observation that the intersection of medial balls implies the adjacency of power cells (in the sense of the power crust). We further elaborate the selection rules in combination with two closely related applications. One application is to develop an easy-to use ball-stick modeling system that helps non-professional users to quickly build a shape with only balls and wires, but any penetration between two medial balls must be suppressed. The other application is to generate porous structures with convex, compact (with a high isoperimetric quotient) and shape-aware pores where two adjacent spherical pores may have penetration as long as the mechanical rigidity can be well preserved.

Quadrilateral mesh generation plays a crucial role in numerical simulations within Computer-Aided Design and Engineering (CAD/E). Producing high-quality quadrangulation typically requires satisfying four key criteria. First, the quadrilateral mesh should closely align with principal curvature directions. Second, singular points should be strategically placed and effectively minimized. Third, the mesh should accurately conform to sharp feature edges. Lastly, quadrangulation results should exhibit robustness against noise and minor geometric variations. Existing methods generally involve first computing a regular cross field to represent quad element orientations across the surface, followed by extracting a quadrilateral mesh aligned closely with this cross field. A primary challenge with this approach is balancing the smoothness of the cross field with its alignment to pre-computed principal curvature directions, which are sensitive to small surface perturbations and often ill-defined in spherical or planar regions. To tackle this challenge, we propose NeurCross , a novel framework that simultaneously optimizes a cross field and a neural signed distance function (SDF), whose zero-level set serves as a proxy of the input shape. Our joint optimization is guided by three factors: faithful approximation of the optimized SDF surface to the input surface, alignment between the cross field and the principal curvature field derived from the SDF surface, and smoothness of the cross field. Acting as an intermediary, the neural SDF contributes in two essential ways. First, it provides an alternative, optimizable base surface exhibiting more regular principal curvature directions for guiding the cross field. Second, we leverage the Hessian matrix of the neural SDF to implicitly enforce cross field alignment with principal curvature directions, thus eliminating the need for explicit curvature extraction. Extensive experiments demonstrate that NeurCross outperforms the state-of-the-art methods in terms of singular point placement, robustness against surface noise and surface undulations, and alignment with principal curvature directions and sharp feature curves.