CIRCNET: MESHING 3D POINT CLOUDS WITH CIRCUMCENTER DETECTION

Abstract

Reconstructing 3D point clouds into triangle meshes is a key problem in computational geometry and surface reconstruction. Point cloud triangulation solves this problem by providing edge information to the input points. Since no vertex interpolation is involved, it is beneficial to preserve sharp details on the surface. Taking advantage of learning-based techniques in triangulation, existing methods enumerate the complete combinations of candidate triangles, which is both complex and inefficient. In this paper, we leverage the duality between a triangle and its circumcenter, and introduce a deep neural network that detects the circumcenters to achieve point cloud triangulation. Specifically, we introduce multiple anchor priors to divide the neighborhood space of each point. The neural network then learns to predict the presences and locations of circumcenters under the guidance of those anchors. We extract the triangles dual to the detected circumcenters to form a primitive mesh, from which an edge-manifold mesh is produced via simple post-processing. Unlike existing learning-based triangulation methods, the proposed method bypasses an exhaustive enumeration of triangle combinations and local surface parameterization. We validate the efficiency, generalization, and robustness of our method on prominent datasets of both watertight and open surfaces. The code and trained models are provided at https://github.com/Ruitao-L/CircNet.

1. INTRODUCTION

Point cloud triangulation (Cazals & Giesen 2004) aims at reconstructing triangle meshes of object surfaces by adding edge information to their point cloud representations. The input point clouds are usually produced by either scanning sensors (e.g., LiDAR) or surface sampling methods. Compared to implicit surface reconstruction (e.g., Kazhdan et al. 2006) , explicit triangulation has the advantages of preserving the original input points and fine-grained details of the surface. Moreover, it does not require oriented normals which are difficult to obtain in practice. Recent advances in geometric deep learning have seen widespread applications of neural functions for surface representations (e.g., Park et al. 2019; Sitzmann et al. 2020b; a; Erler et al. 2020; Gropp et al. 2020; Atzmon & Lipman 2020a; b; Ben-Shabat et al. 2022; Ma et al. 2021; 2022) . In comparison, only a few methods have been proposed to directly learn triangulation of point clouds by using neural networks. This is probably attributed to the combinatorial nature of the triangulation task, hindering the uptake of learning-based methods. The existing works have to enumerate combinations of candidate triangles around each input point, and use neural networks to predict their existence in the triangle mesh (Sharp & Ovsjanikov 2020; Liu et al. 2020) . Figure 1 (a) illustrates the local complexity of those combinatorial methods using a point with four neighboring points. Typically, for a point with K neighbors, the combinatorial methods propose K 2 or O(K 2 ) candidate triangles. Different from these methods, we propose to exploit the duality relationship between a triangle and its circumcenter to implement point cloud triangulation. That is, each vertex of a triangle is equally distant to its circumcenter. We use this characteristic to find triangle triplets from their circumcenters. Figure. 1(b) shows the duality based on the same example of Fig. 1(a) . Our method recovers the vertex triplets of triangle (p, q 1 , q 3 ) based on its circumcenter c and the equidistant characteristic, i.e. ∥p -c∥ = ∥q 1 -c∥ = ∥q 3 -c∥. To obtain circumcenters for point cloud triangulation, we introduce a neural network that is able to detect the circumcenters of all triangles in a mesh. To the best of our knowledge, this is the first single-shot detection architecture for point cloud triangulation. We are inspired by the one-stage methods in object detection (e.g., Liu et al. 2016) . Figure 1 : An example of a point p with four neighboring points q 1 , q 2 , q 3 , q 4 . (a) The combinatorial methods propose all of the six triangles incident to p as candidate triangles. They are (p, q 1 , q 2 ), (p, q 1 , q 3 ), (p, q 1 , q 4 ), (p, q , q 3 ), (p, q 2 , q 4 ), (p, q 3 , q 4 ). The neural network has to classify the targeted triangle (p, q 1 , q 3 ) out of the six candidates. (b) The proposed method eliminates the candidate proposals by detecting a circumcenter c and exploiting its duality with the triangle (p, q 1 , q 3 ) to identify the targeted triangle. Unlike previous combinatorial methods, the proposed method removes the requirement for candidate triangles. Specifically, we detect circumcenters in the neighborhood space of each point, under the guidance of a group of anchor priors. The neural network predicts whether a circumcenter exists in the reconstructed mesh, and where it is. We extract triangles induced by the detected circumcenters to obtain a primitive mesh. The final surface mesh is produced by enforcing edgemanifoldness and filling small holes on the primitive mesh. • We introduce the first neural architecture that triangulates point clouds by detecting circumcenters. The duality between a triangle and its circumcenter is exploited afterwards to extract the vertex triplets of each triangle in the mesh. • The proposed neural network is able to reconstruct primitive meshes in milliseconds, due to its single-shot detection pipeline and its removal of candidate proposals. Normals are not required. • The proposed method casts no restriction on the surface topology, e.g., being watertight. Instead, it allows the surfaces to be open and have genus ('hole') larger than one. • The method generalizes well to unseen point clouds including those of large-scale scenes. It is robust to non-uniform and noisy data. These indicate its promise for real-world applications. 



To validate the proposed method, we train the detection neural network on the ABC dataset (Koch et al. 2019). The trained model is evaluated on the ABC and other datasets, including FAUST (Bogo et al. 2014), MGN (Bhatnagar et al. 2019), and Matterport3D (Chang et al. 2017). The method not only reconstructs meshes in high quality, but also outperforms the previous learning-based approaches largely in efficiency. It generalizes well to unseen, noisy and non-uniform point clouds. Our main contributions are summarized below:

Point cloud triangulation and implicit surface functions are two important research directions surface reconstruction. Their major difference is that the former preserves input points, while the latter does not. Alpha shapes (Edelsbrunner & Mücke 1994) and the ball pivoting algorithm(Bernardini et al.  1999)  are representatives among the traditional methods in point cloud triangulation. The Poisson surface reconstruction (Kazhdan et al. 2006; Kazhdan & Hoppe 2013) is a classical approach in implicit surface functions, but it depends on oriented normals of the input points for good performance. Marching Cubes (Lorensen & Cline 1987) and Dual Contouring (Ju et al. 2002) are restricted to extract the triangle meshes of isosurfaces from their signed distance fields. We refer interested readers to surveys Berger et al. 2017; Cheng et al. 2013; Newman & Yi 2006 for more in depth discussions of the traditional surface reconstruction methods. 2.1 IMPLICIT NEURAL FUNCTIONS Implicit neural functions are techniques in geometric deep learning. They exploit the approximation power of neural networks to represent shapes and scenes as level-sets of continuous functions (Atzmon et al. 2019). Existing literature trains networks to learn either occupancy functions (Mescheder et al. 2019; Peng et al. 2020) or distance functions (Park et al. 2019; Sitzmann et al. 2020b; Atzmon & Lipman 2020a). Since the introduction of DeepSDF (Park et al. 2019), major advances in this research field include simplifying requirements on the ground-truth (e.g., signs of distances) (Atzmon & Lipman 2020a;b), exploring the high-frequency features (Sitzmann et al. 2020b; Tancik et al. 2020), and improving loss functions for better surface representation (Gropp et al. 2020; Ben-Shabat

