Anisotropic Gauss Reconstruction for Unoriented Point Clouds

TL;DR

This work extends the Gauss-based surface reconstruction for unoriented point clouds by introducing an anisotropic form of the fundamental solution through a convection term in the Laplace equation, enabling richer linear systems that better capture directional information. By discretizing an anisotropic Gauss formula and solving under- and over-determined systems with a memory-efficient blocking scheme, the method simultaneously obtains normals and reconstructs surfaces, with an adaptive velocity-vector strategy that improves handling of thin structures and small holes. Across diverse datasets, including noisy and real-world scans, the approach achieves state-of-the-art orientation and reconstruction performance, while reducing sensitivity to regularization and maintaining competitive efficiency. The work provides a practical, robust framework for unoriented surface reconstruction with explicit guidance on velocity selection, iso-surface extraction, and computational considerations, and it highlights avenues for further efficiency and local-geometry adaptation.

Abstract

Unoriented surface reconstructions based on the Gauss formula have attracted much attention due to their elegant mathematical formulation and excellent performance. However, the isotropic characteristics of the formulation limit their capacity to leverage the anisotropic information within the point cloud. In this work, we propose a novel anisotropic formulation by introducing a convection term in the original Laplace operator. By choosing different velocity vectors, the anisotropic feature can be exploited to construct more effective linear equations. Moreover, an adaptive selection strategy is introduced for the velocity vector to further enhance the orientation and reconstruction performance of thin structures. Extensive experiments demonstrate that our method achieves state-of-the-art performance and manages various challenging situations, especially for models with thin structures or small holes. The source code will be released on GitHub.

Figures (14)

• Figure 1: We propose a novel anisotropic formulation by introducing a convection term in the original Laplace equation. By choosing different velocity vectors, we can fully take advantage of the anisotropic feature. Our method establishes and solves equation systems by the inputting unoriented points, simultaneously accomplishing orientation and reconstruction tasks. The color of the output point cloud in the top right-hand corner of the figure represents the normal information. Extensive experiments demonstrate that our method achieves state-of-the-art performance in both orientation and surface reconstruction for unoriented point clouds.
• Figure 2: By introducing anisotropy, our method not only improves the quality of surface reconstruction but also stimulates the potential of the Gauss formula for orientation.
• Figure 3: Under the same input point cloud with 500 points and regularization value, our method reduces the singularity of matrix $B$ and increases the number of effective equations compared to PGR.
• Figure 4: The qualitative comparison of our method with PGR, iPSR, and GCNO+SPR on reconstruction. Our method can reconstruct surfaces with higher quality.
• Figure 5: The qualitative and quantitative comparison of equation quantity in our method. Choosing more velocity vectors can further slightly improve the performance.

Theorems & Definitions (2)

• Theorem 1: Anisotropic Gauss Formula
• Lemma 2