Neural implicit representation is a promising approach for reconstructing surfaces from point clouds. Existing methods combine various regularization terms, such as the Eikonal and Laplacian energy terms, to enforce the learned neural function to possess the properties of a Signed Distance Function (SDF). However, inferring the actual topology and geometry of the underlying surface from poor-quality unoriented point clouds remains challenging. In accordance with Differential Geometry, the Hessian of the SDF is singular for points within the differential thin-shell space surrounding the surface. Our approach enforces the Hessian of the neural implicit function to have a zero determinant for points near the surface. This technique aligns the gradients for a near-surface point and its on-surface projection point, producing a rough but faithful shape within just a few iterations. By annealing the weight of the singular-Hessian term, our approach ultimately produces a high-fidelity reconstruction result. Extensive experimental results demonstrate that our approach effectively suppresses ghost geometry and recovers details from unoriented point clouds with better expressiveness than existing fitting-based methods.

翻译：神经隐式表示是从点云重建表面的一种有前途的方法。现有方法结合多种正则化项（如Eikonal和Laplacian能量项），迫使学习的神经函数具备有符号距离函数（SDF）的特性。然而，从质量低劣的无向点云中推断底层表面的真实拓扑和几何结构仍然具有挑战性。根据微分几何，SDF的黑塞矩阵在表面周围的微分薄壳空间内的点是奇异的。我们的方法强制神经隐式函数在表面附近点的黑塞矩阵行列式为零。该技术使近表面点及其表面投影点的梯度对齐，在仅数轮迭代内即可生成粗略但保真的形状。通过退火奇异黑塞矩阵项的权重，我们的方法最终产生高保真度的重建结果。大量实验结果表明，与现有基于拟合的方法相比，我们的方法有效抑制了幽灵几何，并从无向点云中恢复了细节，具有更强的表现力。