探索神经隐式中的微分几何
Exploring Differential Geometry in Neural Implicits
AI总结：
该研究提出一种结合微分几何的神经隐式框架，通过含高阶导数项的损失泛函与基于曲率的非均匀采样，在保证几何精度的同时提升训练效率，并可估计点采样曲面的微分测度。
摘要：
我们提出一种神经隐式框架，利用神经网络的可微特性与点采样曲面的离散几何，将其近似为神经隐式函数的水平集。\n为训练神经隐式函数，我们提出一种损失泛函，该泛函近似有符号距离函数，且支持包含高阶导数的项（如曲率主方向之间的对齐），以学习更多几何细节。训练过程中，我们采用基于点采样曲面曲率的非均匀采样策略，优先采样几何细节更丰富的点。与现有方法相比，这种采样方式在保证几何精度的同时实现了更快的学习速度。\n我们还利用神经隐式函数的解析导数，估计底层点采样曲面的微分测度。
英文摘要：
We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches. We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.
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