Recent advancements in visualizing deep neural networks provide insights into their structures and mesh extraction from Continuous Piecewise Affine (CPWA) functions. Meanwhile, developments in neural surface representation learning incorporate non-linear positional encoding, addressing issues like spectral bias; however, this poses challenges in applying mesh extraction techniques based on CPWA functions. Focusing on trilinear interpolating methods as positional encoding, we present theoretical insights and an analytical mesh extraction, showing the transformation of hypersurfaces to flat planes within the trilinear region under the eikonal constraint. Moreover, we introduce a method for approximating intersecting points among three hypersurfaces contributing to broader applications. We empirically validate correctness and parsimony through chamfer distance and efficiency, and angular distance, while examining the correlation between the eikonal loss and the planarity of the hypersurfaces.

翻译：近年来，深度神经网络可视化技术的进展为理解其结构提供了新视角，并实现了从连续分段仿射（CPWA）函数中提取网格。同时，神经表面表征学习的发展引入了非线性位置编码，以解决频谱偏差等问题；然而，这给基于CPWA函数的网格提取技术应用带来了挑战。本文以三线性插值方法作为位置编码，提出了理论见解与分析性网格提取方法，揭示了在程函约束条件下超曲面在三线性区域内向平坦平面的变换过程。此外，我们提出了一种近似三个超曲面交点的方法，以拓展其应用范围。通过倒角距离与效率、角度距离对方法的正确性与简洁性进行实证验证，同时考察了程函损失与超曲面平面性之间的关联。