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.

基于深度神经网络的三线性插值方法，我们提出了理论洞见和分析性网格提取方法，展示了在黎曼约束下超曲面向平面的转化，同时介绍了一种逼近三个超曲面相交点的方法，从而在正确性、简洁性和效率等方面进行了经验验证，同时研究了黎曼损失与超曲面平面化之间的相关性。