Polyhedral Complex Derivation from Piecewise Trilinear Networks
Jin-Hwa Kim
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.