Polyhedral Complex Derivation from Piecewise Trilinear Networks
This work addresses a domain-specific problem for researchers in neural surface representation and visualization, offering incremental improvements by adapting existing mesh extraction methods to handle non-linear encodings.
The paper tackled the challenge of applying mesh extraction techniques to neural networks with non-linear positional encoding, specifically trilinear interpolation, by providing theoretical insights and an analytical method for extracting meshes and approximating intersecting points. The results were validated empirically using metrics like chamfer distance and angular distance, showing correctness and efficiency.
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.