  Grid-based structures are commonly used to encode explicit features for
graphics primitives such as images, signed distance functions (SDF), and neural
radiance fields (NeRF) due to their simple implementation. However, in
$n$-dimensional space, calculating the value of a sampled point requires
interpolating the values of its $2^n$ neighboring vertices. The exponential
scaling with dimension leads to significant computational overheads. To address
this issue, we propose a simplex-based approach for encoding graphics
primitives. The number of vertices in a simplex-based structure increases
linearly with dimension, making it a more efficient and generalizable
alternative to grid-based representations. Using the non-axis-aligned
simplicial structure property, we derive and prove a coordinate transformation,
simplicial subdivision, and barycentric interpolation scheme for efficient
sampling, which resembles transformation procedures in the simplex noise
algorithm. Finally, we use hash tables to store multiresolution features of all
interest points in the simplicial grid, which are passed into a tiny fully
connected neural network to parameterize graphics primitives. We implemented a
detailed simplex-based structure encoding algorithm in C++ and CUDA using the
methods outlined in our approach. In the 2D image fitting task, the proposed
method is capable of fitting a giga-pixel image with 9.4% less time compared to
the baseline method proposed by instant-ngp, while maintaining the same quality
and compression rate. In the volumetric rendering setup, we observe a maximum
41.2% speedup when the samples are dense enough.
