Implementation Note 2: Approximating a Smooth Three-Dimensional Function on a Coarse Lattice

arXiv cs.GR details a compact method for approximating smooth 3D functions on a coarse lattice, using a precomputed lookup table and tetrahedral interpolation. The practical pitch is simple: sample once offline, then answer queries with a small table and a weighted sum over four cube corners.

For a common color-conversion case, the paper shows that a 16-cells-per-axis table (17^3 nodes, 16-bit values) fits in 38 KiB, while a full 8-bit table would take 64 MiB. The experimental results report a mean error of 0.08% ink per channel for the smaller table, which is a strong tradeoff for memory-constrained pipelines.

The implementation is also designed for integer arithmetic at runtime. For 8-bit input, it avoids floating point and division entirely, and all intermediate values stay within 32 bits; floating point is only needed once when building the table offline. That makes the approach attractive for embedded hardware, mobile devices, and any real-time system where cache pressure and ALU cost matter.

The broader takeaway for game developers is that this is another reminder that many “expensive” transforms can be pushed into preprocessing and represented with surprisingly small data. Even if the paper’s example is color conversion, the same pattern is relevant anywhere you need fast evaluation of smooth 3D mappings without paying for a dense volume or heavy math at runtime.