the split sum approximation

The Split Sum Approximation is a technique used in real-time PBR rendering to render IBL. It use pre-filtered importance sampling to avoids expensive Monte Carlo integration process.

The split sum approximation isolates approximate integral of product as product of integral. Specifically, it separates the material and lighting components from the rendering equation:

Environment Mapping

For the environment mapping, we can pre-generating a set of differently filtered environment lighting (with mipmapping and trilinear interpolation): At runtime we query the pre-filtered environment lighting at the r (mirror reflected) direction.

BRDF

A typical microfacet theory still have too many parameters to be used in a look-up table.

However, we can use the Schlick’s approximation for the Fresnel term which only depends on “base color” and the half angle , and when writing out explicitly we can even extract the “base color” out of the integral as a constant!

Now our integrals only have two variables (roughness and angle ), so we can precompute a 2D lookup texture:

References

• Real Shading in Unreal Engine 4
• [Alternative Take on the Split Sum Approximation for Cubemap Pre-filtering // Zero Radiance](https://zero-radiance.github.io/post/split-sum/
• Games 202 > Lecture 5 Real-time Environment Mapping