Tizian Zeltner

Slope-Space Integrals for Specular Next Event Estimation

In Transactions on Graphics (Proceedings of SIGGRAPH Asia 2020)

We address the problem of importance sampling light paths involving specular or near-specular reflection (a) or refraction (b). Our approach finds triangles that reflect or refract light from a point on a light source towards a particular shading point in the scene. Our technique can be used in addition to standard next event estimation and handles specular-diffuse-specular (“SDS”) sub-paths that are a well-known failure case of standard uni- and bidirectional path tracers. The images were rendered in 30m (a) and 5m (b) using an unidirectional path tracer together with the proposed sampling technique.


Monte Carlo light transport simulations often lack robustness in scenes containing specular or near-specular materials. Widely used uni- and bidirectional sampling strategies tend to find light paths involving such materials with insufficient probability, producing unusable images that are contaminated by significant variance.

This article addresses the problem of sampling a light path connecting two given scene points via a single specular reflection or refraction, extending the range of scenes that can be robustly handled by unbiased path sampling techniques. Our technique enables efficient rendering of challenging transport phenomena caused by such paths, such as underwater caustics or caustics involving glossy metallic objects.

We derive analytic expressions that predict the total radiance due to a single reflective or refractive triangle with a microfacet BSDF and we show that this reduces to the well known Lambert boundary integral for irradiance. We subsequently show how this can be leveraged to efficiently sample connections on meshes comprised of vast numbers of triangles. Our derivation builds on the theory of off-center microfacets and involves integrals in the space of surface slopes.

Our approach straightforwardly applies to the related problem of rendering glints with high-resolution normal maps describing specular microstructure. Our formulation alleviates problems raised by singularities in filtering integrals and enables a generalization of previous work to perfectly specular materials. We also extend previous work to the case of GGX distributions and introduce new techniques to improve accuracy and performance.