Anisotropic Gaussian Mutations for Metropolis Light Transport
through Hessian-Hamiltonian Dynamics

 Tzu-Mao Li   Jaakko Lehtinen   Ravi Ramamoorthi   Wenzel Jakob   Frédo Durand 
MIT CSAIL Aalto University
University of California, San Diego ETH Zürich MIT CSAIL


The simulation of light transport in the presence of multi-bounce glossy effects and motion is challenging because the integrand is high dimensional and areas of high-contribution tend to be narrow and hard to sample. We present a Markov Chain Monte Carlo (MCMC) rendering algorithm that extends Metropolis Light Transport by automatically and explicitly adapting to the local shape of the integrand, thereby increasing the acceptance rate. Our algorithm characterizes the local behavior of throughput in path space using its gradient as well as its Hessian. In particular, the Hessian is able to capture the strong anisotropy of the integrand. We obtain the derivatives using automatic differentiation, which makes our solution general and easy to extend to additional sampling dimensions such as time.

However, the resulting second order Taylor expansion is not a proper distribution and cannot be used directly for importance sampling. Instead, we use ideas from Hamiltonian Monte-Carlo and simulate the Hamiltonian dynamics in a flipped version of the Taylor expansion where gravity pulls particles towards the high-contribution region. Whereas such methods usually require numerical integration, we show that our quadratic landscape leads to a closed-form anisotropic Gaussian distribution for the final particle positions, and it results in a standard Metropolis-Hastings algorithm. Our method excels at rendering glossy-to-glossy reflections on small and highly curved surfaces. Furthermore, unlike previous work that derives sampling anisotropy with pen and paper and only considers specific effects such as specular BSDFs, we characterize the local shape of throughput through automatic differentiation. This makes our approach very general. In particular, our method is the first MCMC rendering algorithm that is able to resolve the anisotropy in the time dimension and render difficult moving caustics.


Tzu-Mao Li, Jaakko Lehtinen, Ravi Ramamoorthi, Wenzel Jakob, Frédo Durand. Anisotropic Gaussian Mutations for Metropolis Light Transport through Hessian-Hamiltonian Dynamics.
ACM Transactions on Graphics 34(6) (Proceedings of ACM SIGGRAPH Asia 2015)


See here for errata of the equations in the paper.


SIGGRAPH Asia 2015 paper
Supplemental images
SIGGRAPH Asia 2015 presentation slides (Keynote)
SIGGRAPH Asia 2015 presentation slides (Powerpoint)


Talk video


We are grateful to the anonymous reviewers for their valuable comments. This work is funded by NSF grants 1451830 and Academy of Finland grant 277833. Wenzel Jakob was supported by an ETH/Marie Curie fellowship.