Path Space Partitioning and Guided Image Sampling for MCMC

TL;DR

The paper tackles inefficiencies in rendering via a single path-space integral by partitioning the path space into $K$ discrete subspaces, each with its own estimator, and guiding perturbations using image-space information. A Monte Carlo pre-pass identifies partitions and denoises their contributions to build a guidance distribution, while an image-plane MCMC proposal explores these partitions through candidate prefixes $Y'$ evaluated by $S^{*}$. The authors demonstrate improved image quality and variance reduction across multiple scenes, with modest memory overhead and a controllable number of partitions and proposal samples. This approach provides a practical route to accelerate MCMC-based rendering by exploiting partition-specific structure and local image-space guidance.

Abstract

Rendering algorithms typically integrate light paths over path space. However, integrating over this one unified space is not necessarily the most efficient approach, and we show that partitioning path space and integrating each of these partitioned spaces with a separate estimator can have advantages. We propose an approach for partitioning path space based on analyzing paths from a standard Monte Carlo estimator and integrating these partitioned path spaces using a Markov Chain Monte Carlo (MCMC) estimator. This also means that integration happens within a sparser subset of path space, so we propose the use of guided proposal distributions in image space to improve efficiency. We show that our method improves image quality over other MCMC integration approaches at the same number of samples.

Figures (8)

• Figure 1: Our approach splits path space into a discrete set of partitions, each of which can be integrated by a separate estimator. As these are now integrating over sparser spaces, we propose a guided image plane sampling approach based on an analysis of the acceptance probability for image plane perturbations and accelerated by denoising the information used to create the partitions. This image shows the bathroom scene showing variance is reduced using our approach (on the right) and Metropolis Light Transport (in the middle) computed at the same number of samples.
• Figure 2: Our approach partitions the whole of path space (the left image) into a series of partitions, discussed in Section \ref{['sec:partitioning']}, each corresponding to a different subset of path space. We propose splitting on the interaction types, as illustrated in the boxes on the right, with the rightmost box illustrating the contribution from the complementary partition (Equation \ref{['eq:fullpartitions']}). Each of these partitions are constructed by performing a Monte Carlo sampling pre-pass to find the contribution of each found interaction type (top images, and discussed in Section \ref{['sec:PartitioningPractical']}). These contributions are denoised and are used to guide MCMC sampling (proposals illustrated in the denoised bottom images), see Section \ref{['sec:GIS']}.
• Figure 3: Example in 1D of using partitions. a) and b) show MCMC integration with and without partitioning (the partition is shown by the vertical line in b)). Using partitioning decreases variance significantly by allowing two chains to explore the lower and higher contributions separately, each of which uses a different normalizing constant (the horizontal line).
• Figure 4: Illustration of different image space proposal distributions for a region in a partition from Figure \ref{['fig:partitions']}, red regions indicate higher and blue lower values of the proposal distribution. Isotropic approaches (\ref{['fig:kiso']}) do not use any information and can propose paths that are likely to be rejected. Anisotropic proposals (\ref{['fig:kaiso']}) adapt better to the distribution of radiance but are still limited to a parametric distribution. Figure \ref{['fig:kexact']} shows our approach which adapts per-pixel to the estimated lighting based on the denoised estimate of lighting per partition $D_{i}$, and Figure \ref{['fig:ksparse']} shows our sparse approximation.
• Figure 5: Sparse offsets used to compute $Y'$. Green points are the arbitrarily chosen initial points and red are the inverse to guarantee reversibility.