Photon surfaces for robust, unbiased volumetric density estimation

Abstract

We generalize photon planes to photon surfaces : a new family of unbiased volumetric density estimators which we combine using multiple importance sampling. To derive our new estimators, we start with the extended path integral which duplicates the vertex at the end of the camera and photon subpaths and couples them using a blurring kernel. To make our formulation unbiased, however, we use a delta kernel to couple these two end points. Unfortunately, sampling the resulting singular integral using Monte Carlo is impossible since the probability of generating a contributing light path by independently sampling the two subpaths is zero. Our key insight is that we can eliminate the delta kernel and make Monte Carlo estimation practical by integrating any three dimensions analytically, and integrating only the remaining dimensions using Monte Carlo. We demonstrate the practicality of this approach by instantiating a collection of estimators which analytically integrate the distance along the camera ray and two arbitrary sampling dimensions along the photon subpath (e.g., distance, direction, surface area). This generalizes photon planes to curved "photon surfaces", including new "photon cone", "photon cylinder", "photon sphere", and multiple new "photon plane" estimators. These estimators allow us to handle light paths not supported by photon planes, including single scattering, and surface-to-media transport. More importantly, since our estimators have complementary strengths due to analytically integrating different dimensions of the path integral, we can combine them using multiple importance sampling. This combination mitigates singularities present in individual estimators, substantially reducing variance while remaining fully unbiased. We demonstrate our improved estimators on a number of scenes containing homogeneous media with highly anisotropic phase functions, accelerating both multiple scattering and single scattering compared to prior techniques.

A survey on rendering homogeneous participating media

“…• Bias. Monte Carlo based methods produce unbiased results, while the others are mostly biased, except for the method of Bitterli and Jarosz [5], which is unbiased for multiple scattering when using a zero-order estimator, and Deng et al [12] which is also unbiased when using a delta kernel.…”

“…• Storage. Density based approaches and point based approaches use more memory than others, where Bitterli and Jarosz [5] and Deng et al [12] introduced higher-dimensional elements, e.g., photon planes and volumes, reducing memory cost compared to UPBP [1]. Many lights based methods require less memory than density based methods, since they are based on gathering rather than density estimation.…”

“…However, their rendering results have the lowest quality of all these methods. Monte Carlo based methods are the [5] and Deng et al [12] converge faster than UPBP, thanks to their higher-dimensional elements.…”

“…However, their estimators suffer from singularities, and require at least two bounces in the medium past any surface, leaving the remaining transport to other techniques. Deng et al [12] solved these issues by generalizing photon planes to photon surfaces, resulting in different types of estimators, including the new "photon cone", "photon cylinder", "photon sphere", and multiple new "photon plane" estimators. They are combined with multiple importance sampling to increase robustness for arbitrary types of participating media.…”

“…However, there are several limitations in this group of methods. First, most methods in this group are biased, except the zero-order kernel in Bitterli and Jarosz [5] and Deng et al [12]. Second, this group of methods require an extra light pass and extra storage for the light distribution.…”