Photon surfaces for robust, unbiased volumetric density estimation

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.