A Theoretical Analysis of Compactness of the Light Transport Operator

Abstract

Rendering photorealistic visuals of virtual scenes requires tractable models for the simulation of light. The rendering equation describes one such model using an integral equation, the crux of which is a continuous integral operator. A majority of rendering algorithms aim to approximate the effect of this light transport operator via discretization (using rays, particles, patches, etc.). Research spanning four decades has uncovered interesting properties and intuition surrounding this operator. In this paper we analyze compactness, a key property that is independent of its discretization and which characterizes the ability to approximate the operator uniformly by a sequence of finite rank operators. We conclusively prove lingering suspicions that this operator is not compact and therefore that any discretization that relies on a finite-rank or nonadaptive finite-bases is susceptible to unbounded error over arbitrary light distributions. Our result justifies the expectation for rendering algorithms to be evaluated using a variety of scenes and illumination conditions. We also discover that its lower dimensional counterpart (over purely diffuse scenes) is not compact except in special cases, and uncover connections with it being noninvertible and acting as a low-pass filter. We explain the relevance of our results in the context of previous work. We believe that our theoretical results will inform future rendering algorithms regarding practical choices.CCS Concepts: • Computing methodologies → Rendering; • Mathematics of computing → Functional analysis; Integral equations.

TED-Face: Texture-Enhanced Deep Face Reconstruction in the Wild

“…Although the reconstruction fidelity is amazing, NeRF relies on voxel representation, which may not generalize well for face reconstruction. Computer rendering with a simplified shading model can limit the visual perception of estimated faces also through implicit low-pass filtering [ 12 ]. Inspired by the work of [ 19 ], we propose to directly regress radiance residuals from input image to be added on top of computer-generated face rendering.…”

“…Those residuals, as shown in Section 3.4 , are visually sparse and localized, which in theory cannot be captured using low-frequency basis functions of limited bandwidth (the uncertainty principle in Fourier analysis) [ 11 ]. Second, computer rendering of reconstructed faces as an essential step in deriving the reconstruction error to be minimized can bias the estimation because it acts as low-pass filters when a less sophisticated illumination model (e.g., with the spherical harmonic representation of lighting in existing work) is used for light transport computation [ 12 ]. This can produce faces lacking fine colors, too, as shown in Figure 1 .…”