A Theoretical Analysis of Compactness of the Light Transport Operator

Cyril Soler, Ronak Molazem, Kartic Subr

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract / Description of output

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.
Original languageEnglish
Title of host publicationACM SIGGRAPH 2022 Conference Proceedings
EditorsMunkhtsetseg Nandigjav, Niloy J. Mitra, Aaron Hertzmann
Number of pages9
ISBN (Electronic)9781450393379
Publication statusPublished - 5 Aug 2022
EventSIGGRAPH 2022 - Vancouver, Canada
Duration: 8 Aug 202211 Aug 2022


ConferenceSIGGRAPH 2022
Abbreviated titleSIGGRAPH 2022
Internet address

Keywords / Materials (for Non-textual outputs)

  • Light Transport Operator
  • Compactness
  • Fredholm Equations


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