Wave Tracing: Generalizing The Path Integral To Wave Optics

Shlomi Steinberg
Matt Pharr
BibTex 
@article{steinberg_wavetracing_2026,
 	author = {Steinberg, Shlomi and Pharr, Matt},
 	title = {Wave Tracing: Generalizing The Path Integral To Wave Optics},
 	journal = {Computer Graphics Forum},
 	volume = {n/a},
 	number = {n/a},
 	pages = {e70322},
 	keywords = {CCS Concepts, • Computing methodologies → Rendering, Computer graphics, Scientific visualization, • Applied computing → Physics, wave simulations, path tracing, light transport, diffraction, PLT, rendering, coherence, cone tracing, UTD},
 	doi = {https://doi.org/10.1111/cgf.70322},
 	url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/cgf.70322},
 	eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.70322},
 }
EUROGRAPHICS 2026

Publication date: April 8th, 2026
doi: 10.1111/cgf.70322

From ray optics to wave optics. In this paper we analyze the classical path integral formulation of light transport, and rigorously study what wave-optical phenomena can be reproduced by it. We show that some effects, like dispersion and scattering by a restricted class of statistical surface models (rendered in image A), fall under its regime. We generalize the classical path integral to a formulation that is able to account for a much wider variety of wave effects, and based on that generalized path integral present a unified framework that is able to: (B) simulate long-wave radiation and its propagation and diffraction in complex environments, for example to compute its signal strength (visualized color-coded); and (C) render optical wave effects, such as diffraction by arbitrary geometry

Abstract

Modeling the wave nature of light and the propagation and diffraction of electromagnetic fields is crucial for the accurate simulation of many phenomena, yet wave simulations are significantly more computationally complex than classical ray-based models. In this work, we start by analyzing the classical path integral formulation of light transport and rigorously study which wave-optical phenomena can be reproduced by it. We then introduce a bilinear path integral generalization for wave-optical light transport that models the wave interference between paths. This formulation subsumes many existing methods that rely on shooting-bouncing rays or UTD-based diffractions, and serves to give insight into the challenges of such approaches and the difficulty of sampling good paths in a bilinear setting. With this foundation, we develop a weakly-local path integral based on region-to-region transport using elliptical cones that allows sampling individual paths that still model wave effects accurately. As with the classic path integral form of the light transport equation, our path integral makes it possible to derive a variety of practical transport algorithms. We present a complete system for wave tracing with elliptical cones, with applications in light transport for rendering and efficient simulation of long-wavelength radiation propagation and diffraction in complex environments.


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