A Generalized Ray Formulation For Wave-Optics Rendering

TL;DR

The generalized ray is rigorously formulated, which enables linear and weakly-local queries of arbitrary wave-optical distributions of light, and facilitates the application of modern path tracing techniques for wave-optical rendering, with light of any state of coherence and any spectral properties.

Abstract

Under ray-optical light transport, the classical ray serves as a linear and local "point query" of light's behaviour. Linearity and locality are crucial to the formulation of sophisticated path tracing and sampling techniques, that enable efficient solutions to light transport problems in complex, real-world settings and environments. However, such formulations are firmly confined to the realm of ray optics, while many applications of interest -- in computer graphics and computational optics -- demand a more precise understanding of light: as waves. We rigorously formulate the generalized ray, which enables linear and weakly-local queries of arbitrary wave-optical distributions of light. Generalized rays arise from photodetection states, and therefore allow performing backward (sensor-to-source) wave-optical light transport. Our formulations are accurate and highly general: they facilitate the application of modern path tracing techniques for wave-optical rendering, with light of any state of coherence and any spectral properties. We improve upon the state-of-the-art in terms of the generality and accuracy of the formalism, ease of application, as well as performance. As a consequence, we are able to render large, complex scenes, as in Fig. 1, and even do interactive wave-optical light transport, none of which is possible with any existing method. We numerically validate our formalism, and make connection to partially-coherent light transport.

Figures (11)

• Figure 1: The sampling problem. Existing light transport formalisms, like partially-coherent light transport, work by evolving light's properties from the source, through the scene, until light is sensed by a detector. Such formalisms are inherently incompatible with backward (sensor-to-source) models of light transport: they are not able to formulate a light-matter interaction (a BSDF) without knowledge of light's wave properties, however these properties depend on the light source and evolve throughout the scene, hence are very difficult to predict or estimate in a backward model Steinberg_practical_plt_2022. Fundamental path tracing and sampling techniques, like importance sampling of interactions, cannot be applied, greatly hampering the practicality and ability of these formalisms to work with complex, real-world scenes. Solving this sampling problem, i.e. devising a formalism of backward wave-optical light transport, where a wide-range of sampling techniques can be applied, is the primary motivation for this paper.
• Figure 2: Diffraction through double slits. (a) Schematic of Young's double slit experiment. A pair of slits, of width $b$ and spaced a distance $d$ apart, are cut in a thin, conductive plate. A coherent plane wave (illustrated in green) diffracts through the slits, and is observed upon a screen, placed at a distance $z$ from the plate. The superposition of coherent light from both slits results in a rapidly-oscillating phasor $\varphi$ (illustrated in red), producing an interference pattern. (b) The experiment is performed with increasing slit distances $d$, and we compare our method (sampling incident light with generalized rays) with a ground truth (explicitly diffracting the plane wave through the slits). Differences are plotted in the insets at the bottom right of each pattern (intensity of peak fringe is 1). The experiment was performed with wavelength $\lambda=1$ (arbitrary units), $z=10000\lambda$ and $b=40\lambda$.
• Figure 3: Plots of light intensity as a function of $x$ (position on screen) of the experiment in \ref{['fig_twin_slit']}. In red we plot the exact Rayleigh-Sommerfeld (RS) diffraction, i.e. the unobservable interference that arises in singular points. The spatial extent of a detector on the screen (a pixel in each pattern in \ref{['fig_twin_slit_b']}) is illustrated as a cyan bar. Integrating the RS diffraction over that spatial extent of a pixel computes the numeric ground truth, plotted in dashed green. Results obtained with generalized rays are plotted in blue.
• Figure 4: Loss of locality with Cuypers_Haber_Bekaert_Oh_Raskar_2012. Red plot (exact RS diffraction) is as in \ref{['fig_twin_slit_plots']}. Let $\rho_\text{wbsdf}$ be the radius of integration of a diffraction kernel (\ref{['WDF_kernel_def']}) in Cuypers_Haber_Bekaert_Oh_Raskar_2012. Limiting that integration radius produces erroneous results, hence locality is entirely lost with their method: correctness requires integration over the entire scene ($\rho_\text{wbsdf}=\infty$).
• Figure 5: Sample-solve. Our path tracing algorithm (a) uses generalized rays (dotted lines) to sample paths through the scene. Generalized rays are always linear, therefore classical sampling techniques apply essentially unchanged. Once a path is sampled (solid red path), we (b) solve for the partially-coherent light transport, by applying PLT Steinberg_practical_plt_2022 from the light source to the sensor across all intermediate interactions.