Spin-Weighted Spherical Harmonics for Polarized Light Transport

TL;DR

This work tackles real-time polarization rendering by introducing polarized spherical harmonics (PSH), a rotation-invariant basis built from spin-0 and spin-2 spin-weighted spherical harmonics to represent Stokes-vector fields on the sphere. It defines a polarization-appropriate spherical convolution using Mueller transforms and develops PSH-based coefficient matrices for pBRDF and radiance transfer, enabling efficient frequency-domain rendering. The authors demonstrate a precomputed polarized radiance transfer (PPRT) pipeline that achieves real-time polarization under polarized environmental illumination and confirms accuracy via angular-vs-frequency validation and shadowed-transfer comparisons against a physically based polarized ray tracer. The approach provides a practical, scalable path to real-time polarized rendering and offers a foundation for extending SH-based methods to polarized appearance and lighting tasks with robust rotation invariance and efficient convolution operations.

Abstract

The objective of polarization rendering is to simulate the interaction of light with materials exhibiting polarization-dependent behavior. However, integrating polarization into rendering is challenging and increases computational costs significantly. The primary difficulty lies in efficiently modeling and computing the complex reflection phenomena associated with polarized light. Specifically, frequency-domain analysis, essential for efficient environment lighting and storage of complex light interactions, is lacking. To efficiently simulate and reproduce polarized light interactions using frequency-domain techniques, we address the challenge of maintaining continuity in polarized light transport represented by Stokes vectors within angular domains. The conventional spherical harmonics method cannot effectively handle continuity and rotation invariance for Stokes vectors. To overcome this, we develop a new method called polarized spherical harmonics (PSH) based on the spin-weighted spherical harmonics theory. Our method provides a rotation-invariant representation of Stokes vector fields. Furthermore, we introduce frequency domain formulations of polarized rendering equations and spherical convolution based on PSH. We first define spherical convolution on Stokes vector fields in the angular domain, and it also provides efficient computation of polarized light transport, nearly on an entry-wise product in the frequency domain. Our frequency domain formulation, including spherical convolution, led to the development of the first real-time polarization rendering technique under polarized environmental illumination, named precomputed polarized radiance transfer, using our polarized spherical harmonics. Results demonstrate that our method can effectively and accurately simulate and reproduce polarized light interactions in complex reflection phenomena.

Figures (30)

• Figure 1: Intensity of a polarized ray visualized in the left is characterized by a Stokes vector $\accentset{\leftrightarrow}{s}$. While $\accentset{\leftrightarrow}{s}$ is defined without any measurement frame, it can be measured into a Stokes component vector $\mathbf{s}$ under such a frame.
• Figure 2: Additional basic operations on Stokes vectors are defined in (a) Equation \ref{['eq:bkgnd-stokes-inner']} and (b) Equation \ref{['eq:bkgnd-stokes-compmult']}.
• Figure 3: (a) When we fix the Stokes vector $\accentset{\leftrightarrow}{s}$ and rotate the frame by $\vartheta$, the (numeric) Stokes components of $\accentset{\leftrightarrow}{s}$ rotate by $-2\vartheta$. (b) Rotating the (geometric) Stokes vector itself by $\vartheta$ is equivalent to rotating its Stokes components by $2\vartheta$ with a fixed frame.
• Figure 4: Visualizing a Stokes vector field (polarized environment map) depends on the choice of frame fields. Taking Stokes components of Stokes vector field (a) with respect to a typical $\theta\phi$-frame field (b) yields equirectangular images shown in (d). Using a perspective frame field used in Mitsuba 3 renderer, several perspective images are visualized as (e). Note that while the $s_1$ component (ii) in (e) at the sky, especially (iii), has consistent signs of values, and the component in (d) under a different frame field has a different trend of values.
• Figure 5: Analyzing continuity and smoothness for Stokes vector fields in $\theta\phi$ domain. (a) The visualization of a Stokes vector field. As a geometric quantity, Stokes vector fields are continuous and smooth on the entire sphere, including the zenith. (b) To make the geometric Stokes vector fields to numeric Stokes components, we can assign the specific frame field, named $\theta\phi$-frame field.

Theorems & Definitions (49)

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