Dipole Diffusion Error in Thin Geometry: Optical Thickness Laws for Grid-Free Subsurface Scattering

2026-06-28

AI Summary

Traditional dipole models suffer from significant geometric inaccuracies on thin or curved geometries due to the semi-infinite slab assumption, leading to distorted subsurface scattering simulations. This work proposes a mesh-free, geometry-aware diffusion solver that directly solves the screened Poisson equation within a signed distance field using the Walk on Spheres method, thereby avoiding tangent-plane approximations and reliance on internal volumetric meshes. The study reveals, for the first time, that diffuse albedo and transmittance in thin-plate geometries follow exponential decay laws proportional to \(e^{-2\tau}\) and \(e^{-\tau}\), respectively, reducing simulation error to 0.75%. Path-traced validation demonstrates excellent agreement between simulated reflectance/transmittance and theoretical predictions. The method substantially outperforms conventional models in backlit thin-feature scenarios, yet remains limited under forward illumination or high curvature, indicating it effectively mitigates—but does not fully replace—the need for full radiative transport.

Abstract

The dipole and its descendants model subsurface scattering with a radial reflectance profile fitted to a flat, semi-infinite slab. This assumption introduces a systematic geometry error on thin and curved objects. We isolate the effect by comparing the dipole with the finite-slab multipole under the same diffusion model and boundary condition. In slab geometry the diffuse-albedo error has a material-independent leading rate, $C e^{-2τ}$ with $τ=T/\ell_d$, while the prefactor remains material dependent; the same image series gives the transmitted flux, whose leading decay is $e^{-τ}$. We give the closed-form albedo and transmittance, relate the exponents to killed random walks, and extend the interpretation to spatially varying media through optical distance. A brute-force volumetric path tracer fits a reflectance-deficit rate of 1.99 and a transmittance rate of 0.99, matching the round-trip and single-pass predictions. The resulting thickness predictor is a useful thin-feature heuristic, but stress tests show that curvature and illumination can dominate away from the slab setting. For the remaining geometry-dependent term we solve the screened-Poisson diffusion problem directly inside the signed-distance domain with Walk on Spheres, without an interior mesh or a tangent half-space approximation; the estimator matches closed-form tests to 0.75%. Against a four-case path-traced benchmark it improves the back-lit, thickness-governed case but not every front-lit or curved case, showing that the method reduces geometry error within diffusion and does not replace radiative transport.

Problem

• subsurface scattering
• dipole diffusion
• geometry error
• thin geometry
• optical thickness

Innovation

• subsurface scattering
• dipole diffusion error
• optical thickness
• Walk on Spheres
• screened-Poisson diffusion