A Differential Diffusion Theory for Participating Media

We present a novel approach to differentiable rendering for participating media, addressing the challenge of computing scene parameter derivatives. While existing methods focus on derivative computation within volumetric path tracing, they fail to significantly improve computational performance due to the expensive computation of multiply-scattered light. To overcome this limitation, we propose a differential diffusion theory inspired by the classical diffusion equation. Our theory enables real-time computation of arbitrary derivatives such as optical absorption, scattering coefficients, and anisotropic parameters of phase functions. By solving derivatives through the differential form of the diffusion equation, our approach achieves remarkable speed gains compared to Monte Carlo methods. This marks the first differentiable rendering framework to compute scene parameter derivatives based on diffusion approximation. Additionally, we derive the discrete form of diffusion equation derivatives, facilitating efficient numerical solutions. Our experimental results using synthetic and realistic images demonstrate the accurate and efficient estimation of arbitrary scene parameter derivatives. Our work represents a significant advancement in differentiable rendering for participating media, offering a practical and efficient solution to compute derivatives while addressing the limitations of existing approaches.