Kt, AakashShah, IshaanNarayanan, P. J.Bousseau, AdrienDay, Angela2025-05-092025-05-0920251467-8659https://doi.org/10.1111/cgf.70049https://diglib.eg.org/handle/10.1111/cgf70049Single scattering in scenes with participating media is challenging, especially in the presence of area lights. Considerable variance still remains, in spite of good importance sampling strategies. Analytic methods that render unshadowed surface illumination have recently gained interest since they achieve biased but noise-free plausible renderings while being computationally efficient. In this work, we extend the theory of Linearly Transformed Spherical Distributions (LTSDs) which is a well-known analytic method for surface illumination, to work with phase functions. We show that this is non-trivial, and arrive at a solution with in-depth analysis. This enables us to analytically compute in-scattered radiance, which we build on to semi-analytically render unshadowed single scattering. We ground our derivations and formulations on the Volume Rendering Equation (VRE) which paves the way for realistic renderings despite the biased nature of our method. We also formulate ratio estimators for the VRE to work in conjunction with our formulation, enabling the rendering of shadows. We extensively validate our method, analyze its characteristics and demonstrate better performance compared to Monte Carlo single-scattering.CCS Concepts: Computing methodologies → Ray tracing; Reflectance modelingComputing methodologies → Ray tracingReflectance modeling10.1111/cgf.7004911 pages