Abstract: The highest fidelity images to date of complex materials like cloth use extremely high-resolution volumetric models. However, rendering such complex volumetric media is expensive, with brute-force path tracing often the only viable solution. Fortunately, common volumetric materials (fabrics, finished wood, synthesized solid textures) are structured, with repeated patterns approximated by tiling a small number of exemplar blocks. In this paper, we introduce a precomputation-based rendering approach for such volumetric media with repeated structures based on a modular transfer formulation. We model each exemplar block as a voxel grid and precompute voxel-to-voxel, patch-to-patch, and patch-to-voxel flux transfer matrices. At render time, when blocks are tiled to produce a high-resolution volume, we accurately compute low-order scattering, with modular flux transfer used to approximate higher-order scattering. We achieve speedups of up to 12x over path tracing on extremely complex volumes, with minimal loss of quality. In addition, we demonstrate that our approach outperforms photon mapping on these materials.