Differentiable Rendering for Specular Materials via Projected Specular Manifolds

Physics-based differentiable rendering computes derivatives of radiometric measures with respect to scene parameters. A key challenge lies in handling scene discontinuities from geometries, materials, and other factors. These discontinuities lead to a unique boundary integral defined on discontinuity curves. While occlusion-induced discontinuities have been well studied, those caused by materials, especially specular materials, remain largely unexplored. Specular bidirectional scattering distribution functions (BSDFs) contain Dirac delta distributions arising from the laws of perfect reflection or refraction, which constrain light path geometry and restrict the path integral to the specular manifold. The specular manifold is defined by geometric constraints and lacks explicit location, making the evaluation of discontinuity curves challenging. To address this, our key insight is that we can locate the specular manifold without explicit computation by projecting it onto a subregion of a specular object, which we call the projected specular manifold (PSM). After projection, while exact locations of discontinuity curves remain unknown, we can adopt the warped-area reparameterization to convert the boundary integral into an interior integral over the surface, avoiding the need to precisely locate these curves. The key challenge of this reparameterization is constructing an additional velocity field. For this, our key observation is that for a specular path, the distances from vertices to manifold boundaries and associated velocities are known. Therefore, we propose a simple but practical direct-interpolation-based method to construct the velocity. Furthermore, we extend the warped-area sampling to handle visibility boundaries with specular BSDFs and generalize our method to glossy materials. Our computed derivatives match references with minor additional computational cost, and can successfully drive gradient-based optimization tasks like shape and position estimation. Our implementation is available at https://github.com/Rhythm25/PSM.