Unbiased Differential Visibility Using Fixed-Step Walk-on-Spherical-Caps And Closest Silhouettes

Computing derivatives of path integrals under evolving scene geometry is a fundamental problem in physics-based differentiable rendering, which requires differentiating discontinuities in the visibility function. Warped-area reparameterization is a powerful technique to compute differential visibility, and key is construction of a velocity field that is continuous in the domain interior and agrees with defined velocities on boundaries. Robustly and efficiently constructing such fields remains challenging. We present a novel velocity field construction for differential visibility. Inspired by recent Monte Carlo solvers for partial differential equations (PDEs), we formulate the velocity field via Laplace's equation and solve it with a walk-on-spheres (WoS) algorithm. To improve efficiency, we introduce a fixed-step WoS that terminates random walks after a fixed step count, resulting in a continuous but non-harmonic velocity field still valid for warped-area reparameterization. Furthermore, to practically apply our method to complex 3D scenes, we propose an efficient cone query to find the closest silhouettes on a boundary. Our cone query finds the closest point under the geodesic distance on a unit sphere, and is analogous to the closest point query by WoS to compute Euclidean distance. As a result, our method generalizes WoS to perform random walks on spherical caps over the unit sphere. We demonstrate that this enables a more robust and efficient unbiased estimator for differential visibility.