At each shade point, the spherical visibility function encodes occlusion from surrounding geometry, in all directions. Computing this function is difficult and point-sampling approaches, such as ray-tracing or hardware shadow mapping, are traditionally used to efficiently approximate it. We propose a semi-analytic solution to the problem where the spherical silhouette of the visibility is computed using a search over a 4D dual mesh of the scene. Once computed, we are able to semi-analytically integrate visibility-masked spherical functions along the visibility silhouette, instead of over the entire hemisphere. In this way, we avoid the artefacts that arise from using point-sampling strategies to integrate visibility, a function with unbounded frequency content. We demonstrate our approach on several applications, including direct illumination from realistic lighting and computation of pre-computed radiance transfer data. Additionally, we present a new frequency-space method for exactly computing all-frequency shadows on diffuse surfaces. Our results match ground truth computed using importance-sampled stratified Monte Carlo ray-tracing, with comparable performance on scenes with low-to-moderate geometric complexity.