Discontinuous visibility changes at object boundaries remain a persistent source of difficulty in the area of differentiable rendering. Left untreated, they bias computed gradients so severely that even basic optimization tasks fail.

Prior path-space methods addressed this bias by decoupling boundaries from the interior, allowing each part to be handled using specialized Monte Carlo sampling strategies. While conceptually powerful, the full potential of this idea remains unrealized since existing methods often fail to adequately sample the boundary proportional to its contribution.

This paper presents theoretical and algorithmic contributions. On the theoretical side, we transform the boundary derivative into a remarkably simple local integral that invites present and future developments.

Building on this result, we propose a new strategy that projects ordinary samples produced during forward rendering onto nearby boundaries. The resulting projections establish a variance-reducing guiding distribution that accelerates convergence of the subsequent differential phase.

We demonstrate the superior efficiency and versatility of our method across a variety of shape representations, including triangle meshes, implicitly defined surfaces, and cylindrical fibers based on Bézier curves.