A Theory of Topological Derivatives for Inverse Rendering of Geometry
This addresses the sparse signal limitation in prior methods for inverse rendering, enabling more robust shape optimization with topology changes.
The paper tackles the problem of inverse rendering of geometry by introducing a theoretical framework for differentiable surface evolution that enables discrete topology changes through topological derivatives, allowing optimization of image functionals with applications like image vectorization and 3D reconstruction.
We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.