A Theory of Topological Derivatives for Inverse Rendering of Geometry
Ishit Mehta Manmohan Chandraker Ravi Ramamoorthi
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.