Differentiating Through Marching Cubes

You thought you could not differentiate through Marching Cubes? You can, courtesy of the implicit function theorem. In a paper written in collaboration with NeuralConcept, we show this in an upcoming PAMI paper.

Our key insight is that 3D surface samples can . We prove this formally by reasoning about how implicit field perturbations impact 3D surface geometry locally. Specifically, we derive a closed-form expression for the derivative of a surface sample with respect to the underlying implicit field, which is independent of the method used to compute the iso-surface. This lets us extract the explicit surface using a non-differentiable algorithm, such as Marching Cubes, and then perform the backward pass through the extracted surface samples. This yields an end-to-end differentiable surface parameterization that can describe arbitrary topology and is not limited in resolution. We first introduced this approach in the NeurIPs paper that focused on the 0-iso-surface of signed distance functions. In the PAMI paper, we extended it to iso-surface of generic implicit functions, such as occupancy fields by harnessing simple multivariate calculus tools.