A Level Set Theory for Neural Implicit Evolution under Explicit Flows

Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a framework that allows applying deformation operations defined for triangle meshes onto such implicit surfaces. Several of these operations can be viewed as energy-minimization problems that induce an instantaneous flow field on the explicit surface. Our method uses the flow field to deform parametric implicit surfaces by extending the classical theory of level sets. We also derive a consolidated view for existing methods on differentiable surface extraction and rendering, by formalizing connections to the level-set theory. We show that these methods drift from the theory and that our approach exhibits improvements for applications like surface smoothing, mean-curvature flow, inverse rendering and user-defined editing on implicit geometry.