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Representing 3D surfaces as level sets of continuous functions over$\mathbb{R}^3$ is the common denominator of neural implicit representations,which recently enabled remarkable progress in geometric deep learning andcomputer vision tasks. In order to represent 3D motion within this framework,it is often assumed (either explicitly or implicitly) that the transformationswhich a surface may undergo are homeomorphic: this is not necessarily true, forinstance, in the case of fluid dynamics. In order to represent more generalclasses of deformations, we propose to apply this theoretical framework asregularizers for the optimization of simple 4D implicit functions (such assigned distance fields). We show that our representation is capable ofcapturing both homeomorphic and topology-changing deformations, while alsodefining correspondences over the continuously-reconstructed surfaces.

Fluid Dynamics Network: Topology-Agnostic 4D Reconstruction via Fluid Dynamics Priors