Explicit flows for implicit surfaces

Shape deformation for morphing or editing purposes is a central challenge in Computer Graphics. While numerous methods exist, few allow for the explicit evaluation of the deformation at arbitrary times and locations directly, without resorting to intricate advection or interpolation schemes. In this paper, we propose a method that provides an explicit expression of the deformation parameterized as a flow, for continuously deforming shapes defined implicitly. Implicit surfaces are indeed particularly well suited for deformation tasks, since they inherently account for both the surface and the enclosed volume. Our approach leverages invertible neural networks to ensure theoretically that the deformation is a valid flow, while also providing differential quantities useful for geometric regularization. We demonstrate applications of this flow to shape morphing with and without landmarks, shape editing, and pairwise-to-any morphing where we compute pairwise morphings to a canonical shape allowing to deduce transformations between any pair through flow composition. The code for our method is available at https://github.com/camillebnm/explicit_flows_for_implicit_surfaces.