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.

Abstract Shape interpolation is a long standing challenge of geometry processing. As it is ill‐posed, shape interpolation methods always work under some hypothesis such as semantic part matching or least displacement. Among such constraints, volume preservation is one of the traditional animation principles. In this paper we propose a method to interpolate between shapes in arbitrary poses favoring volume and topology preservation. To do so, we rely on a level set representation of the shape and its advection by a velocity field through the level set equation, both shape representation and velocity fields being parameterized as neural networks. While divergence free velocity fields ensure volume and topology preservation, they are incompatible with the Eikonal constraint of signed distance functions. This leads us to introduce the notion of adaptive divergence velocity field, a construction compatible with the Eikonal equation with theoretical guarantee on the shape volume preservation. In the non constant volume setting, our method is still helpful to provide a natural morphing, by combining it with a parameterization of the volume change over time. We show experimentally that our method exhibits better volume preservation than other recent approaches, limits topological changes and preserves the structures of shapes better without landmark correspondences.