Divergence‐Free Shape Correspondence by Deformation

Abstract

We present a novel approach for solving the correspondence problem between a given pair of input shapes with non‐rigid, nearly isometric pose difference. Our method alternates between calculating a deformation field and a sparse correspondence. The deformation field is constructed with a low rank Fourier basis which allows for a compact representation. Furthermore, we restrict the deformation fields to be divergence‐free which makes our morphings volume preserving. This can be used to extract a correspondence between the inputs by deforming one of them along the deformation field using a second order Runge‐Kutta method and resulting in an alignment of the inputs. The advantages of using our basis are that there is no need to discretize the embedding space and the deformation is volume preserving. The optimization of the deformation field is done efficiently using only a subsampling of the orginal shapes but the correspondence can be extracted for any mesh resolution with close to linear increase in runtime. We show 3D correspondence results on several known data sets and examples of natural intermediate shape sequences that appear as a by‐product of our method.

Wassersplines for Stylized Neural Animation

“…Hug et al [2015] regularize the velocity field in dynamical optimal transport and prove existence of minimizers with a velocity gradient regularizer. Eisenberger et al [2019] compute static volume-preserving, velocity fields for mesh registration. Feydy et al [2017] use unbalanced OT to build diffeomorphic registrations in medical imaging.…”

MapTree

“…A third idea would be to take an existing optimization algorithm and start it with multiple dierent initializations. Common approaches for initializing maps include using pointwise descriptors, e.g., [Eisenberger et al 2019;Vestner et al 2017b] or initial segment or landmark correspondences [Ezuz et al 2019;Kleiman and Ovsjanikov 2018;Ren et al 2018]. For these methods it is actually very challenging to modify the initialization in a meaningful way so that the optimization methods would explore dierent parts of the map space.…”

Remixing Functionally Graded Structures: Data-Driven Topology Optimization with Multiclass Shape Blending

“…Since then, many works have improved metamorphosis using function-based operations (Sanchez, 2016;Shapiro, 2007), optimization (Eisenberger et al, 2019), and DL-based interpolation (Oring et al, 2020;Gu et al, 2019). However, these can require intensive user-interaction and computational costs, and are, therefore, intractable or unsuitable for multiscale design.…”