Variational Barycentric Coordinates
Ana Dodik Oded Stein Vincent Sitzmann Justin Solomon
We propose a variational technique to optimize for generalized barycentric
coordinates that offers additional control compared to existing models. Prior
work represents barycentric coordinates using meshes or closed-form formulae,
in practice limiting the choice of objective function. In contrast, we directly
parameterize the continuous function that maps any coordinate in a polytope's
interior to its barycentric coordinates using a neural field. This formulation
is enabled by our theoretical characterization of barycentric coordinates,
which allows us to construct neural fields that parameterize the entire
function class of valid coordinates. We demonstrate the flexibility of our
model using a variety of objective functions, including multiple smoothness and
deformation-aware energies; as a side contribution, we also present
mathematically-justified means of measuring and minimizing objectives like
total variation on discontinuous neural fields. We offer a practical
acceleration strategy, present a thorough validation of our algorithm, and
demonstrate several applications.