Variational Barycentric Coordinates

TL;DR

This paper introduces a neural field-based variational method for optimizing generalized barycentric coordinates, providing greater control and flexibility over existing models, with applications in deformation and smoothness tasks.

Contribution

It presents a novel neural field formulation for barycentric coordinates, enabling optimization with diverse objectives and offering theoretical insights and practical acceleration strategies.

Findings

Flexible neural field model for barycentric coordinates

Ability to optimize with various smoothness and deformation energies

Validated effectiveness through multiple applications

Abstract

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.