# Variational Barycentric Coordinates

arXiv: https://arxiv.org/abs/2310.03861 · HTML: https://mlsanity.com/paper/2310.03861/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 60 references that are in the map.
- First posted on arXiv 5 Oct 2023.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Animatable Neural Radiance Fields for Modeling Dynamic Human Bodies](https://mlsanity.com/paper/2105.02872/) (2021)
- [NeRF-Editing: Geometry Editing of Neural Radiance Fields](https://mlsanity.com/paper/2205.04978/) (2022)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Neural Cages for Detail-Preserving 3D Deformations](https://mlsanity.com/paper/1912.06395/) (2019)
- [Physics informed deep learning for computational elastodynamics without labeled data](https://mlsanity.com/paper/2006.08472/) (2020)
- [Variational Autoencoders for Deforming 3D Mesh Models](https://mlsanity.com/paper/1709.04307/) (2017)
- [Neural Subdivision](https://mlsanity.com/paper/2005.01819/) (2020)
- [Neural Jacobian Fields: Learning Intrinsic Mappings of Arbitrary Meshes](https://mlsanity.com/paper/2205.02904/) (2022)
- [NiLBS: Neural Inverse Linear Blend Skinning](https://mlsanity.com/paper/2004.05980/) (2020)
- [Natural Boundary Conditions for Smoothing in Geometry Processing](https://mlsanity.com/paper/1707.04348/) (2017)
- [Octahedral Frames for Feature-Aligned Cross-Fields](https://mlsanity.com/paper/2007.09740/) (2020)
- [A Smoothness Energy without Boundary Distortion for Curved Surfaces](https://mlsanity.com/paper/1905.09777/) (2019)
- [Symmetric Volume Maps: Order-Invariant Volumetric Mesh Correspondence with Free Boundary](https://mlsanity.com/paper/2202.02568/) (2022)

## Built on by (most cited first, 6 of 6)

- [Image Sculpting: Precise Object Editing with 3D Geometry Control](https://mlsanity.com/paper/2401.01702/) (2024)
- [MeshUp: Multi-Target Mesh Deformation via Blended Score Distillation](https://mlsanity.com/paper/2408.14899/) (2024)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing](https://mlsanity.com/paper/2405.12895/) (2024)
- [Barycentric Neural Networks and Length-Weighted Persistent Entropy Loss: A Green Geometric and Topological Framework for Function Approximation](https://mlsanity.com/paper/2509.06694/) (2025)
- [PoseAlign: Sculpting Pose-Consistent Meshes via Text-Guided Deformation](https://mlsanity.com/paper/2607.10560/) (2026)

## Abstract (arXiv)

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.