# Natural Boundary Conditions for Smoothing in Geometry Processing

arXiv: https://arxiv.org/abs/1707.04348 · HTML: https://mlsanity.com/paper/1707.04348/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2017.

## Built on by (most cited first, 5 of 5)

- [Variational Barycentric Coordinates](https://mlsanity.com/paper/2310.03861/) (2023)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [PoissonNet: A Local-Global Approach for Learning on Surfaces](https://mlsanity.com/paper/2510.14146/) (2025)
- [Developability Approximation for Neural Implicits through Rank Minimization](https://mlsanity.com/paper/2308.03900/) (2023)
- [Variational Green's Functions for Volumetric PDEs](https://mlsanity.com/paper/2602.12349/) (2026)

## Abstract (arXiv)

In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction bakes in low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenious norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy's natural boundary conditions (those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.