Natural Boundary Conditions for Smoothing in Geometry Processing
ACM Transaction on Graphics 2018

Oded Stein        Eitan Grinspun        Max Wardetzky        Alec Jacobson

Columbia University       University of Toronto       Universität Göttingen

Smoothing a noisy function with the Laplacian energy and common low-order boundary conditions (zero Neumann) introduces a bias at the boundary: isolines exit perpendicularly. We propose using a different smoothness energy, the Hessian energy, whose natural boundary conditions avoid this bias.


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 Frobenius 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.




 author = {Stein, Oded and Grinspun, Eitan and Wardetzky, Max and Jacobson, Alec},
 title = {Natural Boundary Conditions for Smoothing in Geometry Processing},
 journal = {ACM Trans. Graph.},
 issue_date = {May 2018},
 volume = {37},
 number = {2},
 month = may,
 year = {2018},
 issn = {0730-0301},
 pages = {23:1--23:13},
 articleno = {23},
 numpages = {13},
 url = {},
 doi = {10.1145/3186564},
 acmid = {3186564},
 publisher = {ACM},
 address = {New York, NY, USA},
 keywords = {Geometry processing, Hessian energy, Laplacian energy, denoising, smoothing},


We would like to thank Ladislav Kavan, Denis Zorin, and Jinshuo Dong for early discussions; Yu Wang for sharing source code; Paul Kry for hosting the 2017 Bellairs workshop; and organizers of the Dagstuhl Seminar #17232.

This work is funded in part by the Binational Science Foundation (US-Israel) Award 2012376, the National Science Foundation Award IIS-14-09286, the NSERC Discovery Grants (RGPIN-2017-05235 & RGPAS-2017-507938), a Canada Research Chair award, the Connaught Fund, and a gift from Adobe Systems Inc.