Natural Boundary Conditions for Smoothing in Geometry Processing

TL;DR

This paper introduces a new smoothness energy based on the squared Frobenius norm of the Hessian for geometry processing, which naturally enforces high-order boundary conditions and avoids boundary bias in shape optimization.

Contribution

The authors propose using the squared Hessian energy with natural high-order boundary conditions, improving shape processing without boundary bias compared to traditional Laplacian-based methods.

Findings

The squared Hessian energy models free boundaries effectively.

Discretizations using finite differences and finite elements are developed.

The approach demonstrates improved shape regularization in various tasks.

Abstract

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.