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.