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 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.
Biography: Oded Stein is a PhD student at Columbia University supervised by Prof. Eitan Grinspun. He received his BSc and MSc in mathematics from ETH Zurich. At ETH Zurich he worked on the boundary element method for solving elliptic PDEs. He currently works on diverse topics in geometry processing and discrete geometry including data processing, computer animation, and fabrication.His interests lie in discrete geometry, geometry processing and PDEs on manifolds.
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