Technical Report MSC-2017-14

Title: On elliptic operators and non-rigid shapes
Authors: Yoni Choukroun
Supervisors: Ron Kimmel
PDFCurrently accessibly only within the Technion network
Abstract: Many shape analysis methods treat the geometry of an object as a metric space captured by the Laplace-Beltrami operator. In this thesis we present an adaptation of a classical operator from quantum mechanics to shape analysis where we suggest to integrate a scalar function through a unified elliptical Hamiltonian operator. We study the addition of a potential function to the Laplacian as a generator for dual spaces in which shape processing is performed.

After exploration of the decomposition of this operator, we evaluate the resulting spectral basis for different applications.

First, we present a general optimization approach for solving variational problems involving the basis defined by the Hamiltonian using perturbation theory for eigenvectors.

Then, we propose an iteratively-reweighted L2 norm for sparsity promoting problems such as the compressed harmonics where solution is reduced to a sequence of simple eigendecomposition of the Hamiltonian. Physically understandable, they do not require non-convex optimization on Stiefel manifolds and produce faster, stable and more accurate results.

We then suggest a new framework for mesh compression using a Hamiltonian based dictionary where regions of interest are enhanced through the proposed operator. By sparsely encoding the geometry of the shape using the proposed data-dependent basis, we improve compression performance compared to previous results that use the standard Laplacian and spectral graph wavelets.

Finally, we propose to apply the Hamiltonian for shape matching where information such as anchor points, corresponding features, and consistent photometry or inconsistent regions can be considered through a potential function for improving the performance in finding correspondence between surfaces.

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