Pixel Club: On Elliptic Operators and Non-rigid Shapes

Yoni Choukroun (EE, Technnion)
Tuesday, 25.4.2017, 11:30
Room 337 Taub Bld.

Many shape analysis methods treat the geometry of an object as a metric space captured by the Laplace-Beltrami operator. 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 will present a general optimization approach for solving variational problems involving the Hamiltonian basis using perturbation theory for eigenvectors. Then, we will 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 for improved results in spectral mesh compression. 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 performances in finding correspondence between surfaces.

​*​Supervised by Professor Ron Kimmel

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