|M.Sc Student||Feld Tal Michael|
|Subject||NonLinear Spectral Analysis for Grouping Problems|
|Department||Department of Electrical Engineering||Supervisors||Professor Guy Gilboa|
|Dr. Noam Kaplan|
|Full Thesis text|
Perceptual grouping problems in vision can often be reformulated as balanced partitioning of a graph, in which every object is a node in the graph and similarities between objects are the edge weights. Although NP-hard, balanced partitioning is usually relaxed to the minimization of a corresponding energy, which under weaker conditions is equivalent to a balanced criteria. The current leading balanced criterion - the Cheeger cut, corresponds to the minimization of the Total variation over the L1 norm, which is a particular case of a Rayleigh quotient of two absolutely one-homogeneous functionals. We examine the problem of minimizing generalized Rayleigh quotients of the form J(u)/H(u), where both J and H are absolutely one-homogeneous functionals. This can be viewed as minimizing J where the solution is constrained to be on a generalized sphere with H(u) = 1, where H is any norm or semi-norm. The solution admits a nonlinear eigenvalue problem, based on the subgradients of J and H. We examine several flows which minimize the ratio. This is done both by time-continuous flow formulations and by discrete iterations. We focus on a certain flow, which is easier to analyze theoretically, following the theory of Brezis on flows with maximal monotone operators. A comprehensive theory is established, including convergence of the flow. We then turn into a more specific case of minimizing graph total variation on the L1 sphere, which approximates the Cheeger-cut problem. Experimental results show the applicability of such algorithms for clustering and classification of images. Finally, we propose a multi-class version of our flow that generates statistically independent binary eigenfunctions. We motivate the use of this set of eigenfunctions by presenting a new reformulation of the spectral hashing problem which is experimentally tested as well.