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 *L*1
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.