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# Mathematics Colloquium: Theory and applications of Spectral Graph Wavelets for defining multiscale transforms on graphs

2012-02-23

4:10pm Neill 5W (refreshments in the Hacker Lounge at 3:30pm)

David Hammond, University of Oregon

Wavelet analysis has proved to be a very successful tool for signal
analysis and processing. However, many interesting data sets are
defined on domains with complicated network-like structure to which
classical wavelets are not suitable. In the first part of my talk I
will describe the Spectral Graph Wavelet Transform (SGWT), a novel
approach for constructing wavelet transforms for data defined on the
vertices of a finite weighted graph.These wavelets are defined using
the eigenvectors and eigenvalues of the discrete graph Laplacian $L$,
which provide an analogue of the Fourier transform for data on the
vertices of graphs. Using a bandpass-shaped generating function $g$
and scale parameter $t$, we define the scaled wavelet operator
$T^t_g=g(tL)$. The spectral graph wavelets are then formed by
localizing this operator by applying it to indicator functions at each
vertex, and wavelet frames can be designed through sampling of the
scale parameter.

In the second half of my talk I will discuss applications of the SGWT
to two distinct problems: image denoising and electroencephalogram
(EEG) source estimation. The image denoising application is based upon
constructing a nonlocal image graph, whose vertices correspond to the
original image pixels and whose edges measure similarity between
(potentially spatially distant) image patches. I will describe an
image model based on assuming sparsity in a frame of nonlocal image
wavelets generated by applying the SGWT to the nonlocal image graph,
and show that this model leads to an effective denoising procedure.

EEG source estimation is an ill-posed inverse problem consisting of
determining current sources inside the brain on the basis of
electrical voltages measured on the scalp. I will describe an approach
to solving this problem based on building a prior penalty on the
cortical sources, using a weighted graph description of brain
connectivity. Applying the SGWT to the brain connectivity graph gives
a frame of cortical graph wavelets. Assuming the desired cortical
sources have a sparse expansion in the cortical graph wavelets leads
to a novel source estimation algorithm which compares favorably with a
standard minimum-norm approach.