Local Basis Expansions for Inverse Problems

Partha Mitra, Ph.D.

Cold Spring Harbor Labs

Ill-posed linear inverse problems, which can be thought of as linear equations of the form Y = K X, where K is a low rank matrix, occur in a number of application domains ranging from geophysics to brain imaging. Y corresponds to a finite set of noisy measurements from an array of sensors, and X is the physical variable to be estimated, typically a scalar or vector field defined over a bounded domain (the source space). A number of approaches exist to estimating X. We have developed an approach based on expansions in terms of basis functions which are approximately localized in a region of interest in the source space. The method is inspired by the multitaper approach to spectral analysis, and the localized basis set is obtained by solving a generalized eigenvalue problem. The method is applied to the problem of source localization in magnetoencephalography. As part of this exercise, we study maximally concentrated combinations of vector spherical harmonics on the surface of a sphere.

 

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