Murat Belge

Multiscale and Curvature Methods for the Regularization of the Linear Inverse Problems

Friday, August 13, 1999
1:00 PM
442 Snell

Abstract

In many linear inverse problems in the physical sciences one is interested in determining an unknown object from its linearly transformed and noisy measurements. Such inverse problems arise quite naturally in a variety of applications such as image restoration and reconstruction, computed tomography (CAT, PET e.t.c.), inverse scattering, geophysical exploration, etc. Usually, the matrix describing the forward transformation is singular or ill-conditioned so that straightforward inversion is either impossible or amplifies the noise to such an extent that the computed solution is useless. A practical way of coping with such problems is to use a regularization procedure where additional prior information, such as the solution should be smooth, is supplied to make the computed solution unique and immune to noise. Typically the prior model contains several parameters which are needed to be tuned to the object of interest.

In this thesis, we explore the prior information specification and hyperparameter selection problems. In the first part of the thesis, we concentrate on the image restoration/reconstruction problems and develop statistically based wavelet domain priors for edge-enhanced restoration. Our prior models include several tuning parameters to adapt the degree of regularization to the scale and orientation varying features of the image. This adaptation is achieved through a data-driven choice of the tuning parameters and leads to the L-hypersurface method for the selection of multiple regularization parameters. The L-hypersurface is a multi-variate extension of the popular L-curve, which is a plot of the size of the solution (measured by an appropriate norm) against the size of the data mismatch for all valid regularization parameters. In order to solve the visualization and interpretation problems associated with the L-hypersurface we propose using the Gaussian curvature of the L-hypersurface as an aid for locating good regularization parameters. Then we deal with the problem of decreasing the computational effort associated with the practical implementation of the generalized L-curve method.

Thesis Committee:
Prof. Eric L. Miller (advisor)
Prof. Dana H. Brooks
Prof. Hanoch Lev-Ari
Prof. Misha E. Kilmer