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

In many applications, the recorded data will almost certainly be a degraded version of the original object that is desired, due to the imperfections of physical measurement systems and the particular physical limitations imposed in every application where data are recorded. The situation becomes more complex due to random noise, which is inevitably mixed with the data and may originate from the measurement process, the transmission medium or the recording process. In many practical situations, the degradation can be adequately modeled by a linear system and an additive white Gaussian noise process. The objective is to estimate the original object given the degraded measurements and the matrix describing the forward transformation. Examples of linear inverse problems include image restoration and reconstruction, inverse scattering, seismic analysis, non-destructive testing, etc. In many cases, the forward transformation acts as a smoothing agent and destroys many high-frequency features in the original object. Mathematically, this means that the system matrix is either ill-conditioned or singular in which case a straightforward inversion is either impossible or results in a solution which is excessively contaminated by the noise. To stabilize the problem, one usually uses a regularization procedure where additional information about the original object is incorporated; one may force the computed solution to be smooth, for example. Regularization methods always include a parameter, called the regularization parameter, which controls the degree of smoothing or regularization applied to the problem. If the regularization parameter is too small the solution will be noisy, and if it is too large the solution will be over-smooth. The two basic problems in the regularization of discrete linear inverse problems are the speci cation of an appropriate prior model that re ects the properties of the original object as closely as possible and the determination of appropriate regularization parameters that produce the closest approximation to the original object under the assumed prior model. We concentrate our e orts on the solution of the prior speci cation and the regularization parameter selection problems. In the rst part of the thesis, we focus on the image restoration problem. Speci cally, we deal with developing multiscale prior models for images to obtain a highly exible framework for adapting the degree of regularization to the scale and orientation varying features in the image. We demonstrate an e cient half-quadratic algorithm for obtaining the restorations from the observed data. In the second part, we develop a multi-variate generalization of the conventional L-curve method, the L-hypersurface, for the selection of multiple regularization parameters. The L-curve is one of the simplest and most popular methods for selecting a single regularization parameter. It is based on a plot of the residual norm against the solution norm drawn in a log scale. It has been numerically shown that the corner of the L-curve, which is de ned as the point on the L-curve with the maximum curvature, provides a good regularization parameter. We extend the notion of the curvature for plane curves to the notion of Gaussian curvature for hypersurfaces and choose the regularization parameters as those maximizing the Gaussian curvature of the L-hypersurface. Then we deal with the problem of decreasing