| This dissertation focuses on the solution of ill-posed inverse problems which are pervasive in many signal and image analysis domains. First general inverse problems are introduced, then the solutions of linear discrete inverse problems, obtained by matrix inversion, are analyzed. Because these solutions are completely dominated by noise, a useful solution can only be obtained by using additional information. This yields the regularized solution of an inverse problem. Two popular regularization techniques, Tikhonov- and total variation regularization, are reviewed and numerical methods are presented that can be used to compute the regularized solution. An example of an ill-posed inverse problem from seismology is presented, where total variation regularized deconvolution is used for deblurring. As compared to results of other deconvolution techniques, including water level deconvolution which is the standard method in seismology, total variation regularized deconvolution results in cleaner and sharper restorations. However, this example also shows that one of the major shortcomings of total variation deconvolution is that it is only able to restore piecewise constant signals.;Finally, a wavelet approximation of the total variation regularized denoised solution is used to remove noise from positron emission tomography scans. An existing method for denoising two dimensional images is extended to process three dimensional image volumes. The method is computationally very efficient and it is shown that it can be used to increase the signal to noise ratio of positron emission tomography scans which are reconstructed using the expectation maximization algorithm. The dissertation concludes with directions for future research. Software packages to perform the denoising of volume data and the deblurring of seismograms have been developed and can be downloaded from the author's web page.;In this dissertation a method to identify edges in blurred data based on higher order total variation regularization is presented. The variable order total variation regularization method approximates smooth parts of the signal with higher order polynomials while it is able to preserve jump discontinuities in the signal. |
