Analysis of Some Optimization Techniques for Regularization of Inverse Problems.

Abstract

The main objective in inverse problems is to approximate some unknown parameters or attributes of interest, given some measurements that are only indirectly related to these parameters. This type of problem appears in many areas of science, engineering and industry. Examples can be found in medical computerized tomography, groundwater flow modeling, etc. In the process of solving these problems often appears an instability phenomenon known as ill-posedness which requires regularization. Ill-posedness is related to the fact that the presence of even a small amount of noise in the data can lead to enormous errors in the approximated solution. Different regularization techniques have been proposed in the literature. In this thesis our focus is put on Total Variation regularization. We study the total variation regularization for both image denoising and image deblurring problems. Three algorithms for total variation regularization will be analysed, namely the split Bregman algorithms, the Alternating Direction Method of Multipliers and the Rudin Osher Fatemi denoising model on the graph. We experiment these algorithms for different implementation examples and compare their performance for denoising problems. Our observation is that these algorithms are comparable in many cases, often times the Split Bregman algorithm is faster in the sense that it achieves a given number of iterations in a shorter running time, but at the same time even though the ROF model on the graph seems slower, it achieves a desired or a prescribed precision with fewer number of iterations.