Signal and Image Processing Algorithms Using Interval Convex Programming and Sparsity

In this thesis, signal and image processing algorithms based on sparsity and interval convex programming are developed for inverse problems. Inverse signal processing problems are solved by minimizing the ?1 norm or the Total Variation (TV) based cost functions in the literature. A modi?ed entropy functional approximating the absolute value function is de?ned. This functional is also used to approximate the ?1 norm, which is the most widely used cost function in sparse signal processing problems. The modi?ed entropy functional is continuously di?erentiable, and convex. As a result, it is possible to develop iterative, globally convergent algorithms for compressive sensing, denoising and restoration problems using the modi?ed entropy functional. Iterative interval convex programming algorithms are constructed using Bregman?s D-Projection operator. In sparse signal processing, it is assumed that the signal can be represented using a sparse set of coe?cients in some transform domain. Therefore, by minimizing the total variation of the signal, it is expected to realize sparse representations of signals. Another cost function that is introduced for inverse problems is the Filtered Variation (FV) function, which is the generalized version of the Total Variation (VR) function. The TV function uses the di?erences between the pixels of an image or samples of a signal. This is essentially simple Haar ?ltering. In FV, high-pass ?lter outputs are used instead of di?erences. This leads to ?exibility in algorithm design adapting to the local variations of the signal. Extensive simulation studies using the new cost functions are carried out. Better experimental restoration, and reconstructions results are obtained compared to the algorithms in the literature. Keywords: Interval Convex Programming, Sparse Signal Processing, Total Variation, Filtered Variation, D-Projection, Entropic Projection