Abstract
Exploiting sparsity in signal and image analysis has been very fruitful in recent researches. Compressive sensing is the pioneering framework in achieving this. However, there are other fields such as dictionary learning and blind source separation which can benefit from sparsity-inducing methods. In this study some advanced methods are developed to improve the recovery performance in compressive sensing and also to utilize sparsity-inducing algorithms in multichannel blind source separation problem. The main focus is on methodological aspects and variety of applications from communications to biomedical signal processing and computer vision can be considered. Finding optimal projections in compressive sensing has been always of interest to the researchers. This problem is tackled using three approaches. First, a segment-wise strategy for sampling of signals is proposed which requires less memory while achieves similar performance as that obtained in conventional random sampling. Second, an improved version of a previous method in measurement matrix optimization is proposed. Third, a more advanced approach for finding nearly-optimal projections is proposed. It is demonstrated that the proposed methods significantly improve the reconstruction quality compared with conventional random sampling. In addition, the performance of the proposed methods are shown to be superior to those of the other related methods.Dictionary learning is extensively named as a framework which provides sparse representation of the data. Learning incoherent dictionaries and attaining fast algorithms axe two important key successes of dictionary learning methods. These issues axe addressed and two different algorithms are proposed. First algorithm is designed to impose incoherence into the dictionary learning algorithm. In the second method the focus is on achieving low computation cost as well as uncorrelated dictionary elements. Furthermore, the analogy of dictionary learning to multichannel blind source separation is investigated. The proposed approaches are then applied to brain fMRI images to detect the activation regions. The empirical results on both synthetic and real data confirm the effectiveness of the proposed methods. Blind separation of sources with underlying sparse nature, from multichannel observations, requires the knowledge of sparsity domain of each source. This, however, is not always the case and the underlying sparse domain of sources might be hidden. This problem is addressed and a practical solution is provided to find the mixing matrix and the constituent sources as well as their sparse domain. This is achieved by fusing dictionary learning algorithm into the source separation process. The proposed method uses an adaptive scheme for obtaining the sparsifying dictionaries, capturing local patterns of the sources, while reducing the separation error. It also has the ability to denoise the sources during the separation and learning. Further, the proposed method is extended for the case of available global sparsifying transform for each source. The results of simulations on both 1-D and 2-D signals are promising and demonstrate that the proposed method is able to recover (and denoise) the sources without prior knowledge about sparse domain of the sources.