Sparse Signal Modeling : application to Image Compression, Image Error Concealment and Compressed Sensing

Signal models are a cornerstone of contemporary signal and image processing methodology. In this report, two particular signal modeling methods, called analysis and synthesis sparse representation, are studied which have been proven to be effective for many signals, such as natural images, and successfully used in a wide range of applications. Both models represent signals in terms of linear combinations of an underlying set, called dictionary, of elementary signals known as atoms. The driving force behind both models is sparsity of the representation coefficients, i.e. the rapid decay of the representation coefficients over the dictionary. On the other hands, the dictionary choice determines the success of the entire model. According to these two signal models, there have been two main disciplines of dictionary designing; harmonic analysis approach and machine learning methodology. The former leads to designing the dictionaries with easy and fast implementation, while the latter provides a simple and expressive structure for designing adaptable and efficient dictionaries. The main goal of this thesis is to provide new applications to these signal modeling methods by addressing several problems from various perspectives. It begins with the direct application of the sparse representation, i.e. image compression. The line of research followed in this area is the synthesis-based sparse representation approach in the sense that the dictionary is not fixed and predefined, but learned from training data and adapted to data, yielding a more compact representation. A new Image codec based on adaptive sparse representation over a trained dictionary is proposed, wherein different sparsity levels are assigned to the image patches belonging to the salient regions, being more conspicuous to the human visual system. Experimental results show that the proposed method outperforms the existing image coding standards, such as JPEG and JPEG2000, which use an analytic dictionary, as well as the state-of-the-art codecs based on the trained dictionaries. In the next part of thesis, it focuses on another important application of the sparse signal modeling, i.e. solving inverse problems, especially for error concealment (EC), wherein a corrupted image is reconstructed from the incomplete data, and Compressed Sensing recover, where an image is reconstructed from a limited number of random measurements. Signal modeling is usually used as a prior knowledge about the signal to solve these NP-hard problems. In this thesis, inspired by the analysis and synthesis sparse models, these challenges are transferred into two distinct sparse recovery frameworks and several recovery methods are proposed. Compared with the state-of-the-art EC and CS algorithms, experimental results show that the proposed methods show better reconstruction performance in terms of objective and subjective evaluations. This thesis is finalized by giving some conclusions and introducing some lines for future works.