Structural Dictionary Learning and Sparse Representation with Signal and Image Processing Applications

The success of sparse representation as a signal representation mechanism has been well-acknowledged in various signal and image processing applications, leading to the state-of-the-art performances. Flexibility and local adaptivity form the main advantage of this representation. It has been widely acknowledged that dictionary design (number of dictionaries, and the number of atoms in a dictionary) has strong implications on the whole representation process. This thesis addresses sparse representation over multiple learned dictionaries aiming at enhancing the representation quality and reducing the computational complexity. The first contribution in this work is performing dictionary learning and sparse repre-sentation in the wavelet domain, merging the desirable attributes of wavelet transform with the representation power of learned dictionaries. Simulations conducted over the problem of single-image super-resolution show that this representation framework is able to improve the representation quality while reducing the computational cost. Our second contribution is a variable patch size sparse representation paradigm. In this setting, the size of the patch is adaptively determined to enhance the quality of sparse representation. The third contribution is a strategy for designing directionally-structured dictionaries via subspace projections. Experimental results show that this strategy improves the quality of sparse representation at reduced computational complexity. The fourth and major contribution is a strategy for residual component-based multiple structured dictionary learning. In this work, we show that a signal and its residual components subject to a sparse coding algorithm do not necessarily follow the same model, as commonly assumed in the multiple dictionary approaches in the literature so far. Accordingly, we propose a mechanism whereby training signal can potentially contribute to the learning of several dictionaries, based on the structure of each of its residual components. This strategy is shown to significantly improve the representation quality while using compact dictionaries. The final contribution in this thesis aims at improving the representation quality of a learned dictionary by performing a second dictionary learning pass over the residual components of the training set. Simulations show that this learning strategy improves the quality of sparse representation. Keywords: Sparse representation, dictionary learning, multiple dictionaries, residual components, structured dictionaries.