Image super-resolution is defined as a class of techniques that enhance the spatial resolution of images. Super-resolution methods can be subdivided in single and multi image methods. This thesis focuses on developing algorithms based on mathematical theories for single image super-resolution problems. Indeed, in order to estimate an output image, we adopt a mixed approach: i.e., we use both a dictionary of patches with sparsity constraints (typical of learning-based methods) and regularization terms (typical of reconstruction-based methods). Although the existing methods already perform well, they do not take into account the geometry of the data to: regularize the solution, cluster data samples (samples are often clustered using algorithms with the Euclidean distance as a dissimilarity metric), learn dictionaries (they are often learned using PCA or K-SVD). Thus, state-of-the-art methods still suffer from shortcomings. In this work, we proposed three new methods to overcome these deficiencies. First, we developed SE-ASDS (a structure tensor based regularization term) in order to improve the sharpness of edges. SE-ASDS achieves much better results than many state-of-the-art algorithms. Then, we proposed AGNN and GOC algorithms for determining a local subset of training samples from which a good local model can be computed for reconstructing a given input test sample, where we take into account the underlying geometry of the data. AGNN and GOC methods outperform spectral clustering, soft clustering, and geodesic distance based subset selection in most settings. Next, we proposed aSOB strategy which takes into account the geometry of the data and the dictionary size. The aSOB strategy outperforms both PCA and PGA methods. Finally, we combine all our methods in a unique algorithm, named G2SR. Our proposed G2SR algorithm shows better visual and quantitative results when compared to the results of state-of-the-art methods.
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