Sparse Representation over Multiple Learned Dictionaries via the Gradient Operator Properties with Application to Single-Image Super-Resolution

Single-image super-resolution is an ill-posed inverse problem that requires effective regularization. Super-resolution over learned dictionaries offers a successful framework for efficiently solving this problem exploiting the sparsity as regularizer. It is well acknowledged that the success of sparse representation comes as a direct consequence of the representation power of learned dictionaries. Along this trend, this thesis considers the problem of super-resolution via sparse representation, where representation is done over a set of compact high and low resolution cluster dictionaries. Such an approach inevitably calls for a model selection criteria both in the learning and reconstruction stages. The model selection criteria should have scale-invariance property so that link between low resolution and high resolution feature spaces is properly established. The main contribution in this thesis is to employ two approximately scale-invariant patch measures for the classification of image patches in the learning and reconstruction stages. These are the sharpness measure and the dominant phase angle defined in terms of the magnitude and phase of the gradient operator, respectively. These measures are empirically shown to have acceptable degrees of scale-invariance. i.e. sharpness measure and the dominant phase angle do not significantly change for two consecutive resolution levels. This invariance to a large extent ensures that model selection is correct in the reconstruction stage where one only knows the low resolution patch. Three super-resolution algorithms are proposed based on selective sparse coding over cluster dictionaries with the proposed measures, applied individually and combined together. In each algorithm, training data is clustered and a coupled dictionary pairs are learned for each cluster. In the learning stage any standard coupled dictionary learning algorithm can be used. In the reconstruction stage, the most appropriate dictionary pair is selected for each low resolution patch and the sparse coding coefficients with respect to the low resolution dictionary are calculated. The link between the low and high resolution feature spaces is the fact that the sparse representation coefficients of the high and low resolution patches are approximately equal. For the case of multiple structured dictionaries this link is also strengthened since the dictionaries are learned for structured feature spaces. Imposing this link, a high resolution patch estimate is obtained by multiplying the sparse coding coefficients with the corresponding high resolution dictionary. Quantitative and qualitative experiments conducted over natural images validate that each of the proposed algorithms is superior to the standard case of using a single dictionary pair, and is competitive with the state-of-the-art super-resolution algorithms. From the rate-distortion perspective, it is shown that computational complexity (rate) can be reduced significantly without a significant loss in quality. This is achieved due to the fact that the proposed clustering criterion lends itself nicely for identifying the patches that are un-sharp (with low frequency content). Such patches can be handled effectively using simple algorithm (computationally much less complex) such as bicubic interpolation instead of computationally expensive sparse representation. Specifically for a typical image, 73.03 % of the patches can be handled using bicubic interpolation without significant degradation in quality. Keywords: Single image super-resolution, sparse representation, dictionary learning, sharpness measure, gradient phase angle, coupled dictionaries.