Single Image Super Resolution via Manifold Approximation

Image super-resolution remains an important research topic to overcome the limitations of physical acquisition systems, and to support the development of high resolution displays. Previous example-based super-resolution approaches mainly focus on analyzing the co-occurrence properties of lowresolution and high-resolution patches. Recently, we proposed a novel single image super-resolution approach based on linear manifold approximation of the high-resolution image-patch space [1]. The image super-resolution problem is then formulated as an optimization problem of searching for the best matched high resolution patch in the manifold for a given low-resolution patch. We developed a novel technique based on the `1 norm sparse graph to learn a set of low dimensional affine spaces or tangent subspaces of the high-resolution patch manifold. The optimization problem is then solved based on the learned set of tangent subspaces. In this paper, we build on our recent work as follows. First, we consider and analyze each tangent subspace as one point in a Grassmann manifold, which helps to compute geodesic pairwise distances among these tangent subspaces. Second, we develop a min-max algorithm to select an optimal subset of tangent subspaces. This optimal subset reduces the computational cost while still preserving the quality of the reconstructed high-resolution image. Third, and to further achieve lower computational complexity, we perform hierarchical clustering on the optimal subset based on Grassmann manifold distances. Finally, we analytically prove the validity of the proposed Grassmann-distance based clustering. A comparison of the obtained results with other state-of-the-art methods clearly indicates the viability of the new proposed framework.