Sparse coding means the expression of inferred data by linear combination of a small number of basis vectors, and redundant data with correlation, such as images, are possible to be sparsely coded. It is proved that sparsely coded information can be inferred from less observed data than ever, and it is known as one of the most effective signal processing methods.
In this project, taking as examples (1) restoration of images degraded by noise, and (2) single image super resolution (the increase of resolution realized from only one image), we analytically evaluated their typical performance, averaged over various kinds of images and levels of noise, with the help of the replica method of statistical physics. This clarified the uniqueness of the solution and the optimal relation between sparsity and the aspect ratio of dictionary matrices which minimizes mean squared error and so on. Now, we are making research of the optimal feature extraction method for single image super resolution.