Reconstruction of Compressively Sensed Images using Convex Tikhonov Sparse Dictionary Learning and Adaptive Spectral Filtering

Abstract: Sparse representation using over-complete dictionaries have shown to produce good quality results in various image processing tasks. Dictionary learning algorithms have made it possible to engineer data adaptive dictionaries which have promising applications in image compression and image enhancement. The most common sparse dictionary learning algorithms use the techniques of matching pursuit and K-SVD iteratively for sparse coding and dictionary learning respectively. While this technique produces good results, it requires a large number of iterations to converge to an optimal solution. In this article, we use a closed form stabilized convex optimization technique for both sparse coding and dictionary learning. The approach results in providing the best possible dictionary and the sparsest representation resulting in minimum reconstruction error. Once the image is reconstructed from the compressively sensed samples, we use adaptive frequency and spatial filtering techniques to move towards exact image recovery. It is clearly seen from the results that the proposed algorithm provides much better reconstruction results than conventional sparse dictionary techniques for a fixed number of iterations. Depending inversely upon the number of details present in the image, the proposed algorithm reaches the optimal solution with a significantly lower number of iterations. Consequently, high PSNR and low MSE is obtained using the proposed algorithm for our compressive sensing framework.