  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.
