The choice of an efficient algorithm mainly depends on your application needs (Dictionary Learning, Denoising, Inverse Problem Solving/Image Restoration, etc.).

Neverthless, without any more information, I suggest that you jointly follow these 2 references in order to help you in solving your problem with computational efficiency:

• Abo-Zahhad et al., " Compressive Sensing Algorithms for Signal Processing Applications: A Survey", 2015 - https://file.scirp.org/pdf/IJCNS_2015060910194817.pdf
• Bajwa et al., "Toeplitz-Structured Compressed Sensing Matrices", 2007 - http://www2.ece.umn.edu/users/jdhaupt/publications/ssp07_toep_cs_matrices.pdf

Regards

Thank you Sir for responding to my question. Actually what happens, if I try to use ISTA/FISTA for performing patch based reconstruction of Super-resolution image as done in the paper..Article Image Super-Resolution Via Sparse Representation

it requires much time as for each patch it solves the sparse representation L1-minimization problem. Since I am not working on the whole image at once, so setting a high iteration number in ISTA/FISTA algorithm is not preferred. Is there any other efficient method for such requirements, i.e. solves patch wise L1-minimization for an image with faster result.

Regards

Helal

DCT-based L1-regularization for robust data interpolation might be an efficient solution. Follow:

• Tepper et al., "L1 Splines for Robust, Simple, and Fast Smoothing of Grid Data", 2012 - https://arxiv.org/pdf/1208.2292

In search of efficiency, switching to L2-Regularization could be interesting. To help you choose the best compromise, check out:

• Van Den Doel et al., "The lost honour of L2-based regularization", 2012 - http://math.boisestate.edu/~calhoun/GEMRES/PNWNAS/Ascher_talk.pdf

Regards