I am working Compressive Sensing for my research and would like to understand the basics of CS Reconstruction. Though MatLab codes are available but I would like to write implement it myself.
There is the original 1993 paper by Mallat & Zhang - "Matching Pursuits With Time-Frequency Dictionaries". You might find that one useful.
Matching persuit is good but old technique. So for reconstruction you can use convex optimization. Its toolbox is available in matlab.
http://cvxr.com/cvx/
This is the link to understand the basic of compressive sensing and reconstruction.
If you want the entire process...you can follow the paper
http://link.springer.com/chapter/10.1007/978-981-10-0767-5_2
This might be helpful:
http://www.zess.uni-siegen.de/myoffer/files/file_5_52170211052012.pdf
The following book is very good. "Sparse and Redundant Representations" by Michael Elad.
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