This page is dedicated to the Alternating l1 method for Compressed Sensing
The paper related to this research is here
A python program can be found here: it just implements the alternating l1 method and compares the result with the plain l1. It uses cvxopt. Monte-Carlo experiments comparing the Alternating l1 approach with the Reweighted l1 of Candès, Wakin and Boyd and the IRLS of Chartrand and Yin is available here.
Case where plain l1 and Alternating l1 worked fine
Case where none worked
Case where plain l1 did not work but Alternating l1 did