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Summary form only given. This paper has described an improved algorithm for reconstructing sparse binary signals using compressive sensing. The algorithm is based on the reweighted lq norm optimization algorithm, but with the important additional operation of bounding in each round of the interior-point method iteration, and progressive reduction of q. Experimental results confirm that the algorithm performs well both in terms of the ability to recover an input signal as well as in terms of speed. We also found that both the progressive reduction and the bounding are integral to the improvement in performance. Future work includes extending this approach to Gaussian distributed, as opposed to binary inputs.