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In many applications one may acquire a composition of several signals thatmay be corrupted by noise, and it is a challenging problem to reliably separatethe components from one another without sacrificing significant details. Addingto the challenge, in a compressive sensing framework, one is given only anundersampled set of linear projections of the composite signal. In this paper,we propose using the Dantzig selector model incorporating an overcompletedictionary to separate a noisy undersampled collection of composite signals,and present an algorithm to efficiently solve the model.  The Dantzig selector is a statistical approach to finding a solution to anoisy linear regression problem by minimizing the $\ell_1$ norm of candidatecoefficient vectors while constraining the scope of the residuals. If theunderlying coefficient vector is sparse, then the Dantzig selector performswell in the recovery and separation of the unknown composite signal. In thefollowing, we propose a proximity operator based algorithm to recover andseparate unknown noisy undersampled composite signals through the Dantzigselector. We present numerical simulations comparing the proposed algorithmwith the competing Alternating Direction Method, and the proposed algorithm isfound to be faster, while producing similar quality results. Additionally, wedemonstrate the utility of the proposed algorithm in several experiments byapplying it in various domain applications including the recovery ofcomplex-valued coefficient vectors, the removal of impulse noise from smoothsignals, and the separation and classification of a composition of handwrittendigits.

Separation of undersampled composite signals using the Dantzig selector with overcomplete dictionaries