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

In many applications, one may acquire a composition of several signals that may be corrupted by noise, and it is a challenging problem to reliably separate the components from one another without sacrificing significant details. Adding to the challenge, in a compressive sensing framework, one is given only an undersampled set of linear projections of the composite signal. In this study, the authors propose using the Dantzig selector model incorporating an overcomplete dictionary 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 a noisy linear regression problem by minimising the ℓ1 norm of candidate coefficient vectors while constraining the scope of the residuals. The Dantzig selector performs well in the recovery and separation of an unknown composite signal when the underlying coefficient vector is sparse. They propose a proximity operator-based algorithm to recover and separate unknown noisy undersampled composite signals using the Dantzig selector. They present numerical simulations comparing the proposed algorithm with the competing alternating direction method, and the proposed algorithm is found to be faster, while producing similar quality results. In addition, they demonstrate the utility of the proposed algorithm by applying it in various applications including the recovery of complexvalued coefficient vectors, the removal of impulse noise from smooth signals, and the separation and classification of a composition of handwritten digits.