Compressive sensing using the log-barrier algorithm with complex-valued measurements

Compressive sensing constitutes a series of theories and algorithms that, under certain conditions, allow one to reconstruct a signal from limited linear measurements, based on knowledge about a domain where the signal is sparse. The l(0)-minimization represents the ideal approach for reconstruction, as it searches for the sparsest representation that explains the measurements, but it is an NP-hard procedure. Fortunately, the l(1)-minimization can frequently be used as an approximation to the l(0) approach, and the problem can be solved by some algorithms in polynomial time. One of the optimization problems formulated in this context corresponds to finding the sparsest solution subject to a quadratic constraint, such as in the log-barrier algorithm provided in the well-known l(1)-Magic package. However, in this particular problem, real-valued signals are reconstructed from real-valued data. In this paper, we show how we can reconstruct sparse real-valued signals from noisy complex-valued measurements. The problem is still posed as a second-order cone program by means of the log barrier method. However, new modifications in the Newton's step equations and in the l(1)-Magic codes are necessary to fit the complex-valued data. In addition, in order to evaluate the reconstructions using complex data, we present the results of numerical experimentation and evaluate the performance of the signal reconstruction in terms of signal-to-error ratios. The provided method is well-suited for real applications involving the acquisition of complex-valued data, such as magnetic resonance imaging and computed tomography.