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The MUSIC Algorithm for Sparse Objects: A Compressed Sensing Analysis

The MUSIC algorithm, with its extension for imaging sparse {\em extended} objects, is analyzed by compressed sensing (CS) techniques. The notion of restricted isometry property (RIP) and an upper bound on the restricted isometry constant (RIC) are employed to establish sufficient conditions for the exact localization by MUSIC with or without the presence of noise. In the noiseless case, the sufficient condition gives an upper bound on the numbers of random sampling and incident directions necessary for exact localization. In the noisy case, the sufficient condition assumes additionally an upper bound for the noise-to-object ratio in terms of the RIC and the condition number of objects. Rigorous comparison of performance between MUSIC and the CS minimization principle, Lasso, is given. In general, the MUSIC algorithm guarantees to recover, with high probability, $s$ scatterers with $n=\cO(s^2)$ random sampling and incident directions and sufficiently high frequency. For the favorable imaging geometry where the scatterers are distributed on a transverse plane MUSIC guarantees to recover, with high probability, $s$ scatterers with a median frequency and $n=\cO(s)$ random sampling/incident directions. Numerical results confirm that the Lasso outperforms MUSIC in the well-resolved case while the opposite is true for the under-resolved case. The latter effect indicates the superresolution capability of the MUSIC algorithm. Another advantage of MUSIC over the Lasso as applied to imaging is the former's flexibility with grid spacing and guarantee of {\em approximate} localization of sufficiently separated objects in an arbitrarily fine grid. The error can be bounded from above by $\cO(λs)$ for general configurations and $\cO(λ)$ for objects distributed in a transverse plane.