The MUSIC algorithm for sparse objects: a compressed sensing analysis

The multiple signal classification (MUSIC) algorithm, and its extension for imaging sparse extended objects, with noisy data is analyzed by compressed sensing (CS) techniques. A thresholding rule is developed to augment the standard MUSIC algorithm. 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 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 dynamic range of objects. This bound points to the super-resolution capability of the MUSIC algorithm. Rigorous comparison of performance between MUSIC and the CS minimization principle, basis pursuit denoising (BPDN), is given. In general, the MUSIC algorithm guarantees to recover, with high probability, s scatterers with 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 random sampling/incident directions. Moreover, for the problems of spectral estimation and source localizations both BPDN and MUSIC guarantee, with high probability, to identify exactly the frequencies of random signals with the number of sampling times. However, in the absence of abundant realizations of signals, BPDN is the preferred method for spectral estimation. Indeed, BPDN can identify the frequencies approximately with just one realization of signals with the recovery error at worst linearly proportional to the noise level. Numerical results confirm that BPDN outperforms MUSIC in the well-resolved case while the opposite is true for the under-resolved case, giving abundant evidence for the super-resolution capability of the MUSIC algorithm. Another advantage of MUSIC over BPDN is the former's flexibility with grid spacing and the guarantee of approximate localization of sufficiently separated objects in an arbitrarily refined grid. The localization error is bounded from above by for general configurations and by for objects distributed in a transverse plane, in line with physical intuition.