Sparse recovery with fusion frames and bounded orthonormal systems

In this paper, we consider the recovery of fusion frame sparse signals from incomplete structured measurements. This signal has energy only in a few subspaces of the fusion frame. The sufficient condition for uniform recovery of fusion frame sparse signal is a fusion restricted isometry property (FRIP) which is the fusion frame version of restricted isometry property (RIP) in the standard compressed sensing (CS) settings. Different classes of random matrices are well known to satisfy FRIP, but such matrices do not possess any structure that emerged from the practical sensing systems. In this paper, we give the theoretical results to bridge this gap by proving that a random sampling matrix associated with a bounded orthonormal system satisfies the FRIP with high probability. In detail, we established the uniform recovery guarantees for fusion frames and structured measurements, which may be the beginning of practical applications relating to the fusion frame sparse signal model.