Structured sampling of structured signals

The paper considers structured sampling of structured signals, more specifically, using block diagonal (BD) measurement matrices to sense signals with uniform partitions that share the same sparsity profile. This model arises in distributed compressive sensing systems. In general, the fact that the number of nonzero entries in the measurement matrix is smaller than in a dense matrix leads to the need for more measurements. However, taking advantage of a certain structure in the sparse signal allows one to relax the conditions on the measurement matrix for the restricted isometry property (RIP) to hold, thus allowing for higher compression rate. We systematically provide guarantees for a unique solution, and also an efficient recovery method. The analysis relies on the RIP of the random BD matrix for signals in a particular union of subspaces. Also, we show how our theoretical results can be used to analyze the multiple measurement vector (MMV) problem.