  We present a novel compressed sensing recovery algorithm - termed Bayesian
Optimal Structured Signal Approximate Message Passing (BOSSAMP) - that jointly
exploits the prior distribution and the structured sparsity of a signal that
shall be recovered from noisy linear measurements. Structured sparsity is
inherent to group sparse and jointly sparse signals. Our algorithm is based on
approximate message passing that poses a low complexity recovery algorithm
whose Bayesian optimal version allows to specify a prior distribution for each
signal component. We utilize this feature in order to establish an
iteration-wise extrinsic group update step, in which likelihood ratios of
neighboring group elements provide soft information about a specific group
element. Doing so, the recovery of structured signals is drastically improved.
We derive the extrinsic group update step for a sparse binary and a sparse
Gaussian signal prior, where the nonzero entries are either one or Gaussian
distributed, respectively. We also explain how BOSSAMP is applicable to
arbitrary sparse signals. Simulations demonstrate that our approach exhibits
superior performance compared to the current state of the art, while it retains
a simple iterative implementation with low computational complexity.
