Compressed sensing signal recovery via forward-backward pursuit

Recovery of sparse signals from compressed measurements constitutes an l0 norm minimization problem, which is unpractical to solve. A number of sparse recovery approaches have appeared in the literature, including l1 minimization techniques, greedy pursuit algorithms, Bayesian methods and nonconvex optimization techniques among others. This manuscript introduces a novel two stage greedy approach, called the Forward-Backward Pursuit (FBP). FBP is an iterative approach where each iteration consists of consecutive forward and backward stages. The forward step first expands the support estimate by the forward step size, while the following backward step shrinks it by the backward step size. The forward step size is larger than the backward step size, hence the initially empty support estimate is expanded at the end of each iteration. Forward and backward steps are iterated until the residual power of the observation vector falls below a threshold. This structure of FBP does not necessitate the sparsity level to be known a priori in contrast to the Subspace Pursuit or Compressive Sampling Matching Pursuit algorithms. FBP recovery performance is demonstrated via simulations including recovery of random sparse signals with different nonzero coefficient distributions in noisy and noise-free scenarios in addition to the recovery of a sparse image.