Compressed sensing signal recovery via forward–backward pursuit

• We propose a greedy CS recovery method with iterative forward and backward steps.
• The forward step being larger than the backward one, the support grows iteratively.
• The sparsity level is not required a priori in contrast to SP and CoSaMP.
• Recovery accuracy is improved over BP, SP and OMP, in run times are equal to OMP.
• We obtain better recovery accuracy than BP on a sparse image recovery problem.

Recovery of sparse signals from compressed measurements constitutes an ℓ0ℓ0 norm minimization problem, which is unpractical to solve. A number of sparse recovery approaches have appeared in the literature, including ℓ1ℓ1 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.