Correlated Sparse Bayesian Learning for Recovery of Block Sparse Signals With Unknown Borders

We consider the problem of recovering complex-valued block sparse signals with unknown borders. Such signals arise naturally in numerous applications. Several algorithms have been developed to solve the problem of unknown block partitions. In pattern-coupled sparse Bayesian learning (PCSBL), each coefficient involves its own hyperparameter and those of its immediate neighbors to exploit the block sparsity. Extended block sparse Bayesian learning (EBSBL) assumes the block sparse signal consists of correlated and overlapping blocks to enforce block correlations. We propose a simpler alternative to EBSBL and reveal the underlying relationship between the proposed method and a particular case of EBSBL. The proposed algorithm uses the fact that immediate neighboring sparse coefficients are correlated. The proposed model is similar to classical sparse Bayesian learning (SBL). However, unlike the diagonal correlation matrix in conventional SBL, the unknown correlation matrix has a tridiagonal structure to capture the correlation with neighbors. Due to the entanglement of the elements in the inverse tridiagonal matrix, instead of a direct closed-form solution, an approximate solution is proposed. The alternative algorithm avoids the high dictionary coherence in EBSBL, reduces the unknowns of EBSBL, and is computationally more efficient. The sparse reconstruction performance of the algorithm is evaluated with both correlated and uncorrelated block sparse coefficients. Simulation results demonstrate that the proposed algorithm outperforms PCSBL and correlation-based methods such as EBSBL in terms of reconstruction quality. The numerical results also show that the proposed correlated SBL algorithm can deal with isolated zeros and nonzeros as well as block sparse patterns.