Joint sparse recovery based on variances

Much research has recently been devoted to sparse signal recovery and image reconstruction from multiple measurement vectors. The assumption that the underlying signals or images have some common features with sparse representation suggests that using a joint sparsity approach to recover each individual signal or image can be more effective than recovering each signal or image separately using standard sparse recovery techniques. Joint sparsity reconstruction is typically performed using \ell 2,1-minimization, and although the process yields better results than separate recoveries, the inherent coupling makes the algorithm computationally intensive, since it is difficult to parallelize. In this investigation, we first observe that the elementwise variance of the signals convey information about their shared support. This observation motivates us to introduce a weighted \ell 1-joint sparsity algorithm, where the weights depend on the calculated variance. Specifically, the \ell 1-minimization should be more heavily penalized in regions where the corresponding variance is small, since it is likely there is no signal there. We demonstrate that our new method, which we refer to as variance-based joint sparse recovery, is more accurate and cost efficient. Applications in sparse signal recovery, parallel magnetic resonance imaging, and edge detection are considered.