Compressive sensing signal reconstruction by weighted median regression estimates

In this paper, we address the compressive sensing signal reconstruction problem by solving an ℓ₀-regularized Least Absolute Deviation (LAD) regression problem. A coordinate descent algorithm is developed to solve this ℓ₀-LAD optimization problem leading to a two-stage operation for signal estimation and basis selection. In the first stage, an estimation of the sparse signal is found by a weighted median operator acting on a shifted-and-scaled version of the measurement samples with weights taken from the entries of the projection matrix. The resultant estimated value is then passed to the second stage that tries to identify whether the corresponding entry is relevant or not. This stage is achieved by a hard threshold operator with adaptable thresholding parameter that is suitably tuned as the algorithm progresses.