MUSAI- ${L}_{{1/2}}$ : MUltiple Sub-Wavelet-Dictionaries-Based Adaptively-Weighted Iterative Half Thresholding Algorithm for Compressive Imaging

Compressive sensing (CS) is an effective approach for compressive recovery, such as the imaging problems. It aims at recovering sparse signal or image from a small number of under-sampled data by taking advantage of the sparse signal structure.
$L_{1/2}$
-norm regularization in CS framework has been considered as a typical nonconvex relaxation approach to approximate the optimal sparse solution, and can obtain stronger sparse solution than
$L_{1}$
-norm regularization. However, it is very difficult to solve the nonconvex optimization problem efficiently resulted by
$L_{1/2}$
-norm. In order to improve the performance of
$L_{1/2}$
-norm regularization and extend the application, we propose a multiple sub-wavelet dictionaries-based adaptively-weighted iterative half thresholding algorithm (MUSAI-
$L_{1/2}$
) for sparse signal recovery. In particular, we propose an adaptive-weighting scheme for the regularization parameter to control the tradeoff between the fidelity term and the multiple sub-regularization terms. Numerical experiments are conducted on some typical compressive imaging problems to demonstrate that the proposed MUSAI-
$L_{1/2}$
algorithm can yield significantly improved the recovery performance compared with the prior work.