Algorithm 1:

Spilt augmented lagrangian shrinkage algorithm (SALSA) [41]

AIM: Estimation of $x_d^{(k)}$ which minimizes Eq. 20

INPUT: M, $x_{\mathit{est}}^{(k)}$, λl 1, α, Nit = 5000.

OUTPUT: $x_d^{(k)}$

Initialize: d = 0

1. $x_d^{(k)}={\mathbf{M}}^T{x}_{\mathit{est}}^{(k)}$ and F = M T M + α I

for i = 1, 2,···Nit

2. $v=\text{soft}(x_d^{(k)}+d,(0.5{\lambda }_{l1})/\alpha )-\text{d}$

3. $x_d^{(k)}={\mathbf{F}}^{-1}({\mathbf{M}}^T{x}^{(k)}+\alpha v)$

4. $d=x_d^{(k)}-v$

end