Table 1.

ASALSA-CF Algorithm

Aim: Estimation of x by solving Equation 7

Input: A, b, λ, α, max_iter

Output: x

Initialize ADMM parameter d = 0.

1. Calculate backprojection solution (x = ATb), CF = AT b2

for k = 1, 2, … max_iter

2. Optimize Equation 9: v = soft(x + d, (0.5 × λ)/α)

3. Optimize Equation 8: x1 = ${\mathit{S}}_{\mathit{k}}\mathbf{\left(}\mathbf{(}{\mathit{C}}_{\mathit{k}}^{\mathit{T}}{\mathit{C}}_{\mathit{k}}\hspace{0.17em}\mathit{+}\hspace{0.17em}\mathit{\lambda }{\mathit{I}}_{\mathit{k}}{\mathbf{)}}^{\mathit{-}\mathit{1}}{\mathit{\beta }}_{\mathbf{1}}{\mathit{C}}_{\mathit{k}}^{\mathit{T}}{\mathit{e}}_{\mathbf{1}}\mathbf{\right)}$, where matrix C is obtained by Lanczos bidiagonalization of AT with k iterations and ${\mathit{\beta }}_{\mathbf{1}}$ is the euclidean norm of (ATb + α* (v − d)).

xrecon = ${\mathit{M}}_{\mathit{k}}\mathbf{\left(}\mathbf{(}{\mathit{H}}_{\mathit{k}}^{\mathit{T}}{\mathit{H}}_{\mathit{k}}\hspace{0.17em}\mathit{+}\hspace{0.17em}\mathit{\lambda }{\mathit{I}}_{\mathit{k}}{\mathbf{)}}^{\mathit{-}\mathit{1}}{\mathit{\beta }}_{\mathbf{1}}{\mathit{H}}_{\mathit{k}}^{\mathit{T}}{\mathit{e}}_{\mathbf{1}}\mathbf{\right)}$, where ${\mathit{\beta }}_{\mathbf{2}}$ is the Euclidean norm of x1.

4. Update ADMM parameter: d = d + xrecon − v

end

5. CF = (x2/CF)

6. x = xrecon. × CF