Algorithm 2.

Solver of the de-noising problem defined by Eqn. (60)

Input: y, β

Output: $\widehat{\mathbf{x}}$

1: $[{\mathbf{a}}^{\left(1\right)},{\mathbf{b}}^{\left(1\right)},{\mathbf{c}}^{\left(1\right)}]\leftarrow [0_{(N_1-1)\times N_2\times N_3},0_{N_1\times (N_2-1)\times N_3},0_{N_1\times N_2\times (N_3-1)}][{\mathbf{d}}^{\left(0\right)},{\mathbf{e}}^{\left(0\right)},{\mathbf{f}}^{\left(0\right)}]\leftarrow [0_{(N_1-1)\times N_2\times N_3},0_{N_1\times (N_2-1)\times N_3},0_{N_1\times N_2\times (N_3-1)}]t^{\left(1\right)}=1$

2: for ζ = 1 to Z do

3:    $[{\mathbf{d}}^{\left(\zeta \right)},{\mathbf{e}}^{\left(\zeta \right)},{\mathbf{f}}^{\left(\zeta \right)}]\leftarrow {\mathcal{P}}_p\left\{[{\mathbf{a}}^{\left(\zeta \right)},{\mathbf{b}}^{\left(\zeta \right)},{\mathbf{c}}^{\left(\zeta \right)}]+{\left(6\beta \right)}^{-1}{\mathcal{P}}_l^T\left\{{\mathcal{P}}_c\right\{\mathbf{y}-0.5\beta {\mathcal{P}}_l\{{\mathbf{a}}^{\left(\zeta \right)},{\mathbf{b}}^{\left(\zeta \right)},{\mathbf{c}}^{\left(\zeta \right)}\}\left\}\right\}\right\}$

4:    $t^{(\zeta +1)}\leftarrow 1+0.5\sqrt{1+4{\left(t^{\left(\zeta \right)}\right)}^2}$

5:    $[{\mathbf{a}}^{(\zeta +1)},{\mathbf{b}}^{(\zeta +1)},{\mathbf{c}}^{(\zeta +1)}]\leftarrow (t^{\left(\zeta \right)}-1)∕t^{(\zeta +1)}[{\mathbf{d}}^{\left(\zeta \right)}-{\mathbf{d}}^{(\zeta -1)},{\mathbf{e}}^{\left(\zeta \right)}-{\mathbf{e}}^{(\zeta -1)},{\mathbf{f}}^{\left(\zeta \right)}-{\mathbf{f}}^{(\zeta -1)}]$

6: end for

7: $\widehat{\mathbf{x}}\leftarrow {\mathcal{P}}_c\{\mathbf{y}-\lambda {\mathcal{P}}_l\{{\mathbf{d}}^{\left(Z\right)},{\mathbf{e}}^{\left(Z\right)},{\mathbf{f}}^{\left(Z\right)}\left\}\right\}$