TABLE I.

Ordered-subset pixel-wise separable quadratic surrogate (OS-PWSQS) algorithm outline.

1) Choose πmij factors using (33).

2) Initialize x(0) using the results of the image-domain method [1].

3) For each iteration d = 1, …, Diter

a) For each subset (subiteration) q = 1, …, Qiter

i) n = d + q/Qiter

ii) Compute gradient of the data fidelity term L̇qj.

$${\mathrm{L̇}}_{qj}=\sum _{i\in {𝒮}_q}\sum _{m=1}^{M_0}{a}_{mij}{\nabla ̵}_{{\mathit{s}}_{im}}{t}_{im}({\mathit{s}}_{im}^{(n)}),{\nabla ̵}_{{\mathit{s}}_{im}}{t}_{im}({\mathit{s}}_{im}^{(n)})={\nabla ̵}_{{\mathit{s}}_{im}}{ȳ}_{im}({\mathit{s}}_{im}^{(n)})-Y_{im}{\nabla ̵}_{{\mathit{s}}_{im}}{h}_{im}({\mathit{s}}_{im}^{(n)}),$$

where ∇̵simȳim and ∇̵sim him are given in (50) and (52) respectively, and

$${\mathit{s}}_{im}^{(n)}=(s_{im1}({\mathit{x}}^{(n)}),\dots ,s_{im{L}_0}({\mathit{x}}^{(n)})),s_{iml}({\mathit{x}}^{(n)})=\sum _{j=1}^{N_{\mathrm{p}}}{a}_{mij}{\mathit{x}}_l^{(n)},l=1,\dots ,L_0.$$

iii) Compute gradient of penalty term Ṙqj.

$${Ṙ}_{qj}=\frac{1}{Q}\left({\beta }_1\frac{\partial }{\partial x_{1j}}{R}_1({\mathit{x}}_1){|}_{{\mathit{x}}_1={\mathit{x}}_1^{(n)}},\dots ,{\beta }_{L_0}\frac{\partial }{\partial x_{L_{0}j}}{R}_L({\mathit{x}}_{L_0}){|}_{{\mathit{x}}_{L_0}={\mathit{x}}_{L_0}^{(n)}}\right)$$

$$\frac{\partial }{\partial x_{lj}}{R}_l({\mathit{x}}_l){|}_{{\mathit{x}}_l={\mathit{x}}_l^{(n)}}=\sum _{k\in {𝒩}_{lj}}{\kappa }_{lj}{\kappa }_{lk}{\mathrm{\psi ̇}}_l(x_{lj}^{(n)}-x_{lk}^{(n)}),l=1,\dots ,L_0$$

iv) Compute L0 × L0 curvature matrices ${\mathit{D}}_{qj}^{(n)}$.

$${\mathit{D}}_{qj}^{(n)}={\mathit{D}}_{\mathrm{L̄},qj}^{(n)}+\frac{1}{Q}{\mathit{D}}_{R,j}^{(n)},{\mathit{D}}_{\mathrm{L̄},qj}^{(n)}=\sum _{i\in {𝒮}_q}\sum _{m=1}^{M_0}\frac{a_{mij}^2}{{\pi }_{mij}}{\mathit{C̆}}_{im}^{(n)},$$

where ${\mathit{C̆}}_{im}^{(n)}$ and ${\mathit{D}}_{R,j}^{(n)}$ are defined in (27) and (40) respectively.

v) Compute H and p using (47) and (48), i.e.,

$$\mathit{H}={\mathit{D}}_{qj}^{(n)},\mathit{p}={\mathrm{L̇}}_{qj}+{Ṙ}_{qj}-({\mathit{x}}_j^{(n)})\prime {\mathit{D}}_{qj}^{(n)}.$$

vi) For each tuple ω ∈ Ω

A) Form ${\mathit{x}}_j^{(n)}(\omega )$, H(ω), p(ω) by extracting elements in ${\mathit{x}}_j^{(n)}$, H and p with indexes corresponding to ω respectively.

B) Obtain minimizer x̂j(ω) of the QP problem in (49) using GSMO.

C) Compute and store minimal surrogate function value ${\varphi }_j^{(n)}({\mathit{x̂}}_j(\omega ))$ using (49).

End

vii) Determine optimal ω̂ by comparing all ${\varphi }_j^{(n)}({\mathit{x̂}}_j(\omega ))$, i.e.,

$$\mathrm{\omega ̂}=\underset{\omega \in \mathrm{\Omega }}{\text{arg min}}{\varphi }_j^{(n)}({\mathit{x̂}}_j(\omega )).$$

viii) Obtain x̂j ≡ x̂j(ω̂) with padded zeros for l ∉ ω.

ix) Update all pixels x(n+1/Qiter) = x̂ = (x̂1, …, x̂j, …, x̂Np).

End

x(n+1) = x(n+Qiter/Qiter).

End