Table 1.

Glossary of terms and pseudocode of SPS optimization method for multiresolution PWLS. Ordered-subsets are omitted for simplification.

Definitions
l̂B, l̂N	forward-projected line integrals onto binned and native detector pixels
lB, lN	binned and native preprocessed line integral data
wB, wN	diagonal entries in weighting matrices WB and WN.
c	precomputed PWLS curvature, c = [W]+
1	volume covering the entire reconstruction FOV of all 1's.
L̇C, L̇F	derivatives of data-fit surrogates for voxels of μ̂C and μ̂F.
dC, dF	curvatures of data-fit surrogates for voxels of μ̂C and μ̂F.
ψ̇, ωψk	gradient and curvature of the penalty function ψ.
ṘC, ṘF	derivatives of penalty surrogates for voxels of μ̂C and μ̂F.
rC, rF	curvatures of penalty surrogates for voxels of μ̂C and μ̂F.

Algorithm
Precompute ${\mathrm{d}}_{\mathrm{C}}={\mathrm{A}}_{\mathrm{C}\mathrm{B}}^{\mathrm{T}}{[\mathrm{A}\mathbf{1}\circ \mathrm{c}]}_{\mathrm{B}}+{\mathrm{A}}_{\mathrm{C}\mathrm{N}}^{\mathrm{T}}{[\mathrm{A}\mathbf{1}\circ \mathrm{c}]}_{\mathrm{N}}$ for iteration n=1, …, N	${\mathrm{d}}_{\mathrm{F}}={\mathrm{A}}_{\mathrm{F}\mathrm{N}}^{\mathrm{T}}{[\mathrm{A}\mathbf{1}\circ \mathrm{c}]}_{\mathrm{N}}$
Coarse Grid Update	Fine Grid Update
l̂B = ACBμ̂C	l̂N = ACNμ̂C + AFNμ̂F
ḣB = wB ∘ (l̂B−lB)	ḣN = wN ∘ (l̂N − lN)
${\dot{\mathrm{L}}}_{\mathrm{C}}={\mathrm{A}}_{\mathrm{C}\mathrm{B}}^{\mathrm{T}}{ḣ}_{\mathrm{B}}+{\mathrm{A}}_{\mathrm{C}\mathrm{N}}^{\mathrm{T}}{ḣ}_{\mathrm{N}}$	${\dot{\mathrm{L}}}_{\mathrm{F}}={\mathrm{A}}_{\mathrm{F}\mathrm{N}}^{\mathrm{T}}{ḣ}_{\mathrm{N}}$
ṘCj = Σk∈KC ckjψ̇([C̃μ̂]k), j ∈ coarse voxels	ṘFi = Σk∈KF ckiψ̇([C̃μ̂]k), i ∈ fine voxels
${\mathrm{r}}_{{\mathrm{C}}_{\mathrm{j}}}={\mathrm{\Sigma }}_{\mathrm{k}\in {\mathrm{K}}_{\mathrm{C}}}{\mathrm{c}}_{{\mathrm{k}}_{\mathrm{j}}}^2{\mathrm{\omega }}_{{\mathrm{\psi }}_{\mathrm{k}}}({\left[\tilde{\mathrm{C}}\hat{\mathrm{\mu }}\right]}_{\mathrm{k}}),j\in \text{coarse voxels}$	${\mathrm{r}}_{{\mathrm{F}}_{\mathrm{i}}}={\mathrm{\Sigma }}_{\mathrm{k}\in {\mathrm{K}}_{\mathrm{F}}}{\mathrm{c}}_{\mathrm{k}\mathrm{i}}^2{\mathrm{\omega }}_{{\mathrm{\psi }}_{\mathrm{k}}}({\left[\tilde{\mathrm{C}}\hat{\mathrm{\mu }}\right]}_{\mathrm{k}}),i\in \text{fine voxels}$
${û}_{\mathrm{C}}=\text{max }({û}_{\mathrm{C}}-\frac{{\dot{\mathrm{L}}}_{\mathrm{C}}+{\mathrm{\beta }}_{\mathrm{C}}{Ṙ}_{\mathrm{C}}}{{\mathrm{d}}_{\mathrm{C}}+{\mathrm{\beta }}_{\mathrm{C}}{\mathrm{r}}_{\mathrm{C}}},0)$	${û}_{\mathrm{F}}=\text{max }({û}_{\mathrm{F}}-\frac{{\dot{\mathrm{L}}}_{\mathrm{F}}+{\mathrm{\beta }}_{\mathrm{F}}{Ṙ}_{\mathrm{F}}}{{\mathrm{d}}_{\mathrm{F}}+{\mathrm{\beta }}_{\mathrm{F}}{\mathrm{r}}_{\mathrm{F}}},0)$
end