Table 1.

Pseudocode for iterative solution of the poly-KCR objective

Λ = {λn} registration vector all components
μ(0)= Initial reconstruction (e.g. FBP)
κ(0) = Initial guess for STF coefficients
$\omega (\mathrm{\Lambda })=\left({\mathrm{\Pi }}_{\{n=1\}}^N\mathbf{D}\{T\left({\lambda }^{\{n\}}\right)s^{\{n\}}\}\right)$	% Generate transformed masks
${\mu }_{\ast }^{(0)}(\mathrm{\Lambda })=\omega (\mathrm{\Lambda }){\mu }^{(0)}$,	% Apply mask to background estimate
for all components n ∈ [1 N]
$p^{\{n\}}=\mathit{Gaussian}({\mathbf{A}}^H\left({\lambda }^{\{n\}}\right)b_{I,H}^{\{n\}},\delta )$	% Component path length with blur
end
% Calculate path length for background volume
L* = A1→	% 1→ denotes the all 1s vector
for all components n ∈ [1 N]
L* = L* − p{n}
end
for t = 1 to max_iterations	% [t−1] denotes (t−1)th iteration value
$l=-{\mathrm{\sum }}_{\{n=1\}}^N{\mathrm{\sum }}_{\{k=1\}}^K{\kappa }_k^{\{n\}[t-1]}{\left(p^{\{n\}}\right)}^k$	% Spectral correction to path length
$l_{\ast }=\mathbf{A}{\mu }_{\ast }^{[t-1]}(\mathrm{\Lambda })$	% Forward projection of background
% Background volume update (Appendix 6.2)
for all voxels, j
$${\mu }_{\ast ,j}^{[t]}={\mu }_{\ast ,j}^{[t-1]}+\frac{{\mathrm{\sum }}_i^M{w}_i{a}_{\ast ,ij}\left(log\frac{g_i}{y_i}-(l_{\ast }+l)\right)-R^{\prime }}{{\mathrm{\sum }}_i^M{w}_i{L}_{\ast ,i}{a}_{\ast ,ij}+R^{″}}$$
end
% Spectral coefficient update (Appendix 6.1)
for all components q ∈ [1 N]
for all coefficients l ∈ [1 K]
$${\kappa }_l^{\{q\}[t]}={\kappa }_l^{\{q\}[t-1]}+\frac{{\mathrm{\sum }}_i^M{w}_i{\left(p_i^{\{q\}}\right)}^l\left(l_{\ast ,i}-{\mathrm{\sum }}_{\{n=1\}}^N{\mathrm{\sum }}_{\{k=1\}}^K{\kappa }_k^{\{n\}[t-1]}{\left(p_i^{\{n\}}\right)}^k-log(\frac{g_i}{y_i})\right)}{{\mathrm{\sum }}_i^M{w}_i{\left(p_i^{\{q\}}\right)}^{2l}}$$
end
end
end