Table I.

Outline of the alternating maximization used by the KCR Algorithm

${\mu }_{\ast }^{[0,\cdot ,0]}$ = Initial reconstruction (FBP)

Λ[0,0,·] = Initial guess for registration parameters

H[0,0,·] = σI, initial guess for inverse Hessian

for k = 0 to max_iterations-1,

% Registration Update Block

for r = 1 to P (number of registration updates)

Compute ${\nabla }_{\mathrm{\Lambda }}L\left({\mu }_{\ast }^{[k,\cdot ,0]},{\mathrm{\Lambda }}^{[k,p-1,\cdot ]}\right)$

H[k, p,·] = BFGS update using new gradient

$\widehat{\gamma }$ = line search in ${\mathrm{\Lambda }}^{[k,p-1,\cdot ]}+\gamma {\mathbf{H}}^{[k,p,\cdot ]}{\nabla }_{\mathrm{\Lambda }}L\left({\mu }_{\ast }^{[k,\cdot ,0]},{\mathrm{\Lambda }}^{[k,p-1,\cdot ]}\right)$

${\mathrm{\Lambda }}^{[k,p,\cdot ]}={\mathrm{\Lambda }}^{[k,p-1,\cdot ]}+\widehat{\gamma }{\mathbf{H}}^{[k,p,\cdot ]}{\nabla }_{\mathrm{\Lambda }}L\left({\mu }_{\ast }^{[k,\cdot ,0]},{\mathrm{\Lambda }}^{[k,p-1,\cdot ]}\right)$

end

Compute $d_i^{[k,p,\cdot ]}\left({\mathrm{\Lambda }}^{[k,p,\cdot ]}\right)$

% Image Update Block

for m = 1 to M (number of subsets)

Compute curvatures, $c_i^{[k,\cdot ,m]}\left({\mu }_{\ast }^{[k,\cdot ,m-1]}{\mathrm{\Lambda }}^{[k,p,\cdot ]}\right)$

Compute modified line integrals, $t_i^{[k,\cdot ,m]}\left({\mu }_{\ast }^{[k,\cdot ,m-1]},{\mathrm{\Lambda }}^{[k,P,\cdot ]}\right)$

$${\mu }_{{\ast }_j}^{[k,\cdot ,m]}={\left[{\mu }_{{\ast }_j}^{[k,\cdot ,m-1]}+\frac{M{\displaystyle \sum _{i\in {\mathrm{\Omega }}_m}{b}_{\mathit{ij}}{\dot{g}}_i}\left(t_i^{[k,\cdot ,m-1]};d_i^{[k,P,\cdot ]}\right)-\beta {\dot{R}}_j\left({\mu }_{\ast }^{[k,\cdot ,m-1]}\right)}{M{\displaystyle \sum _{i\in {\mathrm{\Omega }}_m}{b}_{\mathit{ij}}}\left({\displaystyle \sum _j{b}_{\mathit{ij}}}\right)c_i^{[k,\cdot ,m-1]}+2\beta {\ddot{R}}_j\left({\mu }_{\ast }^{[k,\cdot ,m-1]}\right)}\right]}_{+}$$

end

${\mu }_{\ast }^{[k+1,\cdot ,0]}={\mu }_{\ast }^{[k,\cdot ,M]},{\mathrm{\Lambda }}^{[k+1,0,\cdot ]}={\mathrm{\Lambda }}^{[k,P,\cdot ]},{\mathbf{H}}^{[k+1,0,\cdot ]}={\mathbf{H}}^{[k,P,\cdot ]}$

end