TABLE I.

Pseudo-code for the SF-TR forward projector with the A1 amplitude method (SF-TR-A1) and the A2 method (SF-TR-A2)

• Initialize projection view array to zero, i.e., ${\stackrel{‒}{y}}_{\beta }[s_k,t_l]=0$ for  k = 0, … , Ns − 1 and l = 0, … , Nt − 1

• For each row n1 = 0, 1, … , N1 − 1 of $f\left[\overrightarrow{n}\right]$:

1) For each column n2 = 0, 1, … , N2 − 1:

a) Compute trapezoid vertices τ0, τ1, τ2, τ3 in (38) using    (9).

b) Determine indices (sk values) of detector cells that    intersect [τ0, τ3], i.e. $\{k:[s_k-\frac{r_s}{2},s_k+\frac{r_s}{2}]\cap [{\tau }_0,{\tau }_3]\ne \varnothing \}$.

c) Compute transaxial footprint $F_1(s_k;\beta ;\overrightarrow{n})$ using (41)    and (43) for all these sk values and store them.

d) Compute $l_{{\phi }_0}$ using (32) (SF-TR-A2 only)

e) For each slice n3 = 0, 1, … , N3 − 1:

i) Determine indices (tl values) of detector cells that     intersect [t−, t+], i.e., $\{l:[t_1-\frac{r_t}{2},t_l+\frac{r_t}{2}]\cap [t_{-},t_{+}]\ne \varnothing \}$.

ii) For each tl value:

A) Compute $F_2(t_l;\beta ;\overrightarrow{n})$ using (42).

B) For each sk value:

– Compute projection       $p(s_k,t_l;\beta ;\overrightarrow{n})$ where $p=f\left[\overrightarrow{n}\right]F_1(s_k;\beta ;\overrightarrow{n})F_2(t_l;\beta ;\overrightarrow{n})$ for SF-       TR-A1,       $p=f\left[\overrightarrow{n}\right]l_{{\phi }_0}{F}_1(s_k;\beta ;\overrightarrow{n})F_2(t_l;\beta ;\overrightarrow{n})$ for       SF-TR-A2.

– Update projection view ${\stackrel{‒}{y}}_{\beta }[s_k,t_l]\pm p$.

• Scale all the projection view by l1(sk, tl; β) using (36) for  SF-TR-A1 or by lθ(sk, tl) using (35) for SF-TR-A2.