The unknown projection matrix $P$ is written as a product of three matrices $P=KRT$. Here, $K$ describes the virtual projection geometry’s intrinsic parameters, $R$ is a rotation matrix, and $T$ is a translation matrix. $K$ depends on SDD, pitch between virtual detector elements ($sp$), and the coordinates $(uo,vo)$, which define the piercing point on the virtual detector. $R$ depends on three angles $(\theta x,\theta y,\theta z)$ describing the rotations about the three principal axes. $T$ depends on the location of the virtual source point $(xs,ys,zs)$ in the 3-D object coordinate system. Denoting $cj=cos(\theta j)$ and $sj=sin(\theta j)$, the projection matrix $P$ is given by Display Formula
$P=[\u2212uoSDD/sp0\u2212vo0SDD/sp\u2212100][1000cxsx0\u2212sxcx]\u2062[cy0\u2212sy010sy0cy][czsz0\u2212szcz0001][100\u2212xs010\u2212ys001\u2212zs].$(2)
The $P$ matrix is parameterized by a vector, $\xi $, consisting of nine elements, $\xi =[SDD,uo,vo,\theta x,\theta y,\theta z,xs,ys,zs]$. The pitch between virtual detector elements, $sp$, equals 0.23 mm. A calibration phantom containing a known helical configuration of $N$ high-contrast point-like markers is then imaged [Fig. 2(a)]. The virtual detector coordinates $(ui,vi)$ of the projections of the markers are then determined using a center-of-mass technique. The geometric parameters describing the IGCT system are estimated by minimizing the sum-of-squared differences between the measured positions of the markers $(ui,vi)$ and the $P$-matrix-projected marker detector coordinates [$ui(\xi ),vi(\xi )$] Display Formula$\xi ^=arg\u2009min\xi 1N\u2211i=1N{[ui(\xi )\u2212ui]2+\u2009\u2009[vi(\xi )\u2212vi]2}.$(3)