2.1

Notation

Let , and (in 2D, the last component of and is discarded). Let f denote the 3D object density to be reconstructed. The cone-beam projections of f for a circular source trajectory of radius R_A acquired on a flat detector placed at the origin O are defined by

where λ ∈ Λ = [0,2π) and is the set of vertices (cone-beam source locations) of the trajectory (see figure 1).

Figure 1.

The circular acquisition geometry of center O and radius R_A. The point S(λ) is a vertex of the trajectory. A ray passing through the point P on the detector placed at the origin O is identified by the parameters (λ, u, v).

In 2D, the fan-beam projections of f for a circular source trajectory of radius R_A with angular parametrization are defined by

where γ ∈ (–π/2, π/2) is the usual ray-angle measured counterclockwise with respect to the central ray (which is defined by the source and the center of rotation). The parameters u and γ (respectively for equispaced rays and equiangular rays) are linked by u = R_A tan γ so, in the midplane, we have

2.2

Configuration studied

We consider the following configuration. The measured area is the volume inside a cylinder of center O, extended axially without limit since we consider no axial truncation. We assume that the support of the object function is contained within a known ellipsoid which extends outside the measured area (see figure 2 left).

Figure 2.

Left: the ellipsoid object is partially covered by the cylindrical measured area (drawn in red).

Right: situation in the midplane. The circular FOV of center O and radius R_F covers only a part of the elliptic object. The virtual trajectory is the black arc of circle of center O and radius R_V = R_F and the vertical black dashed line is the boundary of its convex hull.

2.3

The VFB formula used in the midplane

In the midplane, the 2D slice of the object has an elliptic support and the FOV has a circular support. The chosen virtual trajectory is the arc of circle at the border of the FOV and outside the object (see figure 2 right). In that case, the area for which the VFB method is mathematically exact is the convex hull of the virtual trajectory.

We now recall our VFB formula from.¹¹ The rebinning relations between two trajectories with different radius can be seen on figure 3. The data are first rebinned from the acquisition geometry of radius R_A to the virtual geometry of radius R_V using

Figure 3.

The parameters (λ_i, γ_i) of a ray for source trajectories of radius R_i with i ∈ {1, 2} are linked through s = R_i sin γ_i and ϕ = λ_i + γ_i.

where

Then, differentiation and Hilbert filtering are performed on the non-truncated projections in the virtual geometry with

where h_H(s) = ∫_ℝ –i sign(σ)e^2iπσsdσ denotes the Hilbert filter and ∂_i corresponds to the partial derivative with respect to the i-th variable. As the virtual trajectory is not a full scan, the redundancy in the filtered projections is handled by applying a weight w^R_v (that we do not detail) to : . Next, the filtered projections in the acquisition geometry are obtained from the filtered projections in the virtual geometry by

where

Finally, the backprojection is performed in the acquisition geometry. For every in the convex hull of the virtual trajectory, we have:

where

We can see that this formula is designed for equiangular data g^R_A (λ, γ). As we consider equispaced data in this paper, an additional rebinning using equation (3) left is required before using the VFB formula above.

2.4

Modifying the VFB method for cone-beam projections

We now detail how the VFB formula above is modified to be used on cone-beam projections. First, we perform a weighting of the cone-beam projections:

For an object that is constant in z, (11) ensures that for all v: , so the exact reconstruction area will be extended axially if each row of the weighted projections is treated as the row in the midplane.

Then, for all the weighted data rows of parameter v fixed, we perform the following steps as if the transaxial plane of height z = v was the source plane, using the same virtual source trajectory as in the midplane:

where we take for all v, and γ = arctan(u/R_A).

Finally, the backprojection is performed in the acquisition geometry to give , the 3D VFB reconstruction :

for in the convex hull of the virtual source trajectory,

3.1

Simulations

The simulations were performed on a 3D version of the Shepp-Logan phantom and on the 3D head Forbild phantom*. The reconstructed image was computed on a cubic grid of size [401, 401, 401] voxels. The data were acquired on a circular trajectory of center O = (0, 0, 0) and radius R_A, using the software RTK.¹² The projections were transversely truncated such that the measured area was a cylinder of center O and radius R_F. The virtual source trajectory radius was R_V = R_F. The acquisition trajectory along [0, 2π) was sampled with Ν_λ vertices and each projection was composed of N_u × N_v ray-lines. The virtual trajectory was composed of N_λvirt virtual segments and each virtual projection was composed of N_γvirt ray-lines. For the Shepp-Logan phantom, we took R_A = 4, R_F = 0.8, N_λ = 1256, N_u = 409, N_v = 517, N_λvirt = 879 and N_γvirt = 1257. For the head Forbild phantom, we took R_A = 45, R_F = 9, N_λ = 1256, N_u = 409, N_v = 603, N_λvirt = 693 and N_γvirt = 1257.

3.2

Results

Figure 4 shows, for three slices of the 3D Shepp-Logan phantom (left) and the head Forbild phantom (right), the reference image, the reconstructed image using the FDK algorithm with non-truncated data, the reconstructed image using our modified 3D VFB method for transversely-truncated data, and the profiles of the lines drawn in white on the reference and the 3D VFB reconstructions. The mathematically exact reconstruction area (convex hull of the virtual source trajectory), which we also call the recoverable area, is delimited by a black dashed line on the 3D VFB reconstructions.

Figure 4.

Left block: 3D Shepp-Logan (SL) phantom. Right block: Forbild head phantom.

For each block: Left column: (x, y) plane at z = 0. Middle column: (y, z) plane at x = 0. Right column: (x, z) plane at y = 0.4. Top row: 2D slices of the reference phantom. Middle row 1: reconstructions using the FDK algorithm without truncation. Middle row 2: reconstructions using our 3D VFB algorithm with truncation. The black dashed lines define the boundary of the possible reconstruction area. The plotting scale is respectively [1.0 (black), 1.04 (white)] for the SL phantom and [1.0 (black), 1.1 (white)] for the head phantom. Bottom row: profiles corresponding to the white lines, plotted respectively with scale [1.005, 1.045] for the SL phantom and with scale [1.025, 1.085] for the head phantom. The reference profiles are plotted in green dashed line and the reconstruction profiles in red.

Looking at figure 4 left, we can see that the 3D VFB reconstruction is excellent in the recoverable area in the midplane (left column). In the planes at x = 0 (middle column) and at y = 0.4 (right column), the reconstruction is still very good when we are close to the midplane. Further away from the midplane, we observe a slow decrease of the intensity when |z| increases, similar to that on the FDK reconstruction, although not exactly the same. There are also slight horizontal streak artefacts, tangent to the white ellipse, which are less marked on the FDK reconstruction.

The 3D VFB reconstruction of the Forbild head phantom (figure 4 right) is good in the recoverable area in the midplane (left column), but far less accurate that what we obtained for the Shepp-Logan phantom in figure 4 left. The difference is that the Forbild phantom consists of many more and finer anatomical structures than the Shepp-Logan phantom, making it a far more challenging phantom to reconstruct. Consequently, we observe that the FDK and 3D VFB reconstructions suffer from many artefacts for planes at x = 0 (middle column) and at y = −1 (right column). The artefacts are stronger for the 3D VFB reconstruction, as we observe for instance with the white area at the right of the black ellipse at plane x = 0 (middle column), and also with the large black horizontal streak covering the top of the two circular structures at plane y = −1 (right column).