Projection Model

In the ray-driven (RD) model, the x-ray beam is modeled as a pencil beam emitted from the point source to the center of a detector pixel. The x-ray beam intersects with several voxels before it arrives the detector, and the intersection length is then used to compute the weighting coefficient for each voxel, which is the element 𝐴𝑖𝑗 in the system matrix. There are several implementations of this RD model, the two classical approaches are Siddon’s algorithm and Joseph’s algorithm, as shown in Figure 4-4(a) and 4-4(b) respectively. In Siddon’s algorithm, the intersection length is used directly as the contribution of each voxel to the attenuation line-integral. In Figure 3-4(a), the contribution of voxels 1 and 2 to the projection line i is computed as:

𝑝_𝑖 += 𝐿₁𝜇₁ + 𝐿₂𝜇₂. (4-7)

In Joseph’s method, the projector coefficient is computed as a product of the intersection length times the interpolation of the nearest voxels. This method is shown schematically in Figure 3-4(b), where the contribution of voxels 1 and 2 to the projection line i is computed as:

𝑝_𝑖 += 𝐿𝑎2𝜇1+ 𝑎1𝜇2

𝑎₁+ 𝑎₂ . (4-8)

Alternative implementations of the RD model have also been proposed, such as a general step- size approach with interpolation using more nearest voxels and the strip integral method.[110], [111] Those algorithms generally improve the smoothness of the reconstruction; however, they are typically more computationally expensive.

Figure 4-4: Illustration of two classical ray-driven models: (a) Siddon’s algorithm with intersection length, (b) Joseph’s linear interpolation method.[57]

3.3.2 Distance-driven Method

In both PD model and the RD model, the forward and back projection are not the transpose of the other. As a result, iteratively apply forward and back projection would amplify errors and cause high-frequency artifacts. Several approaches have been proposed to address this issue, including using an unmatched projection pair,[112] Fourier-based projection,[62], [63], [65] pixel decomposition,[113] and projector with different interpolation methods.[114]

Figure 4-5: Schematic diagram of (a) 2D distance-driven method, and (b) 3D distance-driven method.[57]

De Man proposed a different projection model, called distance-driven (DD) model, which has mathematically transposed forward and back projector.[109], [115], [116] In DD model, the detector pixel and the image voxel are modeled with a finite width, as illustrated in Figure 4-5(a). The weightings are computed by projecting the boundaries of detector pixels and image voxels onto the same axis, and the overlap length is used as the weighting coefficient. In Figure 3-5(a), the contribution of the voxels 1 and 2 to the projection line i is computed as:

𝑝𝑖 += 𝐿1

𝑜₁𝜇₁+ 𝑜₂𝜇₂ 𝑜1+ 𝑜2

. (4-9)

DD method has a low arithmetic cost and a sequential memory access; therefore, it is computationally efficient. DD method is also fairly accurate as an approximation to the true integral method. Reconstruction images using three different projection methods are shown in Figure 4-6 (FBP reconstruction) and Figure 4-7 (MLEM reconstruction).[115] Compared to PD method and RD method, DD method generates smoother images with less high-frequency artifacts.

Figure 4-6: FBP reconstruction of a head section: reference image, pixel-driven FBP, normalized ray-driven FBP, and distance-driven FBP.[115]

Figure 4-7: Iterative reconstruction of a head section using maximum likelihood (ML) algorithm: reference image, pixel-driven ML reconstruction, ray-driven ML reconstruction, and distance- driven ML reconstruction.[115]

4.3.3 Use Distance-driven Model to Remove the Line Artifact

It is known that primitive projection models, such as pixel-driven model and ray-driven model, would introduce the high-frequency artifact in the reconstruction image. Especially for

frequency artifact can be reduced or eliminated by using a more accurate projection models at the cost of more computation. In the conventional 3D cone beam reconstruction, a trade-off is typically required between the accuracy of the projection model and the computation time. In AFVR, the system matrix for each fan volume is only required to compute once, and the result will be stored in the memory. Therefore, a more accurate projection model can be used, and the computational cost will be amortized among all iterations. Here, we demonstrate the reduction of the high- frequency noise by implementing the distance-driven projection model in the AFVR. Figure 4-8 shows the tomosynthesis reconstruction images of a Shepp-Logan phantom using the ray-driven model and the distance driven model. The reconstruction is performed using AFVR and SIRT algorithm with 20 iterations; the s-DBT imaging geometry is used in this simulation. As can be seen from the figure, the reconstruction using RD model has many high-frequency line artifacts; with the DD model, the artifacts are significantly reduced. It should be noted that the high- frequency artifact is the combined result of both the simple projection model and the limited number of projections in tomosynthesis imaging. The artifact also reduces when the number of projection increases.

Figure 4-9 shows reconstruction slices of a patient’s breast imaged by s-DBT. Multiple line artifacts are observed on the reconstruction slices with the RD projector. On the other hand, the reconstruction slice with the DD projector is quite smooth without any noticeable line artifacts. A line profile of the reconstruction slices is shown in Figure 4-10. The spikes shown in the line profile of the RD modeled reconstruction are eliminated in that of the DD modeled reconstruction.

Figure 4-8: Enlarged view of (a) a Shepp-Logan phantom, AFVR-SIRT reconstruction of the Shepp-Logan phantom (b) using ray-driven projection model, and (c) using the distance-driven projection model.

Figure 4-9: Reconstruction slice using (Left) ray-driven projection model, and (Right) distance- driven projection model.

Figure 4-10: Line profile of the reconstruction slice using the ray-driven projector and the distance- driven projector