# Algorithms in Tomography and Related Inverse Problems--A Review
## 2 Evaluation of Backprojection Methods  
It is universally accepted that mammography is the most efficacious tool for the early detection of breast cancer. With traditional mammography, the object is projected onto the detector or film to generate the 2D projection image of the breast. Superimposed objects on the projection images, caused by overlapped anatomical structures, bring limitations to mammography [85; 86], such as 20% false-negative rates and high recall rates, which may result in unnecessary anxiety to the patients and increase medical costs. Compared to standard mammography, the digital breast tomosynthesis (DBT) technique may overcome the limitations by removing the ambiguities of overlapped tissues and providing 3D localization. Since 3D slice images of the breast can be partially reconstructed based on a few limited-angle projection images, DBT has the potential to help decrease recall rates, improve the accuracy of breast cancer detection, and, therefore, reduce the number of women who die from such cancer [86]. In the process of tomosynthesis, sequences of limited-angle 2D projection images are acquired first and then reconstructed into slice images of the breast. A few image reconstruction algorithms have been investigated by various research groups, including the backprojection (BP) reconstruction algorithm [87], filtered backprojection (FBP) algorithm [88], matrix inversion tomosynthesis (MITS) [89], maximum likelihood expectation maximization (MLEM) [90; 91], simultaneous algebraic reconstruction techniques (SART) [92; 93], etc. Figure 1: The organization of the present review paper in sections. The work in [8] focuses on the investigation of BP algorithms. Two BP variants, including \(\alpha\)-trimmed BP and principal component analysis-based (PCA) BP, were proposed. Their performance in improving the conspicuity of lesions and suppressing noise was studied by computer simulations and phantom experiments. The shift-and-add (SAA) tomosynthesis reconstruction algorithm [87] reconstructs the plane at the specified height by lining up each projection image according to its relative shift amount. Objects at different locations above the detector will be projected onto the detector in positions depending on the relative locations of the objects. In order to reconstruct 3D slices of the breast, each projection image should be shifted by an amount appropriate for the plane of reconstruction. The shift amount can be calculated based on projected positions from the central points of each reconstruction plane. The shifted planes are added together to emphasize structures in the in-focus plane and blur out structures in other planes. Figure 2 shows a parallel tomosynthesis imaging geometry. The reconstructed plane S can be derived by taking the average of all the projection images that have undergone the necessary shifts [39; 87]. The SAA algorithm facilitates the acquisition of 3D reconstructed slices. To enhance the reconstruction of a single pixel on a specific plane located at a certain height above the detector, it is imperative to calculate the shift amounts along both the x and y directions for each pixel on the reconstruction plane. This technique is commonly referred to as point-by-point backprojection [87]. Point-by-point backprojection involves the computation of shift values for every individual pixel position within each reconstructed plane, taking into account the 2D projection of the reconstructed objects within those planes. Figure 3 illustrates this process. The pixels resulting from the backprojection process provide estimations of the object's internal structure. In the conventional BP algorithm, the final pixel value at point A is calculated as the mean of the backprojected pixels derived from all N projection images (where N denotes the number of projection images). To leverage the statistical properties inherent in these N values and thereby enhance image quality, two distinct variants have been introduced. Figure 3: Projection geometry.