# Algorithms in Tomography and Related Inverse Problems--A Review
## 2 Evaluation of Backprojection Methods
The X-ray source projects the point A onto B (detector plane). Figure 2: Parallel X-ray breast tomosynthesis imaging geometry. The \(\alpha\)-trimmed BP technique involves the removal of extreme values within the back-projected pixels. This process entails sorting all pixel values present in the backprojection images, eliminating the lowest \(\alpha/2\) values and the highest \(\alpha/2\) values, and then computing the mean value. The principal component analysis method is a sophisticated multivariate analysis technique rooted in the concept of eigenvectors. It offers a valuable orthogonal linear transformation that shifts from an initial n-dimensional coordinate system to a novel m-dimensional coordinate system (where \(m<n\)). In the implementation of PCA-based backprojection, a pivotal step involves computing the first principal eigenvectors. These eigenvectors are derived from a matrix comprised of \(N\) backprojected pixel values. They serve as the foundational components used to generate the reconstructed image. In the following, the SAA tomosynthesis reconstruction algorithm is explained, and two variants of the BP technique are provided for enhancing the reconstruction of 3D slices. A summary of the key steps follows:  
Step 1. SAA Tomosynthesis Reconstruction:  
The SAA algorithm reconstructs a plane (Figure 2) at a specified height by aligning each projection image based on its relative shift amount. The shift amount, \(shift_{i}(S)\), is calculated using the relationship \(shift_{i}(S)=shift(O)=\frac{H}{STD-H}\cdot(R_{x}-O_{x})\), where \(H\) is the height, \(STD\) is the source-to-image distance, \(R_{x}\) is the pixel position on the detector, and \(O_{x}\) is the central point of the reconstruction plane. The reconstructed plane \(S\) is obtained by averaging all projection images that have undergone the necessary shifts. Step 2. Point-by-Point Backprojection:  
To enhance the reconstruction of a single pixel on a specific plane, point-by-point backprojection is employed. The shift amounts along both the x and y directions (Figure 3) for each pixel on the reconstruction plane are calculated using the mathematical relationship described by equations \(B_{x}=R_{x}+\frac{R_{x}}{R_{x}-A_{x}^{*}}\cdot(A_{x}-R_{x})\) and \(B_{y}=R_{y}+\frac{R_{x}}{R_{x}-A_{z}}\cdot(A_{y}-R_{y})\). Backprojection provides estimations of the object's internal structure. Step 3. Backprojection Variants. \(\alpha\)-Trimmed BP Technique:  
The technique involves removing extreme values within the backprojected pixels by sorting and eliminating the lowest and highest \(d/2\) values. The final pixel value is calculated as the mean of the remaining values (\(s=\frac{1}{N-d}\sum_{i=d/2+1}^{N-d/2}I(B_{i})\)). Parameter \(d\) controls the degree of trimming, ranging from 0 to \(N-1\). PCA-based BP utilizes PCA, a multivariate analysis technique, to transform the coordinate system for enhanced reconstruction. It involves computing the first principal eigenvectors from a matrix of backprojected pixel values. In summary, the 3D reconstructed images produced through the examined reconstruction techniques, including conventional BP, \(\alpha\)-trimmed BP, and PCA-based BP, effectively unveiled masses and micro-calcifications. Among these methods, \(\alpha\)-trimmed BP show-cased superior noise reduction capabilities and proficiently addressed out-of-plane artifacts, thereby enhancing the visibility of in-plane objects. Additionally, it commendably preserved the contours of objects situated near the boundaries. When integrated with FBP as the backprojection step, \(\alpha\)-trimmed BP demonstrated the potential to enhance the overall image quality of the reconstructed slices.