# Algorithms in Tomography and Related Inverse Problems--A Review
## 4 Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware
To introduce finer surface details, images or textures representing the desired intricacies are often applied or mapped onto these polygons during the rendering phase. This method, known as texture mapping, offers an efficient way to enhance surface detail without necessitating an increase in the object's polygon count. Importantly, graphics hardware is finely tuned for rapid texture mapping, even when confronted with perspective distortion [117]. The graphics pipeline comprises three fundamental stages, as depicted in Figure 5: the _geometry processing stage_, the _polygon rasterization stage_, and the _fragment processing stage_. In [49], the work of [PERSON] [118] was referenced, where a volume is represented as an assemblage of point samples positioned at grid points. In this model, values at positions between grid points are approximated by interpolation using a specific kernel function. Linear functions are chosen for this purpose, a choice that has also gained  
Figure 5: Pipeline for rendering graphics using hardware acceleration. extensive popularity in backprojectors and is amenable to efficient implementation in graphics hardware. The provided text discusses the utilization of GPUs for computations within various common CT algorithms. It introduces a standardized notation and then delves into the specifics of imaging modalities, such as transmission and emission X-rays. The text further describes mathematical formulations for the CT process, including projection and backprojection operations. Three reconstruction methods--the Feldkamp algorithm, SART (Simultaneous Algebraic Reconstruction Technique), and OS-EM (Ordered Subsets Expectation Maximization)--are explained using the established notation. Here is a breakdown of the four main points:  
1. Notation and Imaging Modalities: * A volumetric object is defined by its attenuation function \(\mu(x,\ y,\ z)\). * Two imaging modalities are considered: transmission X-ray (external source) and emission X-ray (metabolic sources within the object). * Mathematical formulations \(C_{\varphi}^{Q}(u,v)=Q_{0}\cdot e^{-\int_{0}^{L}\mu(t)dt}\) and \(C_{\varphi}^{E}(u,v)=\int_{0}^{L}E(s)\cdot e^{-\int_{0}^{s}\mu(t)dt}ds\) for recording intensity values on a 2D detector for both modalities are provided. 2. Vector Processing for CT: * Introduction of vector processing for CT using a standardized notation (\(C_{i}^{Q}=C_{\varphi}^{Q}(u,v)\), \(C_{i}^{E}=C_{\varphi}^{E}(u,v)\), \(q_{i}=\sum_{j=0}^{N^{3}-1}\mu_{j}w_{ij}\), \(e_{i}(s)=\sum_{j=0}^{N^{3}-1}E_{j}w_{ij}(s)\), etc.). * The shift from pixel-centric to voxel-centric representation for transmission X-ray. * Formulation of voxel-centric representation for emission X-ray. 3. Projection and Backprojection Operators: * Introduction of projection (\(P_{\varphi}\)) and backprojection (\(B_{\varphi}\)) operators as matrices. * Dynamic computation of elements using interpolators integrated into rasterization hardware. 4. Reconstruction Methods: * Feldkamp algorithm * Depth correction factor (\(w_{ij}(d)=w_{ij}\frac{\pi^{2}}{\left(a+\sqrt{\left(v_{j}\right)+Z(v_{j})}\cos (\varphi-\varphi_{r})\right)^{2}}\)) during backprojection. Grid update equation expressed in condensed notation. * SART (Simultaneous Algebraic Reconstruction Technique) Grid update equation for SART \(V=V+\frac{B_{\varphi}\left(\frac{A_{1}-P_{\varphi}(V)}{P_{\varphi}(V)}\right)} {B_{\varphi}(V)}\) involving a relaxation factor (\(\lambda\)). * OS-EM (Ordered Subsets Expectation Maximization) Grid update equation for OS-EM algorithm \(V=\frac{V}{\sum_{\varphi\in OS}B_{\varphi(a)}(W)}\sum_{\varphi\in OS}B_{\varphi (a)}\left(\frac{I_{\varphi}}{P_{\varphi(a)}(V)}\right)\). The text concludes by emphasizing the potential of technological advancements in GPU capabilities for computed tomography. The provided information outlines the mathematical formulations and algorithms used in CT reconstruction, showcasing the significance of leveraging GPU capabilities for efficient computations in these processes.