# Algorithms in Tomography and Related Inverse Problems--A Review
## 4 Accelerating Popular Tomographic Reconstruction Algorithms on Commodity PC Graphics Hardware  
All algorithms designed for 3D computed tomography (CT) share a common challenge, as highlighted by [PERSON] and [PERSON] [49], primarily involving a series of backprojection operations that significantly contribute to the computational burden. Moreover, iterative algorithms introduce additional computational overhead through forward projections, which pose similar computational demands. Therefore, to render these operations practical for clinical applications, it is imperative to optimize the efficiency of both backprojections and projections. Each projection and backprojection operation inherently possesses a complexity that scales with the volume dataset's size, often denoted as \(O\big{(}N^{3}\big{)}\). In the context under discussion, straightforward projection and backprojection in the spatial domain are considered. In such cases, the primary avenue for reducing the actual computational expense lies in diminishing the constant factor, denoted as \(k\), which relates complexity to computational cost and is responsible for the \(k\cdot N^{3}\) term. Typically, in iterative reconstruction, a widely adopted strategy involves precomputing weight matrices, often referred to as look-up tables. While this approach has demonstrated remarkable acceleration in two-dimensional (2D) reconstruction scenarios, its applicability to 3D reconstruction is hindered by the substantial memory requirements involved. Consequently, for 3D reconstruction, various commercial solutions have emerged, often built on custom hardware, to address these challenges [49]. When considering an appropriate platform, it is important to recognize that the projection and backprojection operations are fundamentally voxel and pixel-based tasks with minimal dependencies. Typically, these operations are computed as array processes within extended loops. An ideal platform for handling such calculations includes vector processors or massively parallel architectures [116]. However, it is worth noting that vector processors like the Cray supercomputer family tend to be expensive. A noteworthy recent development in this domain is the emergence of mainstream computing platforms that share many characteristics with vector processors, notably graphics processors (GPUs). By framing the projection, backprojection, and all other CT computations as stream operations, we can harness the capabilities of these affordable mainstream architectures to achieve rapid CT imaging. In consequence, an outline emerges regarding the adaptation of the most frequently utilized CT algorithms, encompassing filtered backprojection, algebraic methods, and EM methods, to GPU architectures. This transition results in significantly enhanced processing speed while upholding accuracy standards. In a general context, it is evident that previous methods faced a common challenge stemming from the limitations of the graphics hardware they utilized. These hardware systems were confined to integer-arithmetic precision, typically at either 8-bit (PC) or 12-bit (SGI) precision levels. This constraint had a notable impact on their overall accuracy and computational performance. However, a significant stride forward has been achieved with the advent of newer GPU generations. These advanced GPUs introduce a pivotal feature, offering support for floating-point precision at two critical stages within the graphics pipeline. This enhancement carries significant implications. It now allows for the complete reconstruction process to be executed directly within the GPU, performing at the precision levels typically associated with CPUs. Furthermore, the computational tasks handled by the GPU facilitate effortless visualization of the generated data, thus enhancing the overall utility of the technology. Graphics elements are typically constructed using polygon meshes.