Mathematical Principles of X-Ray Computer Tomography

Abstract

The mathematical principles of X-ray computer tomography are implemented by software, using which the object is scanned, the obtained data are collected and processed, and the image is reconstructed. Definitions of computers from the first to fifth generations are given, and a cone-beam tomography, which uses a cone beam of radiation and an array of detectors, is considered. The scanning can be mere rotation, or rotation with translation (over a spiral trajectory), or it can be a complex spatial curve. The more complex the scanning, the more complex the data transformation mathematical description is. The mathematical basis of scanning can be described by the Radon transform. The relationship between the Radon transform and the Fourier transform is shown. Data processing can be algebraic and integral, depending on the stage at which digitization is carried out. In the tomograph, the Fourier transform is mandatorily carried out, using which the cutoff frequency of the function describing the controlled object movement is determined. The determination of the function cutoff frequency allows the functions to be digitized with a minimum error regardless of the stage at which it is performed. Several options for tomographic data reconstruction are considered. Reconstruction methods can be analytical and iterative. Analytical methods include the 2D Fourier transform algorithm, back projection method, and back projection method with filtration. Iterative image reconstruction by the successive approximations method is shown. The advantages and disadvantages of the considered algorithms are shown. The advantage of iterative algorithms is that they are easy to synthesize despite their nonlinearity and do not require the definition of the inverse operator. However, to restore the image by algebraic methods, a fewer number of projections is required. A common disadvantage of iterative algorithms is their low computational efficiency caused by an iterative nature of the calculations. There are no formal approaches as to which particular algorithm should be applied during reconstruction. The questions of when they should be used and how many iterations should be performed are decided on a case-by-case basis proceeding from practical experience.