The Radon Transform and Its Application to Computed Tomography

Abstract

This work studies the mathematical foundations of computerized tomography from the point of view of the Radon transform. Starting from the physical interpretation of X-ray measurements as line integrals, we introduce the main analytical tools needed to formulate the reconstruction problem, including Fourier analysis, distributions, sampling theory and the Fourier slice theorem. We then discuss several inversion procedures, with special emphasis on filtered backprojection and Fourier-based reconstruction methods. Following the approach of Cascante et al., we also describe a gridding method for the numerical recovery of images from discrete projection data. The final part of the work analyzes the computational structure of the algorithm and the role of sampling, interpolation and filtering in practical reconstruction. Overall, the aim is to show how the continuous theory of the Radon transform leads naturally to implementable algorithms for medical image reconstruction.