Fourier Transforms in Tomography: Inversion Techniques

This lecture focuses on the application of Fourier transforms in the context of tomography, particularly the Radon transform and its inverse. The instructor begins by discussing the basic principles of tomography, explaining how x-ray scans model the human body as a function of absorption. The lecture introduces the Radon transform, which is defined as an integral that measures the absorption function along straight lines. The instructor then explains the process of inverting the Radon transform to recover the original function, emphasizing the importance of the Fourier transform in this context. The lecture includes examples and derivations, illustrating how to compute the inverse Radon transform using Fourier techniques. The instructor also discusses the geometric interpretation of the Radon transform and its implications in medical imaging. Throughout the lecture, the instructor highlights the significance of understanding these mathematical methods for practical applications in physics and engineering, particularly in medical imaging technologies.