Radon Transform

About This Series

The Radon transform is an operation that transforms a function into a collection of its line integrals. It was introduced by Johann Radon in 1917 and now serves as the mathematical foundation for image reconstruction techniques such as X-ray CT scanning and seismic exploration.

This series begins with the mathematical definition of the Radon transform and progresses step by step through the Fourier slice theorem, the inverse transform, and filtered back projection.

Definition of the Radon Transform

$$\mathcal{R}f(s, \theta) = \int_{-\infty}^{\infty} f(s\cos\theta - t\sin\theta, s\sin\theta + t\cos\theta) \, dt$$

The line integral of function $f$ along a line at angle $\theta$

Learning by Level

Introduction

High school to first-year university level

  • What is the Radon transform?
  • Geometric meaning of projections
  • The concept of sinograms
  • Connection to CT scanning

Basic

University years 1–2 level

  • Properties of the Radon transform
  • Fourier slice theorem
  • Projection and reconstruction
  • Computing simple examples

Intermediate

University years 3–4 level

  • Inverse Radon transform
  • Filtered back projection
  • Sampling theorem
  • Discretization and artifacts

Advanced

Graduate level

  • Generalized Radon transform
  • Microlocal analysis
  • Incomplete data problems
  • Extension to three dimensions

Learning Path

Introduction Projection concepts Basic Fourier slice Intermediate Inverse & FBP Advanced Generalized & 3D Introduction: Projections, sinograms, CT principles Basic: Fourier slice theorem, fundamental properties Intermediate: Inverse transform, FBP, discretization Advanced: Microlocal analysis, 3D reconstruction

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