Radon Transform

Projections, the Fourier Slice Theorem & CT Reconstruction

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

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

Here $\theta$ is the angle of the normal direction of the line, $s$ is the signed distance from the origin to the line, and $t$ is the coordinate along the line. The arguments of the integrand are simply the parametrization

$$(x, y) = \underbrace{s(\cos\theta, \sin\theta)}_{\text{distance } s \text{ along the normal}} + \underbrace{t(-\sin\theta, \cos\theta)}_{\text{moving along the line}}$$

of that line. In other words, the Radon transform sweeps the line whose normal has angle $\theta$ and whose signed distance from the origin is $s$, and integrates $f$ along it.

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

Frequently Asked Questions

Q1. What is the Radon transform?

The Radon transform is an operation that transforms a two-dimensional function into a collection of its line integrals. It was introduced by Johann Radon in 1917. The function is integrated along the line whose normal direction has angle $\theta$ and whose signed distance from the origin is $s$; arranging the resulting values $\mathcal{R}f(s,\theta)$ as an image on the $(s,\theta)$ plane gives what is called a sinogram. The name comes from the fact that a point-like object traces the sinusoidal curve $s = x\cos\theta + y\sin\theta$.

Q2. In what fields is the Radon transform used?

The most prominent application is X-ray CT scanning. In an idealized model it is not the transmitted intensity measured by the detector itself, but the projection data obtained by taking its logarithm, that corresponds to the Radon transform of the X-ray attenuation coefficient inside the body. Cross-sectional images are reconstructed on the basis of the inverse Radon transform, the best known method being filtered back projection (FBP). It is also used in seismic exploration, electron microscope tomography, and nuclear medicine (PET/SPECT).

Q3. What is the Fourier slice theorem?

The Fourier slice theorem (projection-slice theorem) states that the one-dimensional Fourier transform, with respect to $s$, of the projection data at angle $\theta$ equals the two-dimensional Fourier transform of the original function evaluated on the line through the origin at angle $\theta$. This theorem provides the theoretical foundation for CT scan image reconstruction algorithms such as filtered back projection (FBP).