What Is the Radon Transform?

The Radon transform takes a two-dimensional function $f\left(x,,,y\right)$ and integrates it over straight lines. In its classic formulation, the transform is defined as

$R\left(f,;,\theta ,,,s\right)={\int }_{-\infty }^{\infty }f\left(s\&cos;\theta -t\&sin;\theta ,,,s\&sin;\theta +t\&cos;\theta \right)\mathrm{dt}$

where $\theta$ denotes the projection angle and $s$ represents the signed distance from the origin to the line. This transform underlies computed tomography: X-ray images record line integrals of tissue density, and the original image is reconstructed from these projections.

Although the integral looks daunting, the concept is straightforward. For each angle $\theta$ we slide a line across the plane and sum the function’s values along that line. By repeating the process for many angles, we gather enough data to reconstruct $f$. The beauty of the Radon transform lies in its ability to convert spatial information into a collection of one-dimensional signals.

Why This Calculator?

Students learning about medical imaging or integral geometry often encounter the Radon transform in theoretical form. This calculator demystifies the integral by allowing you to enter a function and parameters, then see the numeric result. For simple functions like Gaussian bumps or indicator regions, the transform has known analytic expressions. Yet it’s instructive to verify these results numerically and experiment with variations.

The integral is approximated here using a symmetric finite interval for $t$ with Simpson’s rule. While true projections run over an infinite line, most reasonable functions decay quickly, so truncating at a moderate bound yields an accurate estimate. You can adjust the range in the script if your function extends farther.

Connections to Tomography

The Radon transform forms the mathematical basis of CT scans. In tomography, X-rays pass through the body at many angles, and detectors measure the resulting attenuation. Each measurement corresponds to an integral of tissue density along the ray path—exactly a Radon transform value. Reconstructing a cross-sectional image requires inverting the transform, often using filtered back-projection or iterative algorithms. Understanding the forward transform is therefore a stepping stone to mastering image reconstruction.

In mathematics, the Radon transform also appears in harmonic analysis and partial differential equations. It links functions on the plane with functions on the set of lines, revealing symmetry properties and enabling new perspectives on the wave equation. Historically, Johann Radon introduced the transform in 1917 while studying integral geometry, decades before its medical applications.

Practical Experimentation

Try computing the transform of a Gaussian $f\left(x,,,y\right)=e^{-x{2}^{}-y{2}^{}}$. Because Gaussian functions are radially symmetric, the transform should also be Gaussian in $s$ and independent of $\theta$. You can confirm this numerically with the calculator by varying $\theta$ and observing that the result does not change significantly.

Alternatively, experiment with a square indicator function $f\left(x,,,y\right)=\begin{array}{ccccc}1& \text{if}& \left|x\right|<1& \text{and}& \left|y\right|<1\\ 0& \text{otherwise}\end{array}$. The transform of this shape reveals piecewise linear behavior as you rotate the line. Visualizing these results deepens intuition for how projections encode geometry.

Broad Perspective

While medical imaging is the most famous application, the Radon transform shows up wherever projections of higher-dimensional data are of interest. In geophysics, seismic waves yield line integrals of Earth’s interior properties. In astronomy, certain instruments capture integrated light along lines of sight. Learning to compute the Radon transform thus unlocks insights across scientific disciplines.

The Radon transform also has an elegant inversion formula involving the Fourier slice theorem. Although beyond the scope of this calculator, the theorem states that the one-dimensional Fourier transform of a projection equals a slice of the two-dimensional Fourier transform of $f$. This powerful result forms the heart of many reconstruction algorithms and demonstrates the deep connections between analysis and geometry.

By exploring the examples and references within this tool, you can build a foundation for more advanced topics. Whether your goal is to understand CT technology, delve into harmonic analysis, or simply appreciate an intriguing integral transform, the Radon transform is a rewarding subject.