## Radon Transform

**Note**

For information about creating projection data from line integrals along paths
that radiate from a single source, called fan-beam projections, see Fan-Beam Projection. To convert
parallel-beam projection data to fan-beam projection data, use the
`para2fan`

function.

The `radon`

function computes
*projections* of an image matrix along specified directions.

A projection of a two-dimensional function *f(x,y)* is a set of
line integrals. The `radon`

function computes the line integrals from
multiple sources along parallel paths, or *beams*, in a certain
direction. The beams are spaced 1 pixel unit apart. To represent an image, the
`radon`

function takes multiple, parallel-beam projections of the
image from different angles by rotating the source around the center of the image. The
following figure shows a single projection at a specified rotation angle.

**Parallel-Beam Projection at Rotation Angle Theta**

For example, the line integral of *f(x,y)* in the vertical
direction is the projection of *f(x,y)* onto the
*x*-axis; the line integral in the horizontal direction is the
projection of *f(x,y)* onto the *y*-axis. The
following figure shows horizontal and vertical projections for a simple two-dimensional
function.

**Horizontal and Vertical Projections of a Simple
Function**

Projections can be computed along any angle *theta* (θ). In
general, the Radon transform of *f(x,y)* is the line integral of
*f* parallel to the *y*´-axis

$${R}_{\theta}\left(x\prime \right)={\displaystyle {\int}_{-\infty}^{\infty}f\left(x\prime \mathrm{cos}\theta -y\prime \mathrm{sin}\theta ,x\prime \mathrm{sin}\theta +y\prime \mathrm{cos}\theta \right)dy\prime}$$

where

$$\left[\begin{array}{c}x\prime \\ y\prime \end{array}\right]=\left[\begin{array}{cc}\text{}\mathrm{cos}\theta & \text{sin}\theta \\ -\mathrm{sin}\theta & \text{cos}\theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

The following figure illustrates the geometry of the Radon transform.

**Geometry of the Radon Transform**

### Plot the Radon Transform of an Image

This example shows how to compute the Radon transform of an image, `I`

, for a specific set of angles, `theta`

, using the `radon`

function. The function returns, `R`

, in which the columns contain the Radon transform for each angle in `theta`

. The function also returns the vector, `xp`

, which contains the corresponding coordinates along the x-axis. The center pixel of `I`

is defined to be `floor((size(I)+1)/2)`

, which is the pixel on the x-axis corresponding to *x' = 0*.

Create a small sample image for this example that consists of a single square object and display it.

I = zeros(100,100); I(25:75,25:75) = 1; imshow(I)

Calculate the Radon transform of the image for the angles 0 degrees and 45 degrees.

[R,xp] = radon(I,[0 45]);

Plot the transform for 0 degrees.

```
figure
plot(xp,R(:,1));
title('Radon Transform of a Square Function at 0 degrees')
```

Plot the transform for 45 degrees.

```
figure
plot(xp,R(:,2));
title('Radon Transform of a Square Function at 45 degrees')
```

### Viewing the Radon Transform as an Image

The Radon transform for a large number of angles is often displayed as an image. In this example, the Radon transform for the square image is computed at angles from 0° to 180°, in 1° increments.

theta = 0:180; [R,xp] = radon(I,theta); imagesc(theta,xp,R); title('R_{\theta} (X\prime)'); xlabel('\theta (degrees)'); ylabel('X\prime'); set(gca,'XTick',0:20:180); colormap(hot); colorbar

**Radon Transform Using 180 Projections**