Radon Transform

Note

For information about creating projection data from line integrals along paths that radiate from a single source, called fan-beam projections, seeFan-Beam Projection. To convert parallel-beam projection data to fan-beam projection data, use thepara2fanfunction.

Theradonfunction computes预测of an image matrix along specified directions.

A projection of a two-dimensional functionf(x,y)is a set of line integrals. Theradonfunction computes the line integrals from multiple sources along parallel paths, orbeams, in a certain direction. The beams are spaced 1 pixel unit apart. To represent an image, theradonfunction 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 off(x,y)in the vertical direction is the projection off(x,y)onto thex-axis; the line integral in the horizontal direction is the projection off(x,y)onto they-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 angletheta(θ). In general, the Radon transform off(x,y)is the line integral offparallel to they´-axis

$R_{θ} (x') = \int_{- \infty}^{\infty} f (x' \cos θ - y' \sin θ, x' \sin θ + y' \cos θ) d y'$

where

$[\begin{matrix} x' \\ y' \end{matrix}] = [\begin{matrix} \cos θ & sin θ \\ - \sin θ & cos θ \end{matrix}] [\begin{matrix} x \\ y \end{matrix}]$

The following figure illustrates the geometry of the Radon transform.

Geometry of the Radon Transform

Plot the Radon Transform of an Image

Open Live Script

This example shows how to compute the Radon transform of an image,I, for a specific set of angles,theta, using theradonfunction. The function returns,R, in which the columns contain the Radon transform for each angle intheta. The function also returns the vector,xp, which contains the corresponding coordinates along the x-axis. The center pixel ofIis defined to befloor((size(I)+1)/2), which is the pixel on the x-axis corresponding tox' = 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)

Figure contains an axes object. The axes object contains an object of type image.

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

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

情节转换for 0 degrees.

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

Figure contains an axes object. The axes object with title Radon Transform of a Square Function at 0 degrees contains an object of type line.

情节转换for 45 degrees.

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

Figure contains an axes object. The axes object with title Radon Transform of a Square Function at 45 degrees contains an object of type line.

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

拉东变换使用180年预测