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Radon Transform


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.



Parallel-Beam Projection at Rotation Angle Theta


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



[xy]=[ cosθsinθsinθcosθ][xy]

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;

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


Plot the transform for 0 degrees.

title('Radon Transform of a Square Function at 0 degrees')

Plot the transform for 45 degrees.

title('Radon Transform of a Square Function at 45 degrees')

Viewing the Radon Transform as an Image


theta = 0:180;
xlabel('\theta (degrees)');

Radon Transform Using 180 Projections