radon

Radon transform

Syntax

R = radon(I)

R = radon(I,theta)

[R,xp] = radon(___)

Description

R = radon(I) returns the Radon transform R of 2-D grayscale image I for angles in the range [0, 179] degrees. The Radon transform is the projection of the image intensity along a radial line oriented at a specific angle.

R = radon(I,theta) returns the Radon transform for the angles specified by theta.

[R,xp] = radon(___) returns a vector xp containing the radial coordinates corresponding to each row of the image.

Examples

Calculate Radon Transform and Display Plot

Make the axes scale visible for this image.

iptsetpref('ImshowAxesVisible','on')

Create a sample image.

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

Calculate the Radon transform.

theta = 0:180;
[R,xp] = radon(I,theta);

Display the transform.

imshow(R,[],'Xdata',theta,'Ydata',xp,'InitialMagnification','fit')
xlabel('\theta (degrees)')
ylabel('x''')
colormap(gca,hot), colorbar

Make the axes scale invisible.

iptsetpref('ImshowAxesVisible','off')

Input Arguments

I — Grayscale image

2-D numeric matrix

Grayscale image, specified as a 2-D numeric matrix.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical

theta — Projection angles

0:179 (default) | numeric scalar | numeric vector

Projection angles in degrees, specified as a numeric scalar or numeric vector.

Data Types: double

Output Arguments

R — Radon transform

numeric column vector | numeric matrix

Radon transform of image I, returned as one of the following.

• If theta is a scalar, then R is a numeric column vector containing the Radon transform for theta degrees.

• If theta is a vector, then R is a matrix in which each column is the Radon transform for one of the angles in theta.

xp — Radial coordinates

numeric vector

Radial coordinates corresponding to each row of R, returned as a numeric vector. The radial coordinates are the values along the x'-axis, which is oriented at theta degrees counterclockwise from the x-axis. The origin of both axes is the center pixel of the image, which is defined as

floor((size(I)+1)/2)

For example, in a 20-by-30 image, the center pixel is (10,15).

Algorithms

The Radon transform of an image is the sum of the Radon transforms of each individual pixel.

The algorithm first divides pixels in the image into four subpixels and projects each subpixel separately, as shown in the following figure.

Each subpixel's contribution is proportionally split into the two nearest bins, according to the distance between the projected location and the bin centers. If the subpixel projection hits the center point of a bin, the bin on the axes gets the full value of the subpixel, or one-fourth the value of the pixel. If the subpixel projection hits the border between two bins, the subpixel value is split evenly between the bins.

References

[1] Bracewell, Ronald N., Two-Dimensional Imaging, Englewood Cliffs, NJ, Prentice Hall, 1995, pp. 505-537.

[2] Lim, Jae S., Two-Dimensional Signal and Image Processing, Englewood Cliffs, NJ, Prentice Hall, 1990, pp. 42-45.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

radon supports the generation of C code (requires MATLAB^® Coder™). For more information, see Code Generation for Image Processing.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Image Processing on a GPU.

Version History

Introduced before R2006a

R2022b: Generate C code using MATLAB Coder

radon now supports the generation of C code (requires MATLAB Coder).

R2022b: Support for thread-based environments

radon now supports thread-based environments.