MATLAB: How to filter/smooth an image that has circular symmetry

filterimage processingMATLABringsymmetry

Hello,

I have a set of 2d data with circular symmetry stored as matrices. I wish to use the symmetry in order to smooth my image by using the other points that are at a comparable distance from the center. This would be in contrast to a filter that uses local neighbouring points to smooth the data, which tends to distort this data.

The image presented here is quite smooth but I have many images with noise.

Thanks, Aurélien

Best Answer

One approach would be to use radon and iradon,

noisyImage= (1+sqrt((-100:100).^2 + (-100:100).'.^2));
noisyImage=noisyImage.*(noisyImage<=100) + 5*randn(size(noisyImage));
theta=0:0.25:180;
R=mean( radon(noisyImage,theta) ,2);
smoothImage=iradon(      repmat(R,1,numel(theta)) ,theta  );
imshowpair(noisyImage,smoothImage,'montage')
colormap hot

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Radon

Radon transform Syntax

R = radon(I, theta) [R,xp] = radon(...)

Description

R = radon(I, theta) returns the Radon transform R of the intensity image I for the angle theta degrees.

The Radon transform is the projection of the image intensity along a radial line oriented at a specific angle. If theta is a scalar, R is a column vector containing the Radon transform for theta degrees. If theta is a vector, R is a matrix in which each column is the Radon transform for one of the angles in theta. If you omit theta, it defaults to 0:179.

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

The radial coordinates returned in xp 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

MATLAB: Transform a radius into a circle

here is the solution:

%%Example data
x = (2:0.1:8);
data = sin(x);
%%Shift the values towards the center, so you get a circle instead of an annulus.
x = x-min(x);
%%Your meshgrid generation code.
rho = x;
theta = (0:0.5:90)*pi/180;
[th, r] = meshgrid(theta, rho);
%%Plotting the values
surf(r.*cos(th), ...
     r.*sin(th), ...
     repmat(data(:), 1, size(th,2)), ...
     'linestyle', 'none'); 
view(2);
axis equal tight;
colorbar;

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