iradon

Inverse Radon transform

Syntax

I = iradon(R, theta)
I = iradon(R,theta,interp,filter,frequency_scaling,output_size)
[I,H] = iradon(...)
[___]= iradon(gpuarrayR,___)

Description

I = iradon(R, theta) reconstructs the image I from projection data in the two-dimensional array R. The columns of R are parallel beam projection data. iradon assumes that the center of rotation is the center point of the projections, which is defined as ceil(size(R,1)/2).

theta describes the angles (in degrees) at which the projections were taken. It can be either a vector containing the angles or a scalar specifying D_theta, the incremental angle between projections. If theta is a vector, it must contain angles with equal spacing between them. If theta is a scalar specifying D_theta, the projections were taken at angles theta = m*D_theta, where m = 0,1,2,...,size(R,2)-1. If the input is the empty matrix ([]), D_theta defaults to 180/size(R,2).

iradon uses the filtered back-projection algorithm to perform the inverse Radon transform. The filter is designed directly in the frequency domain and then multiplied by the FFT of the projections. The projections are zero-padded to a power of 2 before filtering to prevent spatial domain aliasing and to speed up the FFT.

I = iradon(R,theta,interp,filter,frequency_scaling,output_size) specifies parameters to use in the inverse Radon transform. You can specify any combination of the last four arguments. iradon uses default values for any of these arguments that you omit.

interp specifies the type of interpolation to use in the back projection. The available options are listed in order of increasing accuracy and computational complexity.

Value

Description

'nearest'

Nearest-neighbor interpolation

'linear'

Linear interpolation (the default)

'spline'

Spline interpolation

'pchip'Shape-preserving piecewise cubic interpolation
'v5cubic'Cubic interpolation from MATLAB 5. This method does not extrapolate, and it issues a warning and uses'spline' if X is not equally spaced.

filter specifies the filter to use for frequency domain filtering. filter can be any of the strings that specify standard filters.

Value

Description

'Ram-Lak'

Cropped Ram-Lak or ramp filter. This is the default. The frequency response of this filter is | f |. Because this filter is sensitive to noise in the projections, one of the filters listed below might be preferable. These filters multiply the Ram-Lak filter by a window that deemphasizes high frequencies.

'Shepp-Logan'

Multiplies the Ram-Lak filter by a sinc function

'Cosine'

Multiplies the Ram-Lak filter by a cosine function

'Hamming'

Multiplies the Ram-Lak filter by a Hamming window

'Hann'

Multiplies the Ram-Lak filter by a Hann window

'None'No filtering. When you specify this value, iradon returns unfiltered backprojection data.

frequency_scaling is a scalar in the range (0,1] that modifies the filter by rescaling its frequency axis. The default is 1. If frequency_scaling is less than 1, the filter is compressed to fit into the frequency range [0,frequency_scaling], in normalized frequencies; all frequencies above frequency_scaling are set to 0.

output_size is a scalar that specifies the number of rows and columns in the reconstructed image. If output_size is not specified, the size is determined from the length of the projections.

output_size = 2*floor(size(R,1)/(2*sqrt(2)))

If you specify output_size, iradon reconstructs a smaller or larger portion of the image but does not change the scaling of the data. If the projections were calculated with the radon function, the reconstructed image might not be the same size as the original image.

[I,H] = iradon(...) returns the frequency response of the filter in the vector H.

[___]= iradon(gpuarrayR,___) reconstructs the image gpuarrayI from projection data in the gpuArray R. The input image and the return values are 2-D gpuArrays. All other numeric arguments must be a double or a gpuArray of underlying class double. This syntax requires the Parallel Computing Toolbox™.

    Note:   The GPU implementation of this function supports only nearest-neighbor and linear interpolation methods for the back projection.

Class Support

R can be double or single. All other numeric input arguments must be of class double. I has the same class as R. H is double.

R can be a gpuArray of underlying class double or single. All other numeric input arguments must be double or gpuArray of underlying class double. I has the same class as R. H is a gpuArray of underlying class double.

Examples

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Calculate the inverse Radon Transform comparing filtered and unfiltered backprojection

Calculate the inverse Radon transform and compare filtered and unfiltered back projection.

P = phantom(128); 
R = radon(P,0:179);
I1 = iradon(R,0:179);
I2 = iradon(R,0:179,'linear','none');
subplot(1,3,1), imshow(P), title('Original')
subplot(1,3,2), imshow(I1), title('Filtered backprojection')
subplot(1,3,3), imshow(I2,[]), title('Unfiltered backprojection')

Compute the backprojection of a single projection vector. The iradon syntax does not allow you to do this directly, because if theta is a scalar it is treated as an increment. You can accomplish the task by passing in two copies of the projection vector and then dividing the result by 2.

P = phantom(128);
R = radon(P,0:179);
r45 = R(:,46);
I = iradon([r45 r45], [45 45])/2;
imshow(I, [])
title('Backprojection from the 45-degree projection')

Calculate the inverse Radon transform on a GPU

Calculate the inverse Radon transform on a GPU.

P = gpuArray(phantom(128));
R = radon(P,0:179);
I1 = iradon(R,0:179);
I2 = iradon(R,0:179,'linear','none');
subplot(1,3,1), imshow(P), title('Original')
subplot(1,3,2), imshow(I1), title('Filtered backprojection')
subplot(1,3,3), imshow(I2,[]), title('Unfiltered backprojection')

More About

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Algorithms

iradon uses the filtered back projection algorithm to perform the inverse Radon transform. The filter is designed directly in the frequency domain and then multiplied by the FFT of the projections. The projections are zero-padded to a power of 2 before filtering to prevent spatial domain aliasing and to speed up the FFT.

References

[1] Kak, A. C., and M. Slaney, Principles of Computerized Tomographic Imaging, New York, NY, IEEE Press, 1988.

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