iradon
Inverse Radon transform
Description
I = iradon(R,theta)
reconstructs the image I from projection data in
R captured at projection angles
theta.
iradon uses default
values for arguments that you omit.
also returns the frequency response of the filter,
Examples
Compare Filtered and Unfiltered Backprojection
Create an image of the phantom. Display the image.
P = phantom(128);
imshow(P)
title('Original image')
Perform a Radon transform of the image.
R = radon(P,0:179);
Perform filtered backprojection.
Perform unfiltered backprojection.
Display the reconstructed images.
figure
subplot(1,2,1)
imshow(I1,zgdxq)
title('Filtered Backprojection')
subplot(1,2,2)
imshow(I2,zgdxq)
title('Unfiltered Backprojection')
Examine Backprojection at a Single Angle
Create an image of the phantom.
P = phantom(128);
Perform a Radon transform of the image, then get the projection vector corresponding to a projection at a 45 degree angle.
R = radon(P,0:179);
r45 = R(:,46);
Perform the ctdxb transform of this 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.
I = iradon([r45 r45], [45 45])/2;
Display the result.
imshow(I, zgdxq)
title('Backprojection from 45 degrees')
Input Arguments
R — Parallel beam projection data
numeric column vector | numeric matrix
Parallel beam projection data, specified as one of the following.
If theta is a scalar, then specify
R as a numeric column vector containing
the Radon transform for theta
degrees.
If theta is a vector, then specify
R as a 2-D matrix in which each column
is the Radon transform for one of the angles in
theta.
Data Types:single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
theta — Projection angles
numeric vector | numeric scalar | zgdxq
Projection angles (in degrees), specified as one of the
following.
between the angles.
numeric scalarIncremental angle between projections. Projections
are taken at angles m*theta, where
0,1,2,...,size(R,2)-1.
zgdxqAutomatically set the incremental angle between
projections to 180/size(R,2)
Data Types:double
interp — Type of interpolation
'linear' (default) | 'nearest' | 'spline' | 'pchip' | 'v5cubic'
Type of interpolation to use in the back projection, specified as one of
these values, listed in order of increasing accuracy and computational
complexity.
'linear'Linear interpolation
'spline'Spline interpolation
'pchip'Shape-preserving piecewise cubic interpolation
'v5cubic'Cubic convolution used in MATLAB® 5
Data Types:char | string
filter — Filter
'Ram-Lak' (default) | 'Shepp-Logan' | 'Cosine' | 'Hamming' | 'Hann' | 'None'
Filter to use for frequency domain filtering, specified as one of these
values.
ValueDescription'Ram-Lak'Cropped Ram-Lak or ramp filter. 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 de-emphasizes high
frequencies.
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. iradon returns
unfiltered backprojection data.
Data Types:char | string
frequency_scaling — Scale factor
1 (default) | positive number in the range (0, 1]
Scale factor for rescaling the frequency axis, specified as a positive
number in the range (0, 1]. If frequency_scaling is
less than 1, then the filter is compressed to fit into the frequency range
frequencies above frequency_scaling are set to
0.
output_size — Number of rows and columns in the reconstructed image
positive integer
Number of rows and columns in the reconstructed image, specified as a
positive integer. If output_size is not specified, the
size is determined from the length of the projections according to:
output_size = 2*floor(size(R,1)/(2*sqrt(2)))
If you specify output_size, then
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, then the
reconstructed image might not be the same size as the original image.
Output Arguments
I — Grayscale image
numeric matrix
Grayscale image, returned as a numeric matrix. If input projection data
R is data type single, then
I is single; otherwise
I is double.
Data Types:single | double
H — Frequency response
numeric vector
Frequency response of the filter, returned as a numeric vector.
Data Types:double
Algorithms
iradon assumes that the center of rotation is the center point of
the projections, which is defined as ceil(size(R,1)/2).
iradon uses the filtered back projection algorithm to perform the
ctdxb 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
References
Computerized Tomographic Imaging, New York, NY, IEEE Press,
1988.
Extended Capabilities
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
Usage notes and limitations:
The GPU implementation of this function supports only nearest-neighbor and
linear interpolation methods.
For more information, see Image Processing on a GPU.
Introduced before yldmf/p>