The Inverse Radon Transformation
Inverse Radon Transform Definition
iradon function inverts the Radon
transform and can therefore be used to reconstruct images.
R = radon(I,theta);
iradon can then be called to reconstruct the image
I from projection data.
IR = iradon(R,theta);
In the example above, projections are calculated from the original image
Note, however, that in most application areas, there is no original image from
which projections are formed. For example, the inverse Radon transform is commonly
used in tomography applications. In X-ray absorption tomography, projections are
formed by measuring the attenuation of radiation that passes through a physical
specimen at different angles. The original image can be thought of as a cross
section through the specimen, in which intensity values represent the density of the
specimen. Projections are collected using special purpose hardware, and then an
internal image of the specimen is reconstructed by
iradon. This allows for noninvasive
imaging of the inside of a living body or another opaque object.
iradon reconstructs an image from parallel-beam projections. In
parallel-beam geometry, each projection is formed by
combining a set of line integrals through an image at a specific angle.
The following figure illustrates how parallel-beam geometry is applied in X-ray absorption tomography. Note that there is an equal number of n emitters and n sensors. Each sensor measures the radiation emitted from its corresponding emitter, and the attenuation in the radiation gives a measure of the integrated density, or mass, of the object. This corresponds to the line integral that is calculated in the Radon transform.
The parallel-beam geometry used in the figure is the same as the geometry that was described in Radon Transform. f(x,y) denotes the brightness of the image and is the projection at angle theta.
Parallel-Beam Projections Through an Object
Another geometry that is commonly used is fan-beam geometry,
in which there is one source and n sensors. For more
information, see Fan-Beam Projection.
To convert parallel-beam projection data into fan-beam projection data, use the
Improving the Results
iradon uses the filtered back
projection algorithm to compute the inverse Radon transform. This
algorithm forms an approximation of the image
I based on the
projections in the columns of
R. A more accurate result can
be obtained by using more projections in the reconstruction. As the number of
projections (the length of
theta) increases, the
IR more accurately approximates the
I. The vector
contain monotonically increasing angular values with a constant incremental
Dtheta. When the scalar
known, it can be passed to
iradon instead of the array of
theta values. Here is an example.
IR = iradon(R,Dtheta);
The filtered back projection algorithm filters the projections in
R and then reconstructs the image using the filtered
projections. In some cases, noise can be present in the projections. To remove
high frequency noise, apply a window to the filter to attenuate the noise. Many
such windowed filters are available in
iradon. The example
iradon below applies a Hamming window to the filter.
iradon reference page for more
information. To get unfiltered back projection data, specify
'none' for the filter parameter.
IR = iradon(R,theta,'Hamming');
iradon also enables you to specify a normalized frequency,
D, above which the filter has zero response.
D must be a scalar in the range [0,1]. With this option,
the frequency axis is rescaled so that the whole filter is compressed to fit
into the frequency range
[0,D]. This can be useful in cases
where the projections contain little high-frequency information but there is
high-frequency noise. In this case, the noise can be completely suppressed
without compromising the reconstruction. The following call to
iradon sets a normalized frequency value of 0.85.
IR = iradon(R,theta,0.85);
Reconstructing an Image from Parallel Projection Data
The commands below illustrate how to reconstruct an image from parallel projection
data. The test image is the Shepp-Logan head phantom, which can be generated using
phantom function. The phantom image illustrates many of the
qualities that are found in real-world tomographic imaging of human heads. The
bright elliptical shell along the exterior is analogous to a skull, and the many
ellipses inside are analogous to brain features.
Create a Shepp-Logan head phantom image.
P = phantom(256); imshow(P)
Compute the Radon transform of the phantom brain for three different sets of theta values.
R1has 18 projections,
R2has 36 projections, and
R3has 90 projections.
theta1 = 0:10:170; [R1,xp] = radon(P,theta1); theta2 = 0:5:175; [R2,xp] = radon(P,theta2); theta3 = 0:2:178; [R3,xp] = radon(P,theta3);
Display a plot of one of the Radon transforms of the Shepp-Logan head phantom. The following figure shows
R3, the transform with 90 projections.
figure, imagesc(theta3,xp,R3); colormap(hot); colorbar xlabel('\theta'); ylabel('x\prime');
Radon Transform of Head Phantom Using 90 Projections
Note how some of the features of the input image appear in this image of the transform. The first column in the Radon transform corresponds to a projection at 0º that is integrating in the vertical direction. The centermost column corresponds to a projection at 90º, which is integrating in the horizontal direction. The projection at 90º has a wider profile than the projection at 0º due to the larger vertical semi-axis of the outermost ellipse of the phantom.
Reconstruct the head phantom image from the projection data created in step 2 and display the results.
I1 = iradon(R1,10); I2 = iradon(R2,5); I3 = iradon(R3,2); imshow(I1) figure, imshow(I2) figure, imshow(I3)
The following figure shows the results of all three reconstructions. Notice how image
I1, which was reconstructed from only 18 projections, is the least accurate reconstruction. Image
I2, which was reconstructed from 36 projections, is better, but it is still not clear enough to discern clearly the small ellipses in the lower portion of the image.
I3, reconstructed using 90 projections, most closely resembles the original image. Notice that when the number of projections is relatively small (as in
I2), the reconstruction can include some artifacts from the back projection.
Inverse Radon Transforms of the Shepp-Logan Head Phantom