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The `iradon`

function
inverts the Radon transform and can therefore be used to reconstruct
images.

As described in Radon Transform,
given an image `I`

and a set of angles `theta`

,
the `radon`

function can be used
to calculate the Radon transform.

R = radon(I,theta);

The function `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 `I`

.

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 $${R}_{\theta}(x\prime )$$ 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 Data. To convert parallel-beam projection
data into fan-beam projection data, use the `para2fan`

function.

`iradon`

uses the *filtered backprojection* 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
reconstructed image `IR`

more accurately approximates
the original image `I`

. The vector `theta`

must
contain monotonically increasing angular values with a constant incremental
angle `Dtheta`

. When the scalar `Dtheta`

is
known, it can be passed to `iradon`

instead of the
array of theta values. Here is an example.

IR = iradon(R,Dtheta);

The filtered backprojection 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 call to `iradon`

below applies a Hamming
window to the filter. See the `iradon`

reference
page for more information. To get unfiltered backprojection 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);

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 the `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.

`R1`

has 18 projections,`R2`

has 36 projections, and`R3`

has 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`I1`

and`I2`

), the reconstruction can include some artifacts from the back projection.**Inverse Radon Transforms of the Shepp-Logan Head Phantom**

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