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Let's consider how these images of body slices are generated
from the data recorded by the scanner.
The basic mathematics behind this problem
was actually solved in general terms as long ago as 1917.
In many ways it's rather like trying to calculate the positions
of all the trees in a forest,
from photographs taken through the forest in various different directions.
Here's Dr James Cubillo of Coventry University.
The tomographic imaging process begins of course with data acquisition.
This is referred to as the forward projection.
The data, in the form of attenuation profiles,
is obtained at equal intervals as the instrument rotates about the body.
I'll demonstrate that with a simple object.
Here we have a square object with uniform high attenuation.
This is the imaging plane.
If we irradiate the object
with a parallel beam of X-rays from the left here,
you can see that we obtain an attenuation profile here.
Outside of the object area we have zero attenuation.
Where we have the object, we have uniform high attenuation levels.
As we rotate the instrument through some angle,
you can see that we still have zero attenuation outside of the object.
Uniform attenuation here between these two points,
where we have equal path lengths.
Between these two points, the path length is gradually increasing,
and you see we have an attenuation profile which also increases uniformly.
At 45 degrees, you can see that we have maximum attenuation here,
corresponding to the diagonal of the square,
which of course, is maximum for this object.
Between these two points we have uniformly increasing attenuation,
as the path length increases.
And the process continues as we rotate the instrument,
relative to the object.
In these forward projections,
James has actually rotated the object,
and fixed the direction of the beam,
to make it easier to compare one attenuation profile with the next.
However, in an actual scan, and in the reconstructions to come,
it's the instrument that rotates and not the object.
Now, let's look at a slightly more complicated scene,
in which we have two squares,
each of high uniform attenuation.
We irradiate the scene as before,
and you can see that we have rectangular attenuation profiles
for each of the squares.
Rotating the beam through 45 degrees,
we have two triangular profiles,
corresponding to each of these two squares.
At 90 degrees, we're back to the rectangular profiles.
However, at 135 degrees, the two squares are in line,
and therefore we have maximum attenuation,
corresponding to the maximum path length
of these two squares in line.
These attenuation profiles,
and knowledge of the direction of each forward projection,
are all that's required to reconstruct the original object.
This process is known as simple backprojection.
Here we have the original object.
This scale here is known as the hot body scale,
with white and yellow representing high attenuation,
and red and black representing low attenuation.
Planning a backprojection at zero degrees,
we have an equal likelihood,
of the objects occurring anywhere along these paths.
Now if we apply a backprojection at 90 degrees,
we can see that, surprisingly enough,
we have four candidate positions for these squares,
leaving us with an ambiguous situation.
It is only when we add the 45 degree projection,
that we confirm the positions of the two squares.
Going on further to a 135 degree projection,
again the position is reinforced once more.
This ambiguity would not have existed with a single square.
This illustrates a general point.
The more complex the images are,
the more projections are required,
to avoid ambiguity in a reconstructed image.
If we go on further, to reconstructing the whole image,
starting with zero degree projection,
and building up at 15 degree intervals,
we can see the two squares forming with each projection.
And now, with the completed image,
with 24 backprojections,
we can see clearly the two squares that we started with.
However, they do appear slightly blurred.
Convolution of the object with the point spread function,
leads to a blurring effect.
As you can see more clearly
in this three-dimensional mesh plot of the same data.
This problem is inherent to the backprojection process,
and cannot be avoided, even by taking extra projections.
But, as you can see,
a simple object, consisting of two squares,
can still be reconstructed reasonably well.
However, when simple backprojection is applied to real CT data,
the blurring effect seems to be much more serious.
There are two reasons for this -
firstly, the objects imaged in a real CT scan
have a whole range of density values,
whereas the squares in James' model have just one.
Secondly, a much higher spatial resolution is required.
Nevertheless, the technique can be improved,
by filtering the data before carrying out the reconstruction,
and this is known as filtered backprojection.
Here is an infinite ramp filter that can be applied to the reconstruction.
Along the horizontal axis you can seethe spatial frequency,
and on the vertical axis we can see the relative amplitude,
of the signal passed by the filter.
You can see that the filter itself discriminates against low frequencies,
in favour of the high frequencies.
However, with such a filter, we will run into aliasing problems.
So we need to apply the Nyquist criteria,
to cut off the filter at half the maximum spatial sampling frequency,
as we have here.
Now let's see what we get when we apply this filter,
to the backprojections we saw earlier.
Here we have the two original objects,
and here is the filtered backprojection.
As you can see, we get a much clearer, sharper image.
Although it is still not perfect, much of the blurring has been removed.
And here is the mesh plot.
There's obviously a vast improvement over the simple backprojection.
And here we have a comparison of the simple backprojection,
compared with the filtered backprojection,
with their respective mesh plots,
and the improvement is clear to see.
OK, so filtering the data
greatly improves the reconstruction of this simple model,
but how well does it perform on the pelvic slice we saw earlier?
Well, as you can see, we now have a clear sharp image,
in which we can identify the top of the femur, the bladder,
and the spinal cord.
A vast improvement over the blurred image we obtained
with simple backprojection.
Filtered backprojection is the method of reconstruction used
in modern CT imaging.
The only difference from the process you've just seen,
is that the filters are rather more sophisticated
than the cut-off ramp filter James used.