Path:Index:Reconstruction
The most common algorithm used in the tomographic reconstruction
of clinical data is the
filtered backprojection method.
Other methods also exist, please refer to the section on
iterative
reconstruction methods.
1. Data Projection
As a SPECT camera rotates around a patient, it creates a
series of planar images called projections. At
each stop, only photons moving perpendicular to the camera face pass through
the collimator. As many of these photons originate from various depths
in the patient, the result is an overlapping of all
tracer emitting organs along the specific path, much in the same manner
that an X-ray radiograph is a superposition of all anatomical structures
from three dimensions into two dimensions.
A SPECT study consists of many planar images acquired
at various angles. The movie below displays a set of projections
taken of a patient's bone scan.
After all the projections are acquired, they are subdivided
by taking all the projections for a single, thin slice of
the patient at a time.
All the projections for each slice are then ordered into an image called
a
sinogram as shown below. It represents the projection of the tracer distribution in the body into a single slice on the camera at every angle of the acquisition.
The aim of the reconstruction process is to retrieve the radiotracer spacial distribution from the projection data as it is illustrated below. This surface rendered image was reconstructed using a fully 3D OSEM algorithm.
2. Fourier Transform of Data
If the projection sinogram data were reconstructed
at this point, artifacts would appear in the reconstructed
images due to the nature of the subsequent backprojection
operation. Additionally, due to the random nature of radioactivity,
there is an inherent noise in the data that tends to make the reconstructed
images rough. In order to account for both of these effects, it is
necessary to filter the data. When we filter data, we can filter
it directly in the projection space, which means that we convolute the data
by some sort of smoothing kernel.
Convolution is a computationally
intensive task however and so it is useful to avoid using it when possible.
It turns out that the process of convolution in the spatial domain
is equivalent to a multiplication in the frequency domain.
This means that any filtering done by the convolution operation in the
normal spatial domain can be performed by a simple multiplication when
transformed into the frequency domain. To see what is meant by the
spatial domain and the frequency domain consider the following illustrative
example.
Consider a picket fence surrounding Old Lady Fourier's yard.
Since Old Lady Fourier has lived here for a long time and has never looked
after her picket fence, it is rather decrepit. At one time, the pickets
were all evenly spaced apart and there were exactly 33 pickets over the
10 meter width of her yard when expressed in the spatial domain.
We can express this in the frequency domain however by saying that the
picket frequency is 3.3 pickets per m, or 3.3 m-1.
When the fence was new, we can plot
graphically in the spatial domain, the number of pickets vs the length
in the yard. The same plot is shown as plotted in the frequency domain.
In the spatial domain, there is one picket spaced every 0.33 m along the
fence over the entire 10 m length. When transformed, we see that
there is a large peak at the frequency 3.33 m-1, which corresponds
to all the pickets being spaced equally apart.
As some of the pickets disappear, there
is a change in these plots to the ones shown below. Some of the pickets
are missing from the spatial plot, and we see that in the frequency space,
there is a second peak emerging at 1 m-1 as now not all the
pickets are 0.33 meters apart. These pickets are now 1.0 meters apart
and so the frequency has decreased to 1.0 m-1 for these pickets.
This change in the way the same data
is displayed is called a transform. In SPECT imaging
we make a similar transform of the projection data into the frequency space
whereby we can more efficiently filter the data. The transform that
we make use of is called the one dimensional Fourier Transform,
so named after Old Lady Fourier.
3. Data Filtering
Once the data has been transformed to
the frequency domain, it is then filtered in order to smooth out the statistical
noise. There are many different filters available to filter the data
and they all have slightly different characteristics. For instance,
some will smooth very heavily so that there are not any sharp edges, and
hence will degrade the final image resolution. Other filters will
maintain a high resolution while only smoothing slightly. Some typical
filters used are the Hanning filter, Butterworth filter, low pass cosine
filter, Weiner filter, etc. Regardless of the filter used, the end
result is to display a final image that is relatively free from noise and
is pleasing to the eye. The next figure depicts three objects reconstructed
without a filter true (left), without a filter noisy (middle) and with a Hanning filter (right).
4. Inverse Transform of the Data
As the newly smoothed data is now in the
frequency domain, we must transform it back into the spatial domain in
order to get out the x,y,z information regarding spatial distribution.
This is done in the same type of manner as the original transformation
is done, except we use what is called the one dimensional inverse
Fourier Transform. Data at this point is similar to the original (left)
sinogram except it is smoothed as seen below (right).
5. Backprojection
The main reconstruction step involves
a process known as
backprojection. As the original
data was collected by only allowing photons emitted perpendicular to the
camera face to enter the camera, backprojection smears the camera bin data
from the filtered sinogram back along the same lines from where the photon
was emitted from. Regions where backprojection lines from different
angles intersect represent areas which contain a higher concentration of
radiopharmaceutical.
Click on the image below to see an
MPEG movie depicting the backprojection process.
