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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5255_Библиотеки_им_академика_М_И_Перельмана
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Single Photon Emission Tomography (SPECT)152
Figure 9: Different projection angles in Radon transformation
Figure 10: Acquired data in the form of a sinogram
In SPECT studies the gamma camera rotates around the patient and acquires serial planar
images at different angles. Each row of pixels in a planar image can be assumed to be a onedimensional projection of a narrow slice of the patient’s body just over the pixel row. If the
corresponding rows of pixels in different views are placed together side by side serially, an
image is formed that is the Radon transform of the object and usually called a sinogram (Figure
10). This image is in radon domain and is not perceptible by the human brain except in very
simple situations. Therefore it has to be un-transformed into the spatial domain using Inverse
Radon Transformation. The inverse process changes the R (S, ) coordinates into spatial coordinates
g (x, y).
Single Photon Emission Tomography (SPECT) 153
Inverse Radon Transformation
In order to convert a sinogram into a slice image we have to perform the Radon transform in
opposite direction. In Radon transform the projections of an object are acquired at different
angles. In the inverse radon transform we have to back project them at the same angles as they
were acquired. Figure 9 graphically illustrates the basic concept of radon transform. Figure 11
illustrates the basic concept of back projection. In this image only one of the views is back
projected.
Figure 11: Backprojection of acquired data at one given angle
We can continue to back project all the views that overlap at the center of rotation. The
combination of all the back projected views creates an approximation of the real slice (Figure 12).
Figure 12: Backprojection of acquired data from multiple acquired angles
The accuracy (resolution) with which the slice image is obtained depends on the total
number of views used. The more the number of views the more accurate the image is (Figure 13).
Theoretically the number of views should be infinite to get the exact image of the slice. However
this is impossible in practice therefore the reconstructed image quality is not as good as it should
have been.
Single Photon Emission Tomography (SPECT)154
In figure 13 the basic principle of radon transform and back-projection are graphically
reviewed once more. Imagine a cubic shape phantom, filled uniformly with radioactivity. Suppose
32 serial planar images in 32x32 matrices are acquired from 32 angles around the phantom.
Figure 13a shows one of the planar images acquired at an angle = 0 and Figure 13b shows a
single row of pixels from this image. If the corresponding rows from other view are placed
together in order of their angular view, a Radon image is formed (Figure 13R). This image is the
radon transform of a slice of the cube (with an error caused due to digitization). The inverse
Radon transform (back-projection technique) un-transforms this image into spatial domain. The
process is conceptually very simple. Each row of pixels in the Radon image has to be backprojected to make a back projected image (Figure 13C). Back projection of a given pixel means
to make a line of pixels perpendicular to the image surface (line integral) with the same value as
the given pixel. The back projected image has to be rotated by an angle equal to the corresponding
angular view. At last all images have to be added to reconstruct the image of slice in spatial
domain (Figure 13d). The cube slice appears at the intersection of all of the back projections.
The procedure can be repeated for any row of pixels (vertical, horizontal or oblique) to reconstruct
the desired image slice.
Figure 13: Basic principle of Radon transform, (a) planar image acquired at a given angle,
(b) One row of pixels of planar image (R) corresponding rows from other views placed together
in order of their angular view, to form a Radon image. This image is the radon transform of a
slice of the cube (with error caused due to digitization). The backprojected image for data acquired
at q = 0° (c) the backprojected image from various rows (d).
Single Photon Emission Tomography (SPECT) 155
Accuracy of the Inverse Radon Transformation
The simple back-projection technique is actually a discrete implementation of inverse Radon
transform. We have to consider that equation-1 represents the analytical form of the Radon
transform. However in nuclear medicine, data are in discrete form therefore we cannot use the
formula in original form. We acquire the Radon transform of the objects (sinograms) in a
discrete form usually with a very low resolution (64×64 or 128×128 matrix size).
Obviously the reconstructed image cannot have better resolution than the corresponding
sinogram and if the number of views (projections) is not sufficient the resolution of the
reconstructed image becomes worse. Usually the number of the projected views is far less than it
should be and therefore the reconstructed slice image is streaky and visually unappealing. As the
number of projected views increases the streaks become less visible but they are still there to
blur the image.
Figure 14 shows the reconstructed image of a source using different number of views. As can
be seen, in the back projection technique the gross form of the object can be retrieved without
considerable distortion. However the details become unclear due to the presence of the noise.
Figure 14: The original image (A) gets blurred when reconstruction is done by SBP technique (B)
From the mathematical point of view it looks that the reconstructed image is inherently
filtered due to impediment with the back projection technique. Some analytical rummage reveals
the form of this inherent filter as shown in figure 15a. In order to compensate such blurring
effect, an inverse filter that is called the ramp filter must be used (Figure 15b). Therefore the
method is called Filtered Back Projection (FBP) technique.
Single Photon Emission Tomography (SPECT)156
Figure 15: Inherent suppression of high frequency components by SBP technique (a),
RAMP filter recovers high frequency data by amplifying them. High frequency noise
gets enough chance to get amplified.
Filtered Back Projection (FBP)
The inherent filtering of back projection blurs the image and removes the high frequency
component of data. In order to compensate the blurring we have to add the ramp filter into the
procedure, which amplifies the high frequency components.
These concepts are graphically illustrated in figure 16. The first part of figure (16A) represents
the LSF of point source, obtained by drawing a line profile over the actual image (true signal).
The next part (Figure 16B) presents a filter (sinc) function. If these two functions are multiplied
we get a curve with some negative spikes (Figure 16C)
Figure 16: Line spread function (LSF) obtained from the image profile of a line source (A), filter
function with negative spikes with decreasing amplitudes on either side (B). When the acquired
signal is multiplied by the filter function the resultant function looks like C.
Single Photon Emission Tomography (SPECT) 157
This is what the ramp filter actually does. The ramp filter produces negative spikes on either
side of the LSF, which, counteract the effect of the blurring. Figure 17 shows how a ramp filter
modifies the data during back projections.
Figure 17: SBP image on the left and modification of the same by RAMP filter (right)
The Filter back projection technique is not as successful in nuclear medicine as in other
tomographic techniques like computed tomography (CT). The main reason is that the nuclear
medicine data are very much noisy compared to the other imaging modalities.
The ramp filter is actually a high-pass filter. It suppresses the low frequency components and
allows the high frequency components to pass freely. During reconstruction, the high frequency
component of the data (noise) is freely transferred into the final image while low frequency
components (data) are more or less suppressed. Therefore the details of images are submersed in
the noise and become less visible (Figure 18). In order to compensate the effect an extra filter
has to be added into the procedure.
Figure 18: True image on the left. If there is noise in the data the ramp filter passes the noise into
the reconstructed image while the details are obscured as can be seen on the right side image.
Single Photon Emission Tomography (SPECT)158
The filter may be incorporated either before or after backprojection in the following three
ways:
1. Pre-filtering of planar views using a two-dimensional filter (filtering before reconstruction)
2. Combining a one-dimensional filter into the back-projection process (filtering during
reconstruction)
3. Post-filtering of the final reconstructed slice image using a two-dimensional filter (filtering
after reconstruction)
Two-dimensional filtering of many planar images is computationally intensive and time
consuming. Moreover in a SPECT study, among the entire possible slices only few slice images
are usually required. Therefore it is not economical to filter the entire data though it provides
significant advantage. Since adjacent row of pixels in a planar view are rather similar, a filter,
which combines data from neighboring rows, better reduces noise in reconstructed images.
Post- filtering of the final results is not very common. Particularly because filtering after
reconstruction, does not remove the artifacts very efficiently.
The better approach is to apply the filter during the back-projection process that is, on each
pixel before smearing. In other words, each row pixel (a one-dimensional array) is filtered with
a one-dimensional filter kernel to create a set of filtered rows. These filtered rows are then backprojected to provide the reconstructed slice image. The type of filter used in this process is quite
important.
The Window Function: Windows are mathematical functions, which are multiplied by the
filter function in order to modify the amount of amplification of various frequencies. The names
such as Butterworth, Hanning, Hamming and Shepp Logan, familiar in nuclear medicine, are
actually window functions for Ramp filter. All these window functions have a value equal to 1 at
low frequencies but gradually reduce to a minimum with increase in frequency. Multiplication of
such window functions by the ramp filter (in frequency domain) modifies the filter at higher
frequencies (Figure 19). Therefore the high frequency noises are less magnified and beyond
certain frequency they are completely suppressed resulting in a smoother image. The degree of
this elimination depends upon the variables used in the mathematical equation, which defines the
window function. The main variable is usually the cut-off, which determines the threshold
frequency beyond which higher frequency data is significantly reduced. Some window functions,
i.e. Butterworth, also include another variable called the “order”, which determines how quickly
the magnitude of window function drops to zero. The higher the order, the steeper the window
function and the lower the cut-off frequency the smoother the images are.
Single Photon Emission Tomography (SPECT) 159
2
n
c
f
f
f
Order 5
cut off 0.5
a
b
Figure 19: The Ramp filter shows a linear relationship between frequency (x-axis) and amplitude
(y-axis). To suppress the high frequency noise window functions are incorporated in Ramp filter
that allow low frequency components to appear in the image and slowly suppress the high frequency
data (a), the Butterworth filter that has two parameters the order and the cutoff frequency is
shown on the right (b).
Equations for some of the window functions are given below.
Equation for RAMP modified Butterworth filter
F f
( )
where fc is cutoff frequency and n is order.
Equation for RAMP modified Shepp Logan filter
( ) sin
F f
Equation for RAMP modified Hanning filter
1
f
1
2
f
c
f
c
f
2
(8)
(9)
Real data mainly consist of low frequency, which fall off rapidly with increasing frequency.
However the image edge and fine details have high frequencies. Ramp filters promote high
f
F f f
0 5 0 5( ) . . cos
c
(10)
Equation for RAMP modified Hamming filter
f
F f f
0 54 0 46( ) . . cos
c
(11)
Filter selection in nuclear medicine is always a trade-off between resolution and smoothing.
Single Photon Emission Tomography (SPECT)160
( ) ( ) ( ( ) )
X
b
frequency components in order to sharpen the image details. A window function acts here as a
low-pass filter. A low-pass filter reduces the statistical noise and increases the signal to noise
ratio, but at the expense of contrast and resolution. The most difficult task of data filtering is to
choose the desired frequency that needs to be eliminated, suppressed or retained. Correct filtering
can be achieved only when the frequency spectrum of the data and incorporated noise are
precisely known. If the true data and incorporated noise have different frequency spectrum then
removal of the noise is usually not a problem if a proper filter is selected. Actual problem arises
when the true data and noise have common or overlapping frequency components. Any attempt
to filter the noise with common frequencies results in distortion of data and loss of resolution.
The decision about the selection of filter has never been an easy task.
Some of the filters such as Metz and Wiener try to recover the resolution loss at high
frequencies and are called resolution recovery filters. These filters are inverse of the
modulation transfer function (MTF ) and on multiplying with MTF they (hypothetically)
result in an ideal line with unit MTF (Figure 20a). This is however impossible due to noise
in the data. These filters therefore enhance noise significantly at high frequencies. They are
normally used up to an optimum frequency level then rolled off (Figure 20b).
a
Figure 20: Principle of Metz filter (a). The filter is rolled off after amplifying the low frequency
components up to certain frequency (b)
The equation for Metz filter in frequency domain is given as:
1 2
M f MTF f MTF f
Where f is spatial frequency, MTF is modulation transfer function; X is a factor, which
controls the extent to which the inverse filter (first term in the equation on the right side) is
1 1
(12)
Single Photon Emission Tomography (SPECT) 161
Measured projection
mean atten Estimated projection
followed in the frequency domain before the low pass portion of the filter (second term on the
right side) dominates.
Iterative reconstruction technique
This method normally uses iterative procedures to calculate the approximate voxel values
(rather than the exact value) in small steps. The difference between these methods is how and
where the successive corrections start. In algebraic reconstruction algorithm, all the voxels in a
given slice (of image volume) are set to some arbitrary value. An iterative procedure is then used
to gradually change the voxel values to be compatible to the corresponding pixel value in the
planar image. In simple words, a pixel value in a planar image is compared with the sum of the
voxel values along a line pointing to the pixel. If the sum is lower than the pixel value, all the
voxels along the line are increased in value and if the sum is more than the pixel value, the
corresponding voxels are decreased in value. After the first iteration cycle, there will still be an
error between the voxel sums and the pixel values. This is because the changes made for any one
pixel disrupts all the previous corrections of voxel values. The error becomes smaller as number
of iterations is increased.
Iterative techniques are generally used as a means of improving SPECT images but being
slow and computationally intense these techniques could not be used routinely in clinical practice
in the past. With fast progress in computing in terms of speed and data storage capacity, iterative
techniques are gradually taking over the FBP technique in nuclear medicine. The suppliers of the
SPECT systems these days normally provide both the techniques in their systems. Iterative
reconstruction methods use two steps a) forward projection of estimated projections b) back
projection of (error) difference or ratio of the projected projections and observed projections.
One of the most popular techniques is the maximum likelihood expectation maximization
(MLEM) reconstruction technique. The algorithm maximizes the most likelihood source
distribution with more weight assigned to high count region of the profile and less to low count
region. Further it allows incorporation of physics in the algorithm such as attenuation correction,
depth dependent resolution etc. in the estimated projections. The estimated projections are
forward projected and compared with the observed projections. The error (difference or ratio)
from these projections is back projected for making necessary corrections in estimated projections
(Figure 21).
The process (iterations) goes on till the difference or ratios of the estimated and observed
projections converge to an acceptable (predetermined) value. Wallis and Miller (1) used the
following equation for the iteration loop to estimate error by ratio of measured and estimated
projection:
11( ) ( )
k k
ˆ ˆ
( ) ( ) . .
x x Blur Atten
u
_
(13)
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