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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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Image Filtering in Nuclear Medicine182
noise as we deal with random error. Averaging is the simplest method. However there are
many methods of averaging. Before going to methods of averaging we have to answer the
following question.
In the above section we explained the averaging of some measured values of same
sample. In an image (or curve) we have only one value for each pixel. Then what we have to
average in an image? You may immediately say that the average between neighboring pixels.
But we said that error in one pixel is independent of the other pixels. Is it okay to average
some unrelated values? Certainly it is not meaningful to average the unrelated values but
there are two reasons that make the averaging of pixels meaningful.
Remember that pixel values are the counts measured in corresponding voxels in the
patient’s body. The activity in one voxel is not quite unrelated to its neighboring voxels.
Therefore the value in one pixel is not quite unrelated to the neighboring pixels. Moreover
our imaging systems are not quite perfect and all the counts emitted from a voxel (in the
patients’ body) are not put just into a single pixel. Due to error in localization some of
counts are incorrectly placed in neighboring pixels. Therefore the counts in neighboring
pixels are important to some extent. This partial dependency is enough to make the pixel
averaging reasonable. Though pixel values that are a little far away from the pixel under
consideration are quite independent. :
Filtering an image
If you accept that averaging between neighboring pixel values is a method to decrease the
noise in an image then we can go to describe the method. The procedure is not complicated
but there are few steps.
First: We have to define the neighboring pixels. In the simplest form we can assume
that neighboring pixels are those 9 pixels (including the central pixel) that surround a pixel
at the centre.
Second: We have to define the relative contribution of the pixels. The following table is
an example. This table is usually called kernel “
1/16 2/16 1/16
2/16 4/16 2/16
1/16 2/16 1/16
Third: We have calculated the new value for the all pixels one by one as illustrated in
Image Filtering in Nuclear Medicine 183
Figure 6.
Figure 6: Moving average filter
The above operation is called filtering or moving average filtering. In figure 7 the effect
of such filtering is visualized. This type of filters are called smoothening filters.
Figure 7: Smoothing of an image
Different types of filter
In the above example all the filter kernel element had positive values. What about if some of
the filter elements are negative? We can see two examples in figure 8 and figure 9. The
Image Filtering in Nuclear Medicine184
effect of different edge enhancement filters can be appreciated.
Figure 8: An example of edge enhancement filter
Figure 9: Examples of edge enhancement filters
Convolution
The above explained operation of applying a filter on an image is called convolution.
Though it is simple and straightforward but there are some disadvantages using convolution
method for image filtering.
• Sometimes it is difficult to describe the result of the operation.
• It is difficult to design a filter kernel to perform specific tasks.
• The process is very lengthy and time consuming if the filter kernel has too many
elements.
There is however a famous mathematical theorem that is called convolution theorem.
Based on this theorem, convolution of two functions in object domain (time or space) is the
equivalent to their multiplication in frequency domain. It meant that instead of convolution
the operation might be performed in frequency domain. Figure 10 illustrates the operation in
Image Filtering in Nuclear Medicine 185
frequency domain.
Figure 10: Filtering in frequency domain
Filtering in SPECT
In planar gamma camera imaging the smoothing filters are commonly used. Before describing
the SPECT filtering few terms (Concepts) like sampling theorem, Nyquist frequency and cut
off frequency, have to be clearly understood.
Cut-off frequency
The maximum frequency in the object that can be adequately sampled at a given sampling
frequency is termed as the Nyquest frequency. In nuclear medicine images, the Nyquest
(sampling) frequency can therefore be easily determined by the matrix size.
Nyquest frequence (vn) is generally expressed as:
v
n
a = FOV(cm)/matrix size
In gamma camera images obtained with circular detector with 40 cm FOV and 128
matrix size, a = 40/128 = 0.312 cm the Nyquest frequency.
v
n
In case of zoom acquisition (with zoom x2) the FOV reduces to half and thus the value
of the pixel will change accordingly.
Cut-off frequency basically determines the extent of smoothing that is applied in an
image by the given filter. From the knowledge of MTF of the gamma camera system we
know the maximum frequency that the system can resolve. The cut-off frequency has to be
= 1/2 a cycles/cm where a is the pixel size in cm
= 1/2a = 1/(2*0.312) = 1.6 cycles/cm
Image Filtering in Nuclear Medicine186
less than that limit. Lower cut-off makes the image smooth and higher cut-off allows noise
in the image but will give little better resolution. It is normally expressed as fraction of the
Nyquest frequency
Types of filters
Depending upon the frequency that is retained in the image data, filters are characterised by
their cut-off frequencies and the roll-off. The cut-off frequency (as explained before) is the
frequency at which the filter starts attenuating the high frequency components and roll-off
(order ) shows how quickly they drop down as the frequency increases. Filters are normally
classified as:
(a) Low pass filters which allow suppress the high frequency components in the image.
(b) Band pass filters which retain certain range (band) of frequency in the data
(c) High pass filters which allow only high frequency data in the image.
(d) Restoration filters which try to recover the high frequencies have limited use in
nuclear medicine image.
The band pass and high pass filters are hardly of any use in nuclear medicine image
data. The Ramp filter that amplifies high frequency components is used during image
reconstruction.
Reconstruction by simple back projection blurs the reconstructed image data due to 1/r
blurring effect. To counteract this blurring a ramp filter is used which amplifies the signals
in proportion to their frequencies. Because of amplification of high frequency components
the image becomes noisy and is unacceptable in studies with low information densities. The
ramp filter is therefore modified at higher frequency (modified ramp) is such a way that
from low to medium frequency the ramp filter is undisturbed but at higher frequencies a low
pass filter takes over to attenuate the higher frequency components. The ramp filter is
normally combined with low pass filters such as Shepp-Logan, Hanning or Butterworth to
provide the modified ramp filter funciton. Figure-11a shows Ramp and unmodified
Butterworth filter (with different cut-off and order). Figure-11b is the Butterworth filter
modified (combined) with ramp filter.
Pre-filtering in SPECT
Removal of noise from the image data is not an easy task. The Projection images can first be
pre-filtered with a 2D low pass filter or resolution recovery filter to suppress as much high
frequency noise as possible without much loss in resolution and then reconstructed with
unmodified ramp filter. The reconstruction can either be done on the raw data with a modified
ramp filter or on pre-filtered projection image. In pre-filtered projection data a simple ramp
Image Filtering in Nuclear Medicine 187
Figure 11a: Butterworth and ramp filters.
Figure 11b: Modified Butterworth filter.
Image Filtering in Nuclear Medicine188
b
g
filter is used during reconstruction of filtered data.
Theory of filters
There are different types of filter that are used in SPECT imaging eg, Shepp-Logan,
Parzen, Ramp, Wiener, Butterworth, Haming, Hanning and Metz filter. The most commonly
used are the smoothing and enhancement filters. The smoothing filters are low pass filters
and allow the user to select the cut-off frequency which determines the location of the filter
roll-off. An example of this filter type is the Hanning filter. The Hanning filter is defined in
the frequency domain as,
H(v) = 0.5 + 0.5 cos ( v/vc) 0 | v | vc ---- (1)
= 0 othrewise
The cut-off frequency, vc, determines when the function reaches zero gain.
The Butterworth filter is another example of smoothing filter that it is defined in the
frequency domain as,
1
v v
b g
1
/
MTF
1 2
/
2
n
c
v
v
b g
(2)
x
(3)
B(vc) =
The shape of the Butterworth differs from the Hanning in that this filter maintains a
value close to one at low frequencies and has a steeper roll off. The parameter n determines
the steepness of the cut-off.
Another class of filters, which have been used very little in SPECT imaging is the
restoration filters. These filters attempt to recover the resolutin lost in the detection process
by exceeding unit gain over a desired frequency band. Because of the domination of noise at
high frequencies, enhancement filters at some point must also roll-off to zero gain. An
example of a ehancement filter is the Metz filter. This filter is defined in the frequency
domain as,
1 12 MTF
M(v) =
The Metz filter is a combination of deconvolution and smoothing filters. The parameter
x determines the extent to which the filter follows the inverse MTF before rolling off to zero
gain.
Selection of filters
As we have seen that there are number of filters with their parameters that can be used for
Image Filtering in Nuclear Medicine 189
SPECT reconstruction (such as modified ramp, low pass or resolution recovery pre-filters.
For each of them a cut-off and roll-off have to be chosen judiciously. The studies differ in
their count density, total counts and noise present in the data, thus different filters or same
filter with different filter parameters may be used to get optimum results. For example if the
total counts are less, then the cut-off frequency has to be low. It is advisable to standardise
the filter parameters for SPECT studies using standard SPECT phantoms. In an image, large
objects (area of slow change), correspond to low frequencies, whereas small objects (areas
of rapid change), correspond to high spatial frequencies. Most SPECT filter functions allow
the user to control the degree of high frequency suppression by choosing an appropriate cut
off frequency or similar filter parameter, which determines the filter rolling off to a zero
gain. The location of the cut off frequency determines how the filter will affect both image
noise and resolution. Low cut off frequencies provide good noise suppression, but they blur
the image. Higher cut off frequency can preserve the resolution, but do not suppress noise
sufficiently. Theoretically, there should be an optimum cut off frequency for a particular
filter function, which comprises the trade off between noise suppression and spatial resolution
degradation.
The selection of optimum filter is a complex problem because, it depends on various
factors such as the detector response function (spatial resolution), the spatial frequencies of
the object, and the count density of the image. The spatial resolution changes as a function
of distance of the object from the camera so that even with a single patient study, different
areas in the field of view are imaged at different spatial resolution. Therefore, ideally, there
would be a different filter for each patient and each type of examination. The problem is
more compounded because some physicians prefer sharp, crisp images, even in the presence
of visible noise; others prefer noiseless but somewhat blurred images. The choice of a filter
will represent a compromise among image fidelity, noise and image resolution. Thus the
user has to determine the optimum filter parameters for the available filters with their
systems. Hanning, Butterworth, Metz and weiner filters are very commonly used in SPECT
studies. Therefore these filters with their parameters should be checked with varying
acquisition parameters using images of a SPECT phantom. After gaining good experience of
using the filters with varying filter parameters one can make an intelligent guess and try to
get the best results. Such trials are quite easy in phantom studies, which is, though time
consuming but simple. Having selected a filter for a given phantom image one can try to get
good SPECT images in a normal study and finally in an image with pathological conditions.
Unlike the Butterworth filter, which maintains a gain close to one at low frequencies, or
the Metz filter, which exceeds unit gain in the low frequency range, the Hanning filter drops
below unit gain relatively quickly. Because the signal in nuclear medicine is mainly contained
at these low frequencies, Butterworth filter may be more appropriate as discussed below.
The two most common frequency domain filters available for tomography are the Hanning
and Butterworth filters. The Butterworth filter uses two parameters (“frequency cut-off” and
“power factor”) for data manipulation and allows sharper differentiation between passing
Image Filtering in Nuclear Medicine190
“real” data and rejecting noise without subsequent blurring than the one parameter (frequency
cut-off) Hanning filter. The Butterworth filter can be made as sharp as possible to realize its
advantages over the Hanning filter. However, as the Butterworth filter is made “sharper”
when the power factor is greater than 20, a potential exists for this filter to cause artifactual
texture (“ringing”) in the image. Many find the Hanning filter easier to use. This is because
its broad transition of effect between low and high spatial frequencies makes it less sensitive
to small changes in filter specification.
Measurements of SPECT system response can be made to characterize system resolution
at various depths of scatter. This information can be used to guide “resolution-recovery”
methods. Because noise obscures low contrast small lesions in the image, methods to recover
resolution in these areas can fail because they amplify noise as much as real features in the
image.
The objective in most nuclear medicine studies should be to use as high a critical
frequency as possible with better counting statistics but within the constraints of the noise
present. Better counting statistics helps in using higher critical frequency. If the image looks
too noisy, critical frequency may be reduced. Simple experimentation can be used for any
particular image with selected range of possible critical frequency values based on experience.
Fourier Transformation
in Nuclear Medicine
H. Rajabi
Some problems in mathematics are not straightforward and appear to be complex in
nature. Decomposition of them into simple components sometimes solves the problem of
knowing them. Decomposition of complex functions for simplification has important
applications in science and mathematics.
There are innumerable types of functions in mathematics. However, the most simple is
the linear function (y = a + bx). Any complex function can roughly be decomposed into
combinations of some linear functions. This is a cruel example of decomposition in
mathematics. There are number of ways that a signal or a function can be decomposed,
though only few have been of practical applications. The most famous type of decomposition
technique was introduced by Fourier that has significant application in image processing.
Fourier Decomposition
Jean B. Joseph Fourier (1768-1830), the famous French mathematician and physicist, worked
on temperature distributions in which he introduced a new decomposition method. He stated
that any continuous periodic function might be decomposed into sum of properly chosen
sinusoidal function. Among millions of possible decomposition methods, the Fourier
decomposition is the most widely used one presently available due to its few important
properties. The two most important properties of Fourier decomposition are:
(a) Linearity: Fourier decomposition transforms a function into a linear combination
of sinusoidal functions.
(b) Sinusoidal fidelity: A sinusoidal input to a linear system produces a sinusoidal
output of identical frequency, only the amplitude and phase of the signal changes.
Sinusoids are the only function having this property.
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