AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Physics and nuclear medicine |
FILTERS and FILTRATION in nuclear medicine
1. Introduction -
essence and motivation of filtration, spatial and temporal filtration
2. Methodology of
filtration - in spatial and frequency domain, Nyquist frequency
3. Filtration in
back projection in SPECT tomographic scintigraphy
4. Types of filters
in frequency domain
4.1 Low-pass
filters - smoothing
4.2 Bandpass
filters - focusing
5. Filter- mania:
Which filter is the best?
What is
filtration?
By filtration, we generally mean the
process by which a certain part of a given system is leaked and
another part is retained or weakened. The tool for this filtration
procedure is called a filter .
A simple example is the sieving of noodles,
where a strainer-filter passes water and retains the noodles. In
chemistry and other technical applications, different coarse or
fine filters are used for different solutions - up to
microfilters or "molecular sieves", which retain even
the finest particles. In politics and the media, information is
commonly "filtered" (or censored) in order to
manipulate public opinion ... By data
filtering we mean a nonlocal mathematical
process that transforms
data in such a way that it
strengthens structures of a certain character and weakens or
suppresses others. With nonlocality mentioned here generally exists
an inverse ( reverse) transformation ,
which would be capable of fully filtered data to reconstruct the
primary data - there is a loss of information .
Why
filter?
Scintigraphy, in spite of all its advantages, constantly faces
two basic problems:
Scintigraphic images are therefore (compared to, for example, photographic images) relatively poor quality - they are "blurred" and "noisy". Filtration is used to at least partially correct these adverse effects.
Spatial and temporal
filtering
From the spatio-temporal point of view, we can basically perform
two types of filtering of scintigraphic data:
Spatial filtering of scintigraphic images has two basic goals:
2.
Filtration methodology
In terms of the mathematical
procedure of the filtration procedure, we distinguish two basic
methodologies:
Spatial filtering
Each point of the image is "averaged" with its
surrounding points, and the resulting value is stored back to the
starting point. This reduces statistical fluctuations
at each point of the image (given by the square root of the
accumulated number of pulses) - a smoothed image is created.
After smoothing, the value of the number of pulses of a given
middle pixel partially adapts to the values ??in the surrounding
pixels. In this procedure, the contents of the individual
elements of the image are multiplied by appropriate weighting
factors (the starting center point has the highest
weight, the weight of the surrounding points decreases according
to their distance from the center point), all values ??are added
, divided by the sum of weights and stored back
to the default center point. point. Filterthen
we call the mentioned weighting matrix of
averaging. A typical example is the well - known
9-point smoothing with a weight matrix:
1 | 2 | 1 |
2 | 4 | 2 |
1 | 2 | 1 |
Frequency domain
filtering
The use of the frequency domain is based on the Fourier
theorem , according to which each function f (x) can be decomposed
into the sum of cosine and sine harmonic
functions A.cos (2 pn
x) (and similarly for
sine) , where A is the amplitude
harmonic function and n is its frequency (inverse of the
period). The relevant graphs and formulas are shown in Fig. 1a.
2. Filtering in the frequency domain consists of 3 stages :
Fig.1.
Mathematical procedure of filtration in spatial
and frequency domain. Upper part: Methodology of filtration in the spatial domain by convolution with a filter. Lower part: Methodology of filtration in the frequency domain by filter multiplication. |
It can be shown mathematically (this follows from the formulas applied in Fig. 1) that both filtering methods - in the spatial and frequency domain - are equivalent and give an identical result if the frequency domain multiplication filter is a Fourier image of the convolutional filter in the spatial domain. Filtering in the frequency domain, although it is mathematically more complex (but the user does not know it - the computer will take care of it!), Has some advantages, which will result from further explanation. Here we will only state that the filters in the frequency domain can be flexibly changed by means of their form-factors, while according to the shape of the filter curve it is clear what effect it will have - which noises or details in the image it will eliminate.
The frequency domain is sometimes called the Nyquist region, according to the prominent Swedish-American expert Harry Nyquist ( ¶ 7.2.1889 in Sweden, V 4.4.1976 in Texas), who dealt with the issue of filtration in the field of electrical circuits - LRC filters composed of capacitors, inductances and resistors, filtering electrical signals as a function of frequency. |
The transition from the spatial domain to the frequency domain by means of the Fourier transform is shown in more detail in Fig.2. There are schematic images of two different structures: on the left is a large compact structure (lesion) with a gradual round shape, on the right a small structure (lesion) with a sharp profile. Below each of these images, its profile curve is shown. If we perform a Fourier transform, the large structure on the left will be dominated by low spatial frequencies of harmonic decomposition, while high frequencies will be represented only slightly (ie with low amplitude). For a small structure with a sharp profile (right), the relative proportion of higher harmonic frequencies will be much higher.
Fig.2. Transition from
the spatial region to the frequency domain using the
Fourier transform for the case of a large compact lesion (left) and a small sharp lesion (right). |
If we plot graphically the representation of the amplitudes of individual harmonic functions depending on the frequency, a spectrum is created , which in the frequency domain represents the Fourier image of the original structure from the spatial domain.
Frequency
K-space
In the spatial region , the usual Euclidean
space - R-space - the image of the displayed
quantity F is described by a distribution function
, or field, F (x, y, z). In vector notation, by introducing the
spatial vector r , this function is F ( r
). The general Fourier transform gives a new
distribution function ^{^} F ( k ) = ň _{V}
F ( r ) .exp [2 p i ( kr )]
d r , where k = (k _{1} , k _{2} , k _{3}) is a new frequency
vector , scalar product kr = xk _{1} + xk _{2} + xk _{3} ; integrated over the
spatial region V . The distribution function ^{^}
F ( k ) is defined in a new linear 3-dimensional
vector space. Both the spatial F ( k ) and
frequency ^{^} F ( k ) distribution
functions carry the same information and are
related by a direct and inverse Fourier transform.
From a mathematical point of
view, from the common metric Euclidean R-space, the
Fourier transform creates a new "frequency" space,
sometimes referred to asK-space (
K-space - the name originated from the fact that after the
Fourier transform the new independent variable is a
"wave" vector which is used to denote k
, it is generally complex). An abstract K-space is in a sense
"reciprocal" to the usual physical R-space.
Thus, in the lower part of Fig. 2 we see that a large compact lesion has a spectrum ending at low spatial frequencies, while the spectrum of a small sharp lesion also contains high spatial frequencies. This pattern has a general character: the more fine details are in the image, the higher the frequencies of the harmonic functions are represented in the frequency domain. The finest details in a scintigraphic image are statistical fluctuations ( noise ) that change chaotically from pixel to pixel - these correspond to the highest spatial frequencies in the Nyquist region. Therefore, if we use a filter that suppresses high frequencies, we remove disturbing noise from the image - this is smoothing using the so-called low-passfilters. The basic "art" here is to choose a filter that would suppress disturbing noise while preserving as many useful details in the image as possible.
How to achieve such optimal filtration ? The regularities shown in Fig.3 can be a guide to us. The question is answered here: what highest frequencies in the Nyquist region can still express the real details of the distribution of the radio indicator shown by the camera, and what reflect only the interfering noise?
Fig.3. To define the Nyquist frequency for a scintigraphic image. |
The resolution of a scintigraphic image is in principle limited by two factors.
1. Image matrix
The fineness of the used matrix of scintigraphic
image (whether we use the matrix 64x64, 128x128, 256x256, etc.)
limits us how fine details we will be able to display with it. In
the left part of Fig. 3 it is shown that if the pixel size of the
matrix used is d (cm), then the highest frequency of the harmonic
function that can be displayed in such a matrix is n _{max} Ł 1 / 2d.
2. Camera resolution
The basic factor limiting the resolution in scintigraphy is the
spatial resolution of the camera. Fig. 3 in the
upper right shows the profile curves of the LSF image of a point
radioactive source, displayed by a gamma camera with excellent
resolution (dashed line), medium resolution (solid line) and poor
resolution (dotted curve). The FWHM camera resolution is defined
as the half-width of the profile curve of the
PSF image of a point (or line) source. Two points shorter
distances from each other than the FWHM resolution will no longer
be recognized by the camera and will be
displayed as one point in the scintigraphic image. From this
point of view, the highest frequency of harmonic distribution of
radioactivity (model), which the camera would still display,
would be n_{max} Ł ~ 1 / FWHM - at higher frequencies the individual waves of
the (co) sine wave would already merge. The Fourier transform of
the profile curve PSF(x) of the point source image produces a
spectral curve called the modulation transfer function
MTF(n),
which shows with what relative amplitude the camera is able to
display (transmit) the model harmonic distribution of
radioactivity depending on its frequency n . The place where the MTF
drops to zero then defines the maximum spatial frequency n_{max} , which the camera is still able to display.
^{ }
We can therefore make the following statements and definitions at the same time :
For each scintigraphic imaging, there is a certain maximum spatial frequency n_{max} , called the Nyquist frequency , which the system is still able to display. |
Thus, the Nyquist frequency is also
the maximum frequency in the Fourier spectrum of
the image, which still reflects the actual structures of
the object. Frequencies lower than
Nyquist reflect reality in the image , while
frequencies higher than n _{max} no longer have their origin - they are the result of
static fluctuations (noise) and can be removed
without the risk of losing useful details in the image.
The value of the Nyquist
frequency can help us to optimize the "strength" of the
filter: using form-factors (see below), we shape the filter graph
so that it approaches zero just for the value of the Nyquist
frequency. There is no point in maintaining a higher frequency in
the image, because they cannot reflect any real structures, they
only express disturbing noise.
Wavelet ( wavelet
) transform
Fourier harmonic analysis is the basis for
advanced processing and design modifications signals measured
závisloszí and images. Using the Fourier transform
, it decomposes the analyzed signal or image into harmonic
functions sine and cosine with different
amplitudes and frequencies (Fig. 1 and 2), which it then suitably
modifies and subsequently backwards (inversely) transforms. The
Fourier transform provides information about frequencies
which are in the signal (and their representation), but not about
their location - time position at the signal or in the spatial
(coordinate) place at the image. It is therefore especially
suitable for the description of stationary signals or relatively
uniform graphs and images without discontinuities and sharp
fluctuations. For filters to eliminate interfering static
fluctuations (noise), the main problem is the trade-off between
the noise suppression rate while maintaining useful detailed
information in the images (as mentioned
above and will be discussed below in the " Low-frequency
smoothing filters " section) .
Stronger filtering effectively suppresses noise, but also leads
to the risk of smoothing out smaller details in the image.
^{ }The common
denominator of these difficulties is the fact that the basis
functions of the Fourier transform, sine and cosine, have
non-zero periodic values ??in the whole spatial domain ("from - to + infinity") .
Therefore, in Fourier filtering, any change in the frequency
domain is reflected in the whole image (in the spatial domain):
if we try to suppress or cut off certain higher spatial
frequencies during filtering in order to reduce noise at the
desired image location, it can be reflected in deterioration of
spatial resolution throughout image.
Preferred generalization and improvement of standard Fourier
transform is called. Wavelet ( wavelet
) transform^{ }That instead of sine and cosine for the decomposition of
the analyzed signal uses a special basis functions called wavelets
or ripples that are more localized
spatial coordinates and rapidly dwindling into infinity. Base
functions have a limited length - compared to large sine waves
they are only short "ripples" - and with variable
frequency (scale) they can move over the entire spatial area of
??the signal: the analysis can be local with
differently strong filtering at different points in the signal or
image. A number of wavelet functions were created (some of them
are in the picture - b, c, d). E.g. The Morlet wavelet
is a cosine function multiplied by a Gaussian function with a
suitable width.
Above: a) Cosine base
function of Fourier transform. Bottom:
Wavelets: b) "Mexican hat". c)
Morlet. d) Meyer's
^{ }When filtering scintigraphic images, wavelet transforms are so far used only sporadically and experimentally, but with promising results. In the following text, we will focus on standard filtering procedures using the Fourier transform.
3.
Back projection filtration in SPECT tomographic scintigraphy
The general principles mentioned
above have some specifics when applied to SPECT
tomographic scintigraphic images reconstructed by back
projection (treatise " Scintigraphy
", part " Tomographic
scintigraphy ") . Fig. 4 schematically shows in the upper left part how
the back projection of the unfiltered profile of
the point source image projected at several angles (from which
the point source was scanned during SPECT) creates a star
artifact from the projection rays around the resulting
image .
Fig.4 Filtration for SPECT rear projection |
If we apply a suitable filter to
the profile function of a point source in the spatial region such
that at both edges of the curve there is an artificial
"oscillation" into negative values (the magnitude of these negative values ??is directly
proportional to the magnitude and speed of positive growth) , then the rays with the negative edges are superimposed
so that at the intersection they again create a reconstructed
"dot" image, but in its vicinity the negative
half-waves locally disturb the rays of the star
artifact. At greater distances from the point source image, the
traces of the projection beams remain, but in principle it does
not matter - they "mix" with other projection traces
and create a common continuous background.
In the right part of Fig. 4,
this type of filtering is shown in the frequency domain.it
suppresses the star effect locally , here it has a
linear shape and is called a RAMP filter . The
RAMP-filter is a necessary implicit part of every procedure for
SPECT reconstruction by the back projection method *), but at the
same time it amplifies higher spatial frequencies in the image -
fluctuations, noise. If we also apply a user low-pass filter to
suppress statistical fluctuations, the resulting filter is given
by the product of the RAMP filter with the user filter (shown in
the lower part of Fig. 4) - such a filter then suppresses
the star effect and smooths the image .
*) The RAMP filter is not used for the
iterative reconstruction method.
Fig. 5 schematically shows the whole procedure of acquisition, filtration and rear projection in scintigraphy by SPECT method.
Fig.5. Acquisition, filtration and rear
projection procedure in scintigraphy using the SPECT method.
The examined object (patient), whose cross-section has the distribution of the radio indicator A (x, y), is captured by the camera in a series of projections at different angles J , thus creating images of projections p (u). These images are then Fourier transformed into the frequency domain, and the spectrum P ( n )) is multiplied by a filter comprised of a ramp-filter and the filtered spectrum user filtru.Výsledná P _{F} ( n ) is then inverse Fourier transform are converted back into the spatial area (the filtered result projection images p _{F} (u)), after which with back projection (at the same angles J ) it creates the resulting cross-sectional image A´_{f} (x, y).
4.
Types of frequency domain filters
As follows from the previous
explanation, frequency domain image filtering consists in
multiplying the amplitude of each harmonic function (on which the
image has been decomposed) f ( n ) by a certain coefficient,
which decreases or amplifies its amplitude depending on frequency
n . The
set of these coefficients forms a specific filter. Each filter is
implemented by a certain mathematical function F
( n ),
which for each value of the spatial frequency n generates
a coefficient F ( n ), which will multiply the amplitude of the
corresponding harmonic in the image spectrum f ( n ). Functional
prescription of filter F ( n) usually contains certain optional parameters - ie. form-factors
, which together with the mathematical function determine the
specific shape of the filter and thus the strength
of the filter . Each type of filter has its specific
form-factors, but one of the form-factors is common
to all filters : it is a so-called " cutoff
", indicating the maximum frequency from which all higher
harmonics will be cut (canceled) upwards.
4.1 Low-pas filters - smoothing
Scintigraphic images, especially
reconstructed transverse sections of SPECT, often have a large
scatter (fluctuation) in individual voxels - image noise. For a
better evaluation of these images, it is desirable to reduce this
noise - smoothing the image, suppressing
statistical fluctuations. The individual structures in the
scintigraphic image can be simply divided into three groups in
the frequency domain:
- Low frequencies are given by the uniform activity of larger
structures and backgrounds.
- Medium frequencies express changes in the number of pulses
given by different distributions of the radio indicator in the
displayed organs in the range of units up to tens of pixels -
useful diagnostic information.
- The highest spatial frequencies express statistical noise -
random changes in the number of pulses in adjacent pixels.
^{ }In the frequency domain, we achieve image
smoothing (reduction of statistical fluctuations) by
attenuating or suppressing high- frequency
harmonic functions - multiplying the spectrum by a suitable
filter, which for low frequencies has a value close to 1 (leaves changes in the number of pulses corresponding to
the structures of the displayed organ) and
for high frequencies is approaching zero (suppresses
statistical fluctuations in changes in neighboring or nearby
pixels) . Such smoothing filters are called
low-pass- they transmit mainly low frequencies
and reduce (suppress) the amplitude of higher frequencies, ie
statistical fluctuations in the number of pulses in adjacent
pixels. However, in real images, the spatial frequencies of
changes for individual structures usually partially overlap, so
that suppression of high frequency noise can lead to the
smoothing of real fine structures that also have higher spatial
frequencies.
The individual filters are
characterized by a frequency curve, the course of which in
different cities determines their action. Figure 6 shows the
shapes and equations of the most commonly used low-pass filters.
For some of them, a weaker filter is marked with a solid line,
and a thicker filter with a dashed line.
Fig.6. The most commonly used smoothing (low-pass) filters. |
The simplest filter is to simply cut
off (cancel) the harmonic functions of frequencies
higher than a certain maximum frequency n _{N} called "cutoff". The graph of such a filter
has the shape of a rectangle - it leaves all
frequencies up to n _{N} = cutoff with
unchanged amplitude, while all harmonics higher than
"cutoff" are cut off (canceled). The smaller the value
of the form factor "Cutoff" = n _{N is} entered, the stronger the filter effect will be.
The cosine function
filter is also shown , which decreases continuously from the
value F = 1 for the zero frequency n = 0
to the value F = 0 for the maximum frequency n _{N} (value "0" is then assigned to all higher
frequencies). Thickness of the filter is again higher, the lower
the value the form-factor of the "cutoff" = n _{N} enter.
Other occasionally used filters
are Hamming and Parzen (middle
part of Fig. 6). Type filter is a modified Hamming cosine filter,
and besides the parameter "cutoff" has one parameter and which controls
the "steepness" with which the filter is close to zero
in the vicinity of frequencies N ~ N _{N} . A Parzen filter is a combination of two polynomial
functions: a faster descending function is used in the middle of
the range, which is then smoothly followed by a slower descending
part; the form factor is again a "cutoff"._{ }
The most commonly used low-pass
filter is Butterworth shown in the lower part of
Fig.6. It has two form-factors - a cutoff
indicating the spatial frequency of the cut *) and an order
(" order ") regulating how steeply the
filter drops from values close to 1 to zero. It is this high
"ductility" that makes the Butterworth filter so
popular.
Note: The " cutoff
" value for the Butterworrth filter here is not
arithmetically equal to the cut-off frequency, as is the case for
other filters. For n = cutoff, the value of the filter is equal to 0.5, the
Butterworth filter reaches exactly zero even in the limit for n ® Ą . Effectively,
the filter approaches zero for n values= 2 ´ "cutoff",
the higher the "order" of the filter.
Filtration
power^{ }
What are the common patterns of using low-pass
filters? What determines their "strength"? In Fig. 7,
the same scintigraphic image of the brain is filtered by a
progressively thicker and thicker filter.
........ without filter ........................ weak filter .............. ........... medium filter ..................... thick filter .............. ..... very strong filter | ||||
. ........ without filter ........................ Buttw, ord = 13, cutoff = 0.82 . ..... Buttw, ord = 4, cutoff = 0.5 ........ Buttw, ord = 4, cutoff = 0.26 .... Buttw, ord = 4, cutoff = 0.15 | ||||
Fig.7. The result of filtering an image of the brain using filters of different strengths |
The sooner the filter goes to zero (at lower spatial frequencies), the stronger the filtering effect. The following theorem holds :
The filtration strength is inversely proportional to the area below the filter graph in the frequency domain. |
Note that this is exactly the opposite of the spatial region , where the filtration force is directly proportional to the area under the convolution filter graph! Advanced reconstruction algorithms for SPECT and PET also use more sophisticated methods that perform differently strong filtering at different points in the image (cf. " Wavelet transform ") .
4.2 Bandpass filters - focusing
Filtering in the frequency domain allows not only smoothing of
images by reducing higher harmonic frequencies. Conversely , if
we amplify their amplitudes for a certain part
of the higher harmonic frequencies , we can increase the
local contrast and " sharpen
" some details in the image that have been
"blurred" due to imperfect camera resolution. Such bandpass
filters amplify the mid frequencies and attenuate the
high and low frequencies, increasing the contrast of the
objects in the image.
Is it
possible to completely reconstruct reality by artificially
focusing an image?
The image is created by the convolution of the
original distribution (object) and the response function of the
imaging system. From a theoretical point of view, using an
inverse procedure - image deconvolution - it
should be possible to completely reconstruct all
the details of the original object, even those that are smoothed
in the image by imperfect spatial resolution (this procedure is
called " resolution recovery
").
^{ }The following image
shows the testing of this method at our workplace in 1977 on a
Pho Gamma Nuclear Chicago scintillation camera with a Clincom
evaluation device and its own software. Radioactive sources (one
2.5 cm diameter dish and 4 1 cm balls) filled with a ^{99m}
Tc solution with different specific activities were placed under
the scintillation camera - the profile of their actual activity
is in the picture on the top right "Original
distribution". Long-term acquisition (approx. 12
hours) gave a very smooth scintigraphic image (profile on the
right in the middle), without visible statistical fluctuations,
on which the details of the sources are not visible due to poor
resolution (scanned 20 cm from the collimator front). The modulation
transfer function MTF ( _{ }
_{ }_{ }
n) of
the camera and the Fourier transform FT of its inverse value 1 /
MTF ( n ),
the reconstruction filter R (t) in the spatial
region was calculated (bottom left figure). By convolving the
scintigraphic image with this filter R (t) in the spatial area, I
obtained a reconstructed image (bottom right profile), on which
the initial original distribution is almost perfectly
reconstructed in detail. On closer inspection, however, we see
tiny "ripples", especially in flat areas. These do not
correspond to reality - they are artifacts
created by the amplification of hidden statistical fluctuations.
Due to very good "statistics", unattainable in
practice, these artifacts are minor. However, it will be shown
below that in practice these artifacts are a limiting
factor focusing and "reconstruction" of
images.
Filters that, in addition to smoothing statistical fluctuations, are able to focus and highlight details in the image are shown in FIG. and are referred to as bandpass filters (behaving differently in different frequency bands) .
Fig.8. Band focus filters |
They differ from low-pass filters in that they do not decrease from F ( n = 0) = 1 monotonically to zero, but consist of two parts :
The two most commonly used bandpass
filters - the Metz type and the Wiener
type - are largely similar. The ascending part is derived from
the inverse value of the modulation transfer function
MTF and provides according to the theory of scintigraphic imaging
optimal reconstruction ( deconvolution -
image correction to convolutional "blur" by imperfect
camera resolution) and image focus - so-called resolution
recovery RR , " resolution
recovery " - see. treatise
"Scintigraphy", passage " Adverse
effects of scintigraphy "
.
^{ }Form factor " k
" (" order") allows you to
continuously adjust the relative representation of the ascending
(focusing) component of the filter, while the usual
parameter" cutoff "determines the
representation of the descending (smoothing) component of the
filter. part is missing - low value of parameter "k" -
"order", the filter only smoothes similarly to any
other low-pass filter (first part fig.9).
..... Metz, cutoff = 1, ord = 3 . ......... Metz, cutoff = 1, ord = 10 ....... Metz, cutoff = 0.74, ord = 30 ..... Metz, cutoff = 1.8, ord = 90 |
Fig.9. Filtration by a band focus filter of various strengths |
However, the focus of the image
cannot be increased indefinitely - with a high proportion of the
ascending focusing part of the filter, artifacts
begin to appear in the image - false structures
caused by the amplification of statistical fluctuations; when
attempting high focus, the image eventually disintegrates into a
system of islands, many of which are not related to the actual
distribution of the radio indicator (last part of Fig. 9).
We must therefore point out that :
The condition for the success of band focus filters is a quality scintigraphic image with low statistical fluctuations! |
Note: Filters of this type are also used in photo image processing programs (such as Photo Shop) in the "Sharp" function. Statistical fluctuations are usually low here (visible light photons are 3-6 orders of magnitude more than gamma photons), but for example night images may have similar problems as a scintigraphic image and additional focus may unacceptably increase the "graininess" of the image.
5.
Filter - Mania: Which filter is the best ?
In the literature, various authors
often recommend certain filters (filters of certain names), which
gives readers the general impression that some filters are a
priori better than others. They then embark
on laborious experiments with filters and literature searches,
hoping to find some guaranteed best " miracle
" filter that will produce perfect images for them. Here we
show that all this is just a deceptive appearance
.
For an objective
analysis, it is necessary to realize that it does
not filter the name of the filter , or even its
mathematical equation - it only generates coefficients by
which for individual spatial frequenciesn will multiply the amplitude
of the corresponding harmonic function in the spectrum in the
Nyquist region. Therefore, if we set different filters - with
different names and different functional expressions - using
form-factors so that their graphs converge, the result of the
filtering will be identical .
Note: I do not
list the relevant image here - it would be an uninteresting
series of identical images. However, everyone can try that eg
Sheep-Logan-Hamming filters with parameter cutof = 1.0, Parzen
with cutof = 1.2, Hamming with cutof = 0.9, a = 0.5 and Butterwoth filter
with parameters cutof = 0.4, order = 1.5 have almost the same
graphs and give the same results when applied to
scintigraphic images .
^{ }This rule goes even further. In Fig. 10,
the same brain image is filtered by three different filters -
filters of different names and different shapes of its graph -
but chosen so that the area under the filter
curves is approximately the same ; this can be
achieved by targeted tuning of form-factors. We see that the
result of filtration is almost identical ,
despite the considerable differences in functional regulations
and the shape of the graphs of individual filters.
.... Band-Lim, cutoff = 0.51 . ......... Cosine, cutoff = 0.80 .... ....... GenHann, cut = 0.97, a = 0.5 . ... Butt, cutoff = 0.49, ord = 5 |
Fig.10. Filters of different names and shapes give almost the same result if they have the same area |
These interesting laws are not
generally known (nor are they described in
the literature) and seem surprising at
first glance; however, they are consistent with the theoretical
analysis. The lesson is that "filter mania" has
no justification , at least for low-pass filters.
So to the question " Which filter is the best? ", Asked in the title of this paragraph, we can
easily answer: None!
With each of the used low-pass
filters we can achieve practically the same result by
suitable setting of their shaping parameters , it only requires
experience and critical thinking.
And finally, I would like to take
one more piece of advice *): Let's not try to
change the filters often during the practical evaluation of
scintigraphic images! - this would impair the reproducibility of
the images, especially when comparing in repeated examinations.
When starting work with a new system, it is best to start with
certain recommended or previously proven filters (eg Butterworth, order = 3, cutoff = 0.4) , try them out properly (and possibly modify them) and
then use them for a long time .
Only when there is a serious reason (change
in methodology, equipment, sufficient accumulation of experience,
etc.) , it is reasonable to change the
filter used for the examination. And of course we can
purposefully try different filtration when we are looking for
some atypical anomalies in the image ...
*) Perhaps I can afford to give some modest
advice - I am probably the oldest "witness" of the
filtration of scintigraphic images in the field of nuclear
medicine in our country. From a physical-mathematical point of
view, I actively dealt with the issue of filtration around 1976
in connection with the theory of scintigraphic imaging and modulation transfer
functions .
Risks
and pitfalls of image filtering
Once again, a warning about image filtering:
An old Czech proverb says: " Where
there is nothing, the devil doesn't take it ".
For our example, this proverb paraphrase as follows:
" Where there is nothing, there the devil can
deliver - an
artifact !
" - the devil, in the person of excessive
filtering , highlighting the statistical
fluctuations adds artifacts -
And where something is , there's the devil can muster
:
An exaggerated filtration smooth out the details in the
picture !
This work was lectured at seminars and symposia of the Czech Society of Nuclear Medicine and is a regular part of postgraduate courses and seminars for the field of nuclear medicine within the IPVZ at KNM in Ostrava.