Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5255_Библиотеки_им_академика_М_И_Перельмана
.pdf
Fourier series of sine and cosine function
( )
f x
.cos(2. . . . ) .sin(2. . . . ) 1,2,3..
( ).cos(2. . . . ).
( ).sin(2. . . . ).
a f x f n x dx
b f x f n x dx
a.cos( t) - b.sin( t)
a.cos( t) - b.sin( t) = m.cos( t + )
2 2
a b
tan ( )
b
a
The Fourier series of a function f (x) is given by
a a f n x b f n x n
0 0 0
n n
1 1
n n
( ).
0
0
Where
1
=
2
1
a f x dx
0
1
n
1
n
Fourier Transformation in Nuclear Medicine192
(1)
Fourier analysis means to calculate the value constants a0, a1,…. a
,…bn. These are
n, b1
the amplitudes of sine and cosine functions of harmonic frequencies with fundamental
frequency as f0.
Sinusoidal Function
A sinusoidal function is made of a linear combination of a cosine and a sine functions
(Figure 1):
(2)
Where a, b are constants and w is angular frequency. In general, all sinusoidal functions
can be written as a “phase-shifted” sine or cosine function;
(3)
Where
m =
1
=
Fourier series of sinusoidal function
Using the above notation the Fourier series of a function f (x) may be written as:
Fourier Transformation in Nuclear Medicine 193
( )
f x
.cos(2. . . . ) 1,2,3...
i t
t i t e
z cos( t) - sin( t)
= cos( t) - i.sin( t)
Figure 1: A sinusoidal function is made of a linear combination of cosine and sine functions as
shown in this figure.
c c f n x n
=
0 0
n n
1
n
(4)
Complex functions
As per Euler formula, a complex trigonometric function may be written in the form of an
exponential function as
(cos .sin )
Using the above complex notation, it is always possible to show the combination of two
trigonometric numbers as a single complex number as given below.
i t
=|z|. e
Though it is always possible to perform this type of conversion between a couple of real
numbers to a complex number but it is not always meaningful to perform mathematical
operation (summation, multiplications….) on such numbers. In the case of Fourier transform
it was proved that performing the operation in complex domain results in quite similar
results as in real domain. It is quite convenient for mathematicians to use complex forms of
Fourier series because of short notation and fast calculation. The following is one of the
favorite forms of Fourier series for mathematicians.
(5)
Fourier Transformation in Nuclear Medicine194
( )
F f x
i f nx
( )
X n
x k e n N
=
f x e
( ).
2
0
(6)
Fourier transformation
In mathematics the relationship between variables are represented by functions. Curves are
visual representation of functions. A curve may be one, two or multi dimensional. In practice,
a curve may be used to represent the measured value of a physical quantity against time or
space. In such cases the curve is usually called signal. The term signal implies that only the
values of the variables are at hand not the real mathematical relationship between the
variables. Signals may be in continuous or discrete forms. Both signal and function may be
periodic or aperiodic. Fourier decomposition is applicable in all cases however in different
ways.
Fourier transform was initially premised for continuous periodic function and the term
Fourier analysis was used for decomposition of such functions to infinite series of sinusoidal
functions. When this condition is not satisfied the procedure of decomposition is called
Fourier transformation.
Signals in nuclear medicine are always in discrete and aperiodic forms. The procedure
of decomposing such signals is called Discrete Fourier Transform (DFT). In DFT it is
assumed that the signal is periodic but only one single period of the signal is available.
Among all types of Fourier decomposition only DFT is possible to perform on digital
computers. Equation-6 in discrete form can be written as:
=
N
( ). 1,2,3...
1
k
.2 . /
ik n N
(7)
Graphic representation of Fourier transformation
Fourier transformation means to decompose a function into its sinusoidal components. The
number of sinusoidal functions is usually very high. Showing too many sinusoids in a single
graph make it congested and difficult to distinguish (Figure 2)
Fourier Transformation in Nuclear Medicine 195
Figure 2: In this figure the curve on the top (resembling a renogram) is decomposed into sine and
cosine functions (bottom). Only a few of sine and cosine functions are shown as showing too
many sinusoidal curves in a single graph make it congested and undistinguishable.
To avoid the problem of showing so many sine and cosine functions a simple convention
has been made which represents each sine or cosine curve with a bar (Figure 3). Height of
the bar is proportional to the amplitude of the sine or cosine function. The location of bar
along the horizontal axes is proportional to the frequency of the wave.
Figure 3: Each sine or cosine wave is represented by a bar of same amplitude. The frequency of
sine wave is shown on x-axis
Fourier Transformation in Nuclear Medicine196
Using the above mentioned convention, for a one dimensional curve two bar charts are
created. One of the charts shows the amplitudes vs. frequency of the cosine waves. The
other bar chart shows the amplitudes vs. frequency of the sine waves (Figure 4).
Figure 4: Fourier transform converts a triangular function into many cosine and sine functions.
Each cosine and sine function is shown using a bar. Each graph is actually the amplitude vs.
frequency of sine and cosine functions in the new convention.
When using the complex notation, the cosine part is usually regarded as real part and
sine part as imaginary part.
Considering equation-3, it is obvious that instead of sine and cosine components, the
amplitudes and phase of the sinusoidal function may be shown as in figure 5.
Figure 5: This graph is exactly the same graph as shown in figure 4 but instead of cosine and sine
function, their amplitudes and phases are shown separately (refer to equation-3).
One important question arises here. Which type of graphic representation is better? The
answer depends on our intention. The first type of graphic representation of Fourier
transformation is usually called rectangular and the second one is usually called polar
representation. In polar representation, the amplitude vs. frequency reveals the real frequency
distribution of data. While the phase vs. frequency usually does not carry important
information (at least in nuclear medicine) in rectangular representation and it is very difficult
to perceive the frequency distribution of data. Therefore for visual interpretation the polar
representation is preferred. However for mathematical purpose, the rectangular representation
(and notation) is preferred.
Fast Fourier Transformation
There are several ways to calculate the Discrete Fourier Transform (DFT), such as solving
Fourier Transformation in Nuclear Medicine 197
simultaneous linear equations and correlation. However, the calculation is very tedious and
time consuming. Cooley and Tukey in 1965 introduced an algorithm for calculation of
Fourier transformation, which is now called Fast Fourier Transformation (FFT) (1). The
algorithm was initially developed by the great German mathematician Karl Friedrich Gauss
around 1805, which was later, described by Heideman et al (2). The FFT algorithm is very
complicated but incredibly efficient that reduces the computation time by hundreds.
Although the algorithm is very complicated in nature but simple to use. There are so
many computer codes available to perform the FFT on the data. One can get it in almost
every computer language on the Internet very easily. MATLAB provides a very good and
simple function to perform FFT on one or two dimensional data.
Some important points in FFT codes
Number of input data points
In almost all versions of algorithm available, the number of input data points must be a
power of 2 (i.e. 2, 4, 8, 16, 32,….1024). If the number of data points does not match the
requirement, one can use zero padding technique to add required number of zeros to the end
of data. Some algorithms automatically perform zero padding. Zero padding has a little
effect on output data.
Imaginary part of input data
If one is using a code that is based on complex FFT algorithms, it requires both the real and
the imaginary inputs. If the data is composed of only real part (nuclear medicine data), one
must feed the code with the required number of zeros as the imaginary part. In such cases
the imaginary part of output must me ignored.
Number of output data points
If there are N values in the input data, the algorithms provide N values as real and N values
as imaginary. However the second half of both real and imaginary parts are only the
symmetrical values of the first half.
Nyquist sampling theorem
It is quite a familiar fact that for good resolution the sampling interval must be short enough
(3,4). Figure 6 illustrates this fact graphically. If the sampling interval is too long the details
between the sampling points are lost. Therefore the sampled data (discrete data) does not
exactly represent the original continuous data. The details are therefore lost and it appears
as if the data is filtered. This fact is graphically represented in figure 6.
Fourier Transformation in Nuclear Medicine198
Figure 6: The curve on the top represents an original function. Discrete sampling is shown in the
middle part of the figure. The resultant curve (bottom) is not exactly like the original function. If
the sampling interval is not short enough, the details between sampling intervals are lost. Therefore
the discrete data does not exactly represent the real continuous data. It actually represents a
smoothed data as shown in this figure.
Sampling interval is also an important issue in FFT (DFT in general) however with a
different viewpoint. In Fourier transform the data is decomposed into several sinusoidal
functions of harmonic frequencies. One important drawback is that number of data points is
limited. For example a data set composed of 100 sinusoidal functions is represented by only
64 acquired data points (64 data points in frequency domain). The result of transformation
in such a case will not be correct, as 64 points cannot show 100 sinusoidal functions. This
example just shows that there is limitation in the minimum number of input data points that
depend on number of sinusoidals in the data. However we can’t say that number of input
data must be equal to the number of sinusoidals in the data. Actually limitation is imposed
by the sinusoidal that has the highest frequency in the data and not the total number of
sinusoidals. Therefore there is a limitation for sampling interval not the number of data
points.
Fourier Transformation in Nuclear Medicine 199
Based on Shannon-Nyquist sampling theorem; the sampling rate should be at least two
times the highest frequency component in the data. Practically it is difficult to know the
highest frequency in the data. Figure 7 graphically illustrates the error caused due to incorrect
sampling. Roughly we can say that some frequency components are missing. This phenomenon
is called aliasing.
This is due to the fact that incorrect sampling causes loss of details or the details are
smoothened. The details correspond to high frequency sinusoids. Therefore the amplitudes
of high frequency components become smaller than their real values and vice versa whereas
that of low frequency components may become higher than their real values. This fact is
graphically represented in figure 7. To avoid aliasing the sampling rate should be high
enough. If the sampling rate is not high enough then one can filter the data to remove very
high frequency components.
Figure 7: Graphical illustration of aliasing. Left part of the figure is a result of correct sampling.
The bars on the right side of the figure as shown by arrow are the result of incorrect sampling. If
sampling interval is too long the details between consecutive samplings are lost. Therefore the
corresponding sine and cosine representing the details in frequency domain are lost. At the same
time because the details look smoother some incorrect low frequency sine and cosines appear in
the data. Because data is composed of so many frequencies the overall effect is incorrect amplitudes
of sine and cosines.
Sudden abruption in data
One of the most important assumptions of DFT (including FFT) is that the data is periodic.
As mentioned earlier, in DFT we can assume that our data is periodic but we have only one
period of data in hand. This assumption is valid only when the first and last points of data
are identical. Figure 8 graphically illustrates a situation in which this essential condition is
not satisfied. The graph illustrates a rupture in data. Technically rupture means very high
frequency component in data that requires very small sampling intervals or rapid sampling.
As a rule of thumb the slow changes in an image corresponds to low frequency sinusoids
and rapid changes correspond to high frequency sinusoid. In other words the number of
sinusoids to reconstruct a given function depends on the slope of the function. The more the
slope the more will be the number of sinusoids required. The more number of sinusoids
Fourier Transformation in Nuclear Medicine200
required means higher frequency. Sudden abruption in data acquisition usually creates such
situation where aliasing is unavoidable
Figure 8: Sudden abruption in data creates aliasing
To partially handle the error caused by this type of aliasing, a simple method is padding
a very low frequency sinusoid to the end of the data. Though this method solves the problem
of aliasing but it adds a very high amplitude component to the frequency spectrum that
needs precaution.
Two-dimensional FFT
There is not much difference between one-dimensional and two-dimensional FFT algorithms.
If we have one-dimensional code but the data is two dimensional, we just apply the code on
each row and each column of data consecutively.
A simple application of FFT
Before closing the discussion on FFT one of its important application needs to be mentioned.
It is calculation of the Modulation Transfer Function of a nuclear medicine imaging system.
If we draw a line profile over the image of a point source, we get the Line Spread Function
(LSF) that resembles a Gaussian curve (figure 9). The Fourier transform of the LSF results
in a curve that is called Modulation transfer function. The curve shows that amplitude
decreases with increasing frequency. This means beyond certain limit the frequency
components are not reproduced by the imaging system.
Fourier Transformation in Nuclear Medicine 201
Figure 9: A point object is shown at the centre (top). Its ideal image is shown on the left side and
observed (real) image is on the right. Similarly the ideal point spread function is shown on the
left and observed (real) profile is on the right (middle). The Modulation Transfer Function in
ideal imaging system is shown on the left and the observed (real) is on the right (bottom).
References
1. Cooley JW and Tukey JW. An algorithm for the machine calculation of complex Fourier Series”.
Mathematics Computation 1965; 19: 297-301.
2. Heideman MT, Johnson DH, and Burrus CS, “Gauss and the history of the fast Fourier transform”.
IEEE ASSP Magazine 1984; 4: 14–21.
3. H. Nyquist, “Certain topics in telegraph transmission theory”, Trans. AIEE, vol. 47, pp. 617-644,
Apr. 1928 Reprint as classic paper in: Proc. IEEE, Vol. 90, No. 2, Feb 2002.
4. Shannon CE. “Communication in the presence of noise”, Proc. Institute of Radio Engineers, vol. 37,
no.1, pp. 10-21, Jan. 1949. Reprint as classic paper in: Proc. IEEE, Vol. 86, No. 2, Feb 1998.
Соседние файлы в папке Библиотека им академика М.И. Перельмана