Fourier Transform and
Its Medical Application
서울의대 의공학교실
김희찬
강의내용
• Fourier Transform의 수학적 이해
• Fourier Transform과 신호처리
• Fourier Transform과 의학영상 응용
Integral transform
• a particular kind of mathematical
operator (a symbol or function
representing a mathematical operation)
• any transform T of the following form:
Output function Tf
Input function f
Kernel function K of 2 variables
Inverse Kernel function K-1
for inverse transform
<source>http://en.wikipedia.org/wiki/Integral_transform
Integral transform
• Motivation
– manipulating and solving the equation in
the target domain can be much easier than
manipulation and solution in the original
domain.
– The solution is then mapped back to the
original domain with the inverse of the
integral transform.
<source>http://en.wikipedia.org/wiki/Integral_transform
Integral transform
Transform
Symbol
K
t1
Fourier
transform
Laplace
transform
<source>http://en.wikipedia.org/wiki/Integral_transform
t2
K-1
u1
u2
Laplace Transform

F ( s)   f (t )e st dt
Differential Equation
0
where s =  + j is a complex number
L[ f (t )]  F ( s )
Transform differential
equation to
algebraic equation.
L1[ F (s)]  f (t )
• the ability to convert
differential equations to
algebraic forms
• widely adapted to
engineering problems
Solve equation
by algebra.
Determine
inverse
transform.
Solution
Pierre-Simon, marquis
de Laplace (1749-1827)
French Astronomer and
Mathematician
Laplace Transform
Common Transform Pairs
f (t )
1 or u (t )

F ( s )   (1)e  st dt
0
F ( s) 

e  1
e 

0



 s  0

s

 s
 st
0
e  t
sin  t
F ( s )  L[ f (t )]
1
s
1
s 

s 2
2
cos  t
f (t )  e
e  t sin  t
 t


0
0
F ( s )   e  t e  st dt   e  (  s )t dt

e  ( s  )t 
e 0

 0
( s   )  0
( s   )

1
s 
e  t cos  t
t
tn
e  t t n
 (t )
s
s2   2

(s   )2   2
s 
(s   )2   2
1
s2
n!
s n1
n!
( s   ) n 1
1
Laplace Transform
Laplace Transform Operations
f (t )
f '(t )

t
0
f (t )dt
F (s)
sF ( s )  f (0)
F (s)
s
F (s   )
et f (t )
f (t  T )u (t  T )
f (0)
e sT F (s)
lim sF (s)
lim f (t )
lim sF ( s)*
t 
s 
s 0
Laplace Transform
• Ex) Solve a differential equation shown below.
• Sol)
d2y
dy

3
 2 y  24
2
dt
dt
y (0)  10 and y '(0)  0
s 2Y (s)  10s  0  3 sY (s) 10  2Y ( s) 
24
10s  30

s( s 2  3s  2) s 2  3s  2
24
10 s  30


s( s  1)( s  2) ( s  1)( s  2)
Y ( s) 
YF (s) 
12
4
2


s s 1 s  2
yf (t )  12  4et
 2e2t
24
s
Laplace Transform
• Ex) Solve a differential equation shown below.
• Sol)
d2y
dy

2
 5 y  20
2
dt
dt
y (0)  0 and y '(0)  10
s 2Y (s)  0  10  2  sY (s)  0  5Y ( s) 
Y ( s) 
20
s
20
10

s( s 2  2s  5) s 2  2s  5
Y ( s) 
4
4s  8
10
4
4s  2
 2
 2
  2
s s  2 s  5 s  2 s  5 s s  2s  5
Y ( s) 
4
4( s  1)
3(2)


s ( s  1)2  (2) 2 ( s  1) 2  (2) 2
y(t )  4  4et cos 2t  3et sin 2t
Periodic Signal Representation
Time vs Frequency
시간축
주파수축
Fourier Series
Harmonic Analysis : 주기적인 신호는 기본주기와 이의 정수 배
주기를 갖는 sine파(고조파:harmonics)형의 합으로 나타낼 수 있다.
Fundamental
Harmonics
Orthogonal Basis Function
• spectral factorization :
V, I
– expanding a function from its "standard"
representation to a sum of orthonormal basis
functions, suitably scaled and shifted.
– the determination of the amount by which an
individual orthonormal basis function must be
scaled in the spectral factorization of a function,
f, is termed the "projection" of f onto that basis
V, I = Acos(t+)
function.
f = s(t) = Acos(t+)
t
A constant, DC waveform
where t : time,
 : frequency,
A : amplitude,
 : phase angle
A
- / 
-A
An AC, sine waveform
t
T = 1/f
= 2/
Harmonics Analysis
Figure Harmonic coefficients of the aortic
pressure waveform
Figure Harmonic reconstruction of the
aortic pressure waveform.
Effect of Higher Harmonics
Original waveform
N=1
N=3
N=7
N=19
Reconstructed waveform
N=79
abruptly changing points in time
Effect of Higher Harmonics
Effect of Higher Harmonics
Periodic Signal Representation:
The Trigonometric Fourier Series
: fundamental frequency
: harmonics
Joseph Fourier initiated the
study of Fourier series in order
to solve the heat equation.
Fourier Series
• Example Problem
MATLAB Implementation
Figure (a) MATLAB result showing the first 10 terms of
Fourier series approximation for the periodic square wave
of Fig. 10.7a. (b) The Fourier coefficients are shown as a
function of the harmonic frequency.
Compact Fourier Series
• The sum of sinusoids and cosine can be rewritten by a single cosine
term with the addition of a phase constant;
• Example Problem
Exponential Fourier Series
Euler’s formula :
Relationship to trigonometry :
Proofs : using Talyor series,
Exponential Fourier Series
•
Complex exponential functions are directly related to sinusoids and cosines;
•
Euler’s identities:
Meaning of the
negative frequencies?
It requires only
one integration.
• Example Problem
Transition from Fourier Series
to Fourier Transform
Continuous Aperiodic signal’s frequency components.
Fourier Transform
Fourier Series
T→,
0=2/T →0,
m0→
t
Fourier Series

t
Fourier Transform

Aperiodic Signal Representation
Time vs Frequency
Bandwidth
Fourier Transform
• Fourier Integral or Fourier Transform;
– Used to decompose a continuous aperiodic signal into its
constituent frequency components.
– X() is a complex valued function of the continuous frequency, .
– The coefficients cm of the exponential Fourier series approaches
X() as T  .
– Aperiodic function = a periodic function that repeats at infinity
• Example Problem
Properties of
the Fourier Transform
• Linearity
• Time Shifting / Delay
• Frequency Shifting
• Convolution theorem
Discrete Fourier Transform
• DTFT (Discrete Time Fourier Transform) : Fourier
transform of the sampled version of a continuous signal;
– X() is a periodic extension of X’() - Fourier transform of a
continuous signal x(t) ;
• Periodicity :
• Poisson summation formula*:
*which indicates that a periodic extension of function
samples of function
can be constructed from the
• DFT (Discrete Frourier Transform) : Fourier series of a
periodic extension of the digital samples of a continuous
signal;
N-1
Discrete Fourier Transform
• Symmetry (or Duality)
– if the signal is even: x(t) = x(-t)
– then we have
– For example, the spectrum of an even square wave is a sinc
function, and the spectrum of a sinc function is an even square
wave.
• Extended Symmetry
t
Fourier Series


t
Discrete Time Fourier Transform
t
Fourier Transform
t
Discrete Fourier Transform


Discrete Fourier Transform
• fast Fourier transform (FFT) :
– an efficient algorithm to
compute the discrete Fourier
transform (DFT) and its inverse.
– There are many distinct FFT
algorithms.
– An FFT is a way to compute
the same result more quickly:
computing a DFT of N points
in the obvious way, using the
definition, takes O(N2)
arithmetical operations, while
an FFT can compute the same
result in only O(NlogN)
operations.
Figure (a) 100 Hz sine wave. (b)
Fast Fourier transform (FFT) of
100 Hz sine wave.
Figure (a) 100 Hz sine wave
corrupted with noise. (b) Fast
Fourier transform (FFT) of the
noisy 100 Hz sine wave.
Biosignal Representation
Time vs Frequency
biosignals
power spectrum
Biosignal Representation
The occipital EEG recorded
while subject having eyes
closed shows high intensity
in the alpha band (7-13 Hz).
Spectrogram :
a time-varying spectral
representation(forming
an image) that shows
how the spectral
density of a signal
varies with time
Signal Filtering
• Filtering : remove unwanted frequency components
• Low-Pass, High-Pass, Band-Pass, Band-Stop
• via Hardware and/or Software
Signal Filtering using
Fourier Transform
• Selected parts of the frequency spectrum H(f)
Low-pass Filter
Band-pass Filter
Signal Filtering using
Fourier Transform
• Rejection of the selected parts of the frequency
spectrum H(f)
Notch Filter
Heart Rate Variability (HRV)
• Heart rate variability (HRV) is a measure of the
beat-to-beat variations in heart rate.
• Time domain measures
– standard deviation of beat-to-beat intervals
– root mean square of the differences between heart beats
(rMSSD)
– NN50 or the number of normal to normal complexes that
fall within 50 milliseconds
– pNN50 or the percentage of total number beats that fall
with 50 milliseconds.
• Frequency domain measures
– ULF(<0.0033Hz), VLF(0.0033~0.04), LF(0.04~0.15)
– HF (0.15~0.4Hz)
– LF/HF : an index of sympathetic to parasympathetic
balance
HRV Examples
Heart rhythm of a 33-year-old male experiencing anxiety.
The prominent spikes are due to pulses of activity in the
sympathetic nervous system.
Heart rhythm of a heart transplant recipient.
Note the lack of variability in heart rate, due to loss of
autonomic nervous system input to the heart.
Heart rhythm
of a healthy 30-year-old male driving car and
then hiking uphill.
Heart rhythm
of a 44-year-old female with low heart rate variability while
suffering from headaches and pounding sensation in her head.
Heart Rate Variability (HRV)
Pan, J. and Tompkins, W. J. 1985. A real-time QRS detection algorithm.
IEEE Trans. Biomed. Eng. BME-32: 230–36,
A Real-time QRS Detection Algorithm
ECG sampled at 200 samples per second.
Low-pass filtered ECG.
ECG after bandpass filtering and differentiation. ECG signal after squaring function.
Bandpass-filtered ECG.
Signal after moving window integration.
2D Fourier Transform
• Fourier transform can be generalized to
higher dimensions:
2D Fourier Transform
a pure horizontal cosine of
8 cycles and a pure
vertical cosine of 32 cycles
2D cosines with both
horizontal and vertical
components
The FTs also tend to have
bright lines that are
perpendicular to lines in the
original letter. If the letter has
circular segments, then so
does the FT.
Image Processing using
Fourier Transform
• Smoothing LPF operation;
Image Processing using
Fourier Transform
• Sharpening HPF operation;
X-ray computed tomography
• computed tomography (CT scan) or computed
axial tomography (CAT scan), is a medical
imaging procedure that utilizes computerprocessed X-rays to produce tomographic
images or 'slices' of specific areas of the body.
X-ray computed tomography
• 1917: J. Radon, Mathematical basis
• 1963: A. Cormack(Tuffs Univ.) developed the
mathematics behind computerized tomography.
• 1972: G.N. Hounsfield(EMI), built practical scanner
Allan M. Cormack
USA
Tufts University
Medford, MA, USA
1924 - 1998
Sir Godfrey N. Hounsfield, UK
Central Research Laboratories,
EMI, London, UK
1919 -
The Nobel Prize in Physiology or Medicine 1979
"for the development of computer assisted tomography"
X-ray Imaging System
• differential attenuation of x-rays to produce an
image contrast
dI  n Idx
dI / dx  n I
I  I 0e  x
n : atoms per unit volume of the material
I : X-ray intensity at x
I0: incident X-ray intensity
:linear attenuation coefficient[np/cm or cm-1]
X-ray Imaging System
• Linear Attenuation Coefficient
I0
I

I = I0e-x
x
I0
1 2 3 ••• N-1 N
x
x
x
•••
x
I
x
I = I0e-(1+2+3•••+N-1+N)x
i= ln(I0/I)/x
X-ray computed tomography
CT scanner with cover removed to
show internal components.
T: X-ray tube, D: X-ray detectors
X: X-ray beam, R: Gantry rotation
Reconstruction Problem
“Is the problem mathematically solvable?”
1
256
(1) Iterative method
(2) Fourier transform method
(3) Back projection method
65281
c1,c2,c,3,….c256
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C2
C3
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w2,1 w2,2 … w2,65536
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w65536,,1 …w65536,,65536
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2
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Algebraic Reconstruction Technique
N
fijq 1  fijq 
cross section
g j   fijq
i 1
N
Where q=indicator for the iteration #.
fij(calculated element)
gj(measured projection)
N elements per line
Iterative ray-by-ray reconstruction
Object
8
9
7
16
+2
8
8
1
5
6
+2
3
3
10
12
14
11
-.5
Next
Iteration
11
+.5
7.5 8.5
2.5 3.5
11
11
-1.5
st
1 Iteration
7
1
+1.5
9
7
1
5
9
5
Radon Transform
• Radon transform operator performs the line
integral of the 2-D image data along y’
• The function p(x’) is the 1-D projection of f(x,y)
at an angle 
• Properties
– The projections are periodic in  with a period of 2
and symmetric; therefore, p(x’) = p(-x’)
– The Radon transform leads to the projection or central
slice theorem through a 1-D or 2-D Fourier Transform.
– The Radon transform domain data provide a sinogram.
Radon Transform (Cont.)
y
y’
Object
f(x,y)
p ( x ')  R[ f ( x, y )]

x’

x


f ( x, y ) ( x cos   y sin   x ')dxdy




f ( x 'cos   y 'sin  , x 'sin   y 'cos  ) dy '

where
p ( x ')   f ( x, y)dy '
y’
x ' x1
x’
Projection
0
x’=x1
 x '   cos 
 y '    sin 
  
or
 x  cos 
 y    sin 
  
sin    x 
cos    y 
 sin    x ' 
cos    y '
Projection Theorem
• Relationship between the 2-D Fourier transform
of the object function f(x,y) and 1-D Fourier
transform of its Radon transform or the projection
data p(x’).
P ( )  1[ p ( x ')]


 p ( x ') exp(i x ')dx '





f ( x 'cos   y 'sin  , x 'sin   y 'cos  ) exp( i x ') dx ' dy '



f ( x, y ) exp[i ( x cos   y sin  )]dxdy

 F ( cos  ,  sin  )  F ( x ,  y )
 F ( ,  )

Fourier Transform Method
construct
2-D
Spectrum
F(, )
f(x,y)
inverse
2-D
transform
p(x’)
1-D
transform
P()
• A 1-D Fourier transform of the projection data p(x’) at a given
view angle  is the same as the radial data passing through the
origin at a given angle  in the 2-D Fourier transform domain data.