Fourier Transform and Its Medical Application

서울의대 의공학교실

김 희 찬

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: Input function f Output function Tf

<source>http://en.wikipedia.org/wiki/Integral_transform

Kernel function K of 2 variables

Inverse Kernel function K-1 for inverse 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

<source>http://en.wikipedia.org/wiki/Integral_transform

Transform Symbol K t1 t2 K-1 u1 u2

Fourier transform

Laplace transform

Laplace Transform

• the ability to convert differential equations to algebraic forms

• widely adapted to engineering problems

Pierre-Simon, marquis de Laplace (1749-1827) French Astronomer and Mathematician

Differential Equation

Transform differential

equation to

algebraic equation.

Solve equation

by algebra.

Determine

inverse

transform.

Solution

[ ( )] ( )L f t F s

1[ ( )] ( )L F s f t

0( ) ( ) stF s f t e dt

where s = + j is a complex number

Laplace Transform

( )f t ( ) [ ( )]F s L f t

1 or ( )u t 1

s

te 1

s

sin t 2 2s

cos t 2 2

s

s

sinte t 2 2( )s

coste t 2 2( )

s

s

t 2

1

s

nt 1

!n

n

s

t ne t 1

!

( )n

n

s

( )t 1

Common Transform Pairs

0( ) (1) stF s e dt

0

0

1( ) 0

ste eF s

s s s

( )

0 0

( ) 0

0

( )

0( ) ( )

1

t st s t

s t

F s e e dt e dt

e e

s s

s

( ) tf t e

Laplace Transform

( )f t ( )F s

'( )f t ( ) (0)sF s f

0( )

t

f t dt ( )F s

s

( )te f t ( )F s

( ) ( )f t T u t T ( )sTe F s

(0)f lim ( )s

sF s

lim ( )t

f t

*

0lim ( )s

sF s

Laplace Transform Operations

Laplace Transform • Ex) Solve a differential equation shown below.

• Sol)

2

23 2 24

d y dyy

dt dt (0) 10 and '(0) 0y y

2 24( ) 10 0 3 ( ) 10 2 ( )s Y s s sY s Y s

s

2 2

24 10 30( )

( 3 2) 3 2

24 10 30

( 1)( 2) ( 1)( 2)

sY s

s s s s s

s

s s s s s

12 4 2( )

1 2F s

s s s

2( ) 12 4 2t tf t e e

Y

y

Laplace Transform • Ex) Solve a differential equation shown below.

• Sol)

2

22 5 20

d y dyy

dt dt (0) 0 and '(0) 10y y

2 20( ) 0 10 2 ( ) 0 5 ( )s Y s sY s Y s

s

2 2

20 10( )

( 2 5) 2 5Y s

s s s s s

2 2 2

4 4 8 10 4 4 2( )

2 5 2 5 2 5

s sY s

s s s s s s s s

2 2 2 2

4 4( 1) 3(2)( )

( 1) (2) ( 1) (2)

sY s

s s s

( ) 4 4 cos2 3 sin 2t ty t e t e t

Time vs Frequency

주파수축

시간축

Periodic Signal Representation

Fundamental

Harmonics

Fourier Series

Harmonic Analysis : 주기적인 신호는 기본주기와 이의 정수 배 주기를 갖는 sine파(고조파:harmonics)형의 합으로 나타낼 수 있다.

Orthogonal Basis Function • spectral factorization :

– 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 function.

t

V, I

A constant, DC waveform

A

-A

T = 1/f = 2/

t

V, I = Acos(t+)

- /

An AC, sine waveform

f = s(t) = Acos(t+) where t : time, : frequency, A : amplitude, : phase angle

Harmonics Analysis

Figure Harmonic reconstruction of the aortic pressure waveform.

Figure Harmonic coefficients of the aortic pressure waveform

Effect of Higher Harmonics

Original waveform

Reconstructed waveform

N=1 N=3

N=7 N=19

N=79

abruptly changing points in time

Effect of Higher Harmonics

Effect of Higher Harmonics

Joseph Fourier initiated the study of Fourier series in order to solve the heat equation.

: fundamental frequency : harmonics

Periodic Signal Representation: The Trigonometric Fourier Series

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:

• Example Problem

It requires only one integration.

Meaning of the negative frequencies?

Transition from Fourier Series to Fourier Transform

T→, 0=2/T →0,

m0→

Fourier Series Fourier Transform

t

Fourier Series

t

Fourier Transform

Continuous Aperiodic signal’s frequency components.

Bandwidth

Time vs Frequency

Aperiodic Signal Representation

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*:

N - 1

• DFT (Discrete Frourier Transform) : Fourier series of a periodic extension of the digital samples of a continuous signal;

*which indicates that a periodic extension of function can be constructed from the samples of function

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

t t

Fourier Series

Discrete Time Fourier Transform Discrete Fourier Transform

t 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.

Discrete Fourier Transform

power spectrum biosignals

Time vs Frequency

Biosignal Representation

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

• Rejection of the selected parts of the frequency spectrum H(f)

Notch Filter

Signal Filtering using Fourier Transform

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 44-year-old female with low heart rate variability while

suffering from headaches and pounding sensation in her head.

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 33-year-old male experiencing anxiety. The prominent spikes are due to pulses of activity in the

sympathetic nervous system.

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. Bandpass-filtered ECG.

ECG after bandpass filtering and differentiation.

ECG signal after squaring function. Signal after moving window integration.

2D Fourier Transform

• Fourier transform can be generalized to higher dimensions:

2D Fourier Transform

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.

2D cosines with both

horizontal and vertical

components

a pure horizontal cosine of

8 cycles and a pure

vertical cosine of 32 cycles

Image Processing using Fourier Transform

• Smoothing LPF operation;

• Sharpening HPF operation;

Image Processing using Fourier Transform

X-ray computed tomography

• computed tomography (CT scan) or computed axial tomography (CAT scan), is a medical imaging procedure that utilizes computer-processed 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

0

/

x

dI n Idx

dI dx n I

I I e

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

I = I0e-x

I0 I

x

1

I0

2 3 N-1 N

I

x

x

x

x

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

c1,c2,c,3,….c256

1 256

65281

1

2

3

.

.

.

.

65536

C1

C2

C3

.

.

.

.

.

C65536

w1,1 w1,2 … w1,65536

w2,1 w2,2 … w2,65536

.

.

.

.

.

.

w65536,,1 …w65536,,65536

=

“Is the problem mathematically solvable?”

(1) Iterative method

(2) Fourier transform method

(3) Back projection method

Algebraic Reconstruction Technique

cross section

fij(calculated element)

N elements per line

gj(measured projection)

1 1

Nq

j ijq q i

ij ij

g f

f fN

Where q=indicator for the iteration #.

Iterative ray-by-ray reconstruction

11

14

1

8

5

7 9 16

6

12 10

3 3

8 8

3.5 2.5

8.5 7.5

11 11

5

11

1

7 9

5

9 7

1

-1.5 +1.5

1st

Iteration

Object +2

+2

-.5 +.5

Next

Iteration

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.

Object f(x,y)

y

x

x’

y’

Projection x’

y’

0

1'( ') ( , ) ' x xp x f x y dy

x’=x1

Radon Transform (Cont.)

( ') [ ( , )]

( , ) ( cos sin ')

( 'cos 'sin , 'sin 'cos ) '

' cos sin

' sin cos

cos sin '

sin cos '

p x R f x y

f x y x y x dxdy

f x y x y dy

where

x x

y y

or

x x

y 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’).

1( ) [ ( ')]

( ') exp( ') '

( 'cos 'sin , 'sin 'cos )exp( ') ' '

( , ) exp[ ( cos sin )]

( cos , sin ) ( , )

( , )

x y

P p x

p x i x dx

f x y x y i x dx dy

f x y i x y dxdy

F F

F

Fourier Transform Method

1-D transform

f(x,y)

p(x’) P()

construct 2-D

Spectrum F(, )

inverse 2-D

transform

• 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.