fourier transform and its medical application

5/19/2018 Fourier Transform and Its Medical Application

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Fourier Transform and

Its Medical Application


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Fourier Transform

Fourier Transform

Fourier Transform


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Integral transform

a particular kind of mathematicaloperator (a symbol or functionrepresenting a mathematical operation)

any transform T of the following form:Input function fOutput function Tf

http://en.wikipedia.org/wiki/Integral_transform

Kernel function K of 2 variables

Inverse Kernel function K-1

for inverse transform


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Integral transform

Motivation

manipulating and solving the equation inthe target domain can be much easier than

manipulation and solution in the originaldomain.

The solution is then mapped back to the

original domain with the inverse of theintegral transform.



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Integral transform


Transform Symbol K t1 t2 K-1 u1 u2

Fouriertransform

Laplacetransform


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Laplace Transform

the ability to convert

differential equations toalgebraic forms

widely adapted toengineering problems

Pierre-Simon, marquisde Laplace (1749-1827)French Astronomer andMathematician

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

wheres = + jis a complex number


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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

t2

1

s

nt 1

!n

n

s

t ne t

1

!

( )nn

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


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Laplace Transform

( )f t ( )F s

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

0( )

t

f t dt ( )F ss

( )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


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Laplace Transform

Ex) Solve a differential equation shown below.

Sol)

2

2 3 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


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Laplace Transform

Ex) Solve a differential equation shown below.

Sol)

2

2 2 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


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Time vsFrequency

Periodic Signal Representation


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undamental

Harmonics

Fourier Series

Harmonic Analysis : sine(:harmonics).


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Orthogonal Basis Function

spectral factorization : expanding a function from its "standard"

representation to a sum of orthonormal basisfunctions, suitably scaled and shifted.

the determination of the amount by which anindividual orthonormal basis function must bescaled in the spectral factorization of a function,f, is termed the "projection" of fonto 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


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Harmonics Analysis

Figure Harmonic reconstruction of theaortic pressure waveform.

Figure Harmonic coefficients of the aorticpressure waveform


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Effect of Higher Harmonics

Original waveform

Reconstructed waveform

N=1 N=3

N=7 N=19

N=79

abruptly changing points in time


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Joseph Fourier initiated thestudy of Fourier series in orderto solve the heat equation.

: fundamental frequency

: harmonics

Periodic Signal Representation:

The Trigonometric Fourier Series
http://en.wikipedia.org/wiki/Image:Fourier2.jpg


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Fourier Series

Example Problem


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MATLAB Implementation

Figure (a) MATLAB result showing the first 10 terms ofFourier series approximation for the periodic square waveof Fig. 10.7a. (b) The Fourier coefficients are shown as a

function of the harmonic frequency.


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Compact Fourier Series

The sum of sinusoids and cosine can be rewritten by a single cosineterm with the addition of a phase constant;

Example Problem


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Exponential Fourier Series

Eulers formula :

Relationship to trigonometry :

Proofs : using Talyor series,
http://upload.wikimedia.org/wikipedia/commons/7/71/Euler's_formula.svg


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Exponential Fourier Series Complex exponential functions are directly related to sinusoids and cosines;

Eulers identities:

Example Problem

It requires onlyone integration.

Meaning of thenegative frequencies?


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Transition from Fourier Seriesto Fourier Transform

T,0=2/T 0,

m0

Fourier Series Fourier Transform

t

Fourier Series

t

Fourier Transform

ContinuousAperiodicsignals frequency components.


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Bandwidth

Time vsFrequency

Aperiodic Signal Representation


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Fourier Transform

Fourier Integral or Fourier Transform; Used to decompose a continuous aperiodicsignal into its

constituent frequency components.

X() is a complex valued function of the continuous frequency, .

The coefficientscm

of the exponential Fourier series approachesX() as T .

Aperiodic function = a periodic function that repeats at infinity

Example Problem


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Properties ofthe Fourier Transform

Linearity

Time Shifting / Delay

Frequency Shifting

Convolution theorem


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Discrete Fourier Transform

DTFT (Discrete Time Fourier Transform) : Fouriertransform of the sampled version of a continuous signal;

X()is a periodic extension of X() -Fourier transform of acontinuous signal x(t) ;

Periodicity :

Poisson summation formula*:

N - 1

DFT (Discrete Frourier Transform) : Fourier series of aperiodic extension of the digital samples of a continuoussignal;

*which indicates that aperiodic extensionof function can be constructed from thesamples of function


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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 sincfunction, 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


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fast Fourier transform (FFT) : an efficient algorithm to

compute the discrete Fouriertransform (DFT) and its inverse.

There are many distinct FFT

algorithms.

An FFT is a way to computethe same result more quickly:computing a DFT of N pointsin the obvious way, using the

definition, takes O(N2)arithmetical operations, whilean FFT can compute the sameresult in only O(NlogN)operations. Figure (a) 100 Hz sine wave. (b)

Fast Fourier transform (FFT) of100 Hz sine wave.

Figure (a) 100 Hz sine wavecorrupted with noise. (b) FastFourier transform (FFT) of thenoisy 100 Hz sine wave.



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power spectrumbiosignals

Time vsFrequency

Biosignal Representation


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Biosignal Representation

The occipital EEG recordedwhile subject having eyes

closed shows high intensityin the alpha band (7-13 Hz).

Spectrogram:a time-varying spectralrepresentation(formingan image) that shows

how the spectraldensity of a signalvaries with time


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Signal Filtering

Filtering: remove unwanted frequency components Low-Pass, High-Pass, Band-Pass, Band-Stop

via Hardware and/or Software


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Signal Filtering using

Fourier Transform Selected parts of the frequency spectrum H(f)

Low-pass Filter Band-pass Filter


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Rejection of the selected parts of the frequencyspectrum H(f)

Notch Filter

Signal Filtering usingFourier Transform


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Heart Rate Variability (HRV)

Heart rate variability(HRV) is a measure of thebeat-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(


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HRV Examples

Heart rhythmof 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 rhythmof 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.


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Heart Rate Variability (HRV)

Pan, J. and Tompkins, W. J. 1985. A real-time QRS detection algorithm.


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Pan, J. and Tompkins, W. J. 1985. A real time QRS detection algorithm.IEEE Trans. Biomed. Eng. BME-32: 23036,

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.


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2D Fourier Transform

Fourier transform can be generalized tohigher dimensions:


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2D Fourier Transform

The FTs also tend to havebright lines that areperpendicular to lines in theoriginal letter. If the letter hascircular segments, then so

does the FT.

2D cosines with bothhorizontal and verticalcomponents

a pure horizontal cosine of8 cycles and a purevertical cosine of 32 cycles


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Image Processing using

Fourier Transform Smoothing LPF operation;


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Sharpening HPF operation;

Image Processing using

Fourier Transform


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X-ray computed tomography

computed tomography (CT scan) or computedaxial tomography (CAT scan), is a medicalimaging procedure that utilizes computer-

processed X-rays to produce tomographicimages or 'slices' of specific areas of the body.
http://localhost/var/www/apps/conversion/tmp/scratch_8//upload.wikimedia.org/wikipedia/commons/5/50/Computed_tomography_of_human_brain_-_large.pnghttp://localhost/var/www/apps/conversion/tmp/scratch_8//upload.wikimedia.org/wikipedia/commons/1/13/Rosies_ct_scan.jpg


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1917: J. Radon, Mathematical basis

1963: A. Cormack(Tuffs Univ.) developed themathematics behind computerized tomography.

1972: G.N. Hounsfield(EMI), built practical scanner

Allan M. CormackUSATufts UniversityMedford, MA, USA1924 - 1998

Sir Godfrey N. Hounsfield, UK

Central Research Laboratories,EMI, London, UK1919 -

The Nobel Prize in Physiology or Medicine 1979

"for the development of computer assisted tomography"


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X-ray Imaging System

differential attenuation of x-rays to produce animage 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]
http://localhost/var/www/apps/conversion/tmp/scratch_8//upload.wikimedia.org/wikipedia/commons/4/4d/US_Navy_090704-N-6259S-007_Hospital_Corpsman_2nd_Class_Kleinne_Lapid_takes_a_chest_X-ray_of_a_patient_during_a_Continuing_Promise_2009_medical_community_service_project_at_Hospital_Espana_in_Chaminga,_Nicaragua.jpg


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X-ray Imaging System

Linear Attenuation Coefficient

I = I0e

-xI

0 I

x

1I

0

2

3 N-1 NI

x

x

x

x

x

I = I0e-(1+2+3+N-1+N)x

i= ln(I0/I)/x
http://localhost/var/www/apps/conversion/tmp/scratch_8//upload.wikimedia.org/wikipedia/commons/a/a6/Pneumonia_x-ray.jpg


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CT scanner with cover removed toshow internal components.T: X-ray tube, D: X-ray detectors

X: X-ray beam, R: Gantry rotation


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Reconstruction Problem

c1,c2,c,3,.c256

1 256

65281

12

3.

.

.

.

65536

C1

C2

C3

.

.

.

.

.

C65536

w1,1w1,2w1,65536

w2,1w2,2w2,65536

.

.

.

.

.

.

w65536,,1w65536,,65536

=

Is the problem mathematically solvable?

(1) Iterative method

(2) Fourier transform method

(3) Back projection method


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Algebraic Reconstruction Technique

cross section

fij(calculated element)

Nelements per line

gj(measured projection)

1 1

Nq

j ijq q i

ij ij

g f

f fN

Where q=indicator for the iteration #.


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Iterative ray-by-ray reconstruction

11

14

1

8

5

79 16

6

1210

33

88

3.52.5

8.57.5

1111

5

11

1

79

5

97

1

-1.5 +1.5

1stIteration

Object +2

+2

-.5 +.5

Next

Iteration


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Radon Transform

Radon transform operator performs the lineintegral 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 2and symmetric; therefore, p(x) = p(-x)

The Radon transform leads to the projection or centralslice theorem through a 1-D or 2-D Fourier Transform.

The Radon transform domain data provide a sinogram.


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Objectf(x,y)

y

x

x

y

Projectionx

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


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Projection Theorem Relationship between the 2-D Fourier transform

of the object function f(x,y) and 1-D Fouriertransform of its Radon transform or the projectiondata 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


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Fourier Transform Method

1-Dtransform

f(x,y)

p

(x) P

(

)

construct

2-DSpectrumF(, )

inverse2-D

transform

A 1-D Fourier transform of the projection data p(x) at a givenview angle is the same as the radial data passing through the

fourier transform and its medical application

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