Fourier Transform and Its Medical Application 서울의대의공학교실 김희찬

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1 Fourier Transform and Its Medical Application 서울의대의공학교실 김희찬

2 강의내용 Fourier Transform 의수학적이해 Fourier Transform 과신호처리 Fourier Transform 과의학영상응용

3 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 <source>http://en.wikipedia.org/wiki/integral_transform Inverse Kernel function K -1 for inverse transform

4 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

5 Integral transform Transform Symbol K t 1 t 2 K -1 u 1 u 2 Fourier transform Laplace transform <source>http://en.wikipedia.org/wiki/integral_transform

6 Laplace Transform 0 st F( s) f ( t) e dt where s = + j is a complex number L[ f ( t)] F( s) L 1 [ F( s)] f ( t) Differential Equation Transform differential equation to algebraic equation. 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 ( ) French Astronomer and Mathematician

7 Laplace Transform st F( s) (1) e dt 0 st 0 e e 1 Fs ( ) 0 s s s 0 f () t e t t st ( s) t F() s e e dt e dt 0 0 ( s ) t 0 e e 0 ( s) ( s) 0 1 s f () t F( s) L[ f ( t)] 1 or ut ( ) 1 s e t 1 sin t cos t e e t n t t t sint cost Common Transform Pairs s 2 2 s s 2 2 s 2 2 ( s ) s 2 2 ( s ) 1 2 s n! n 1 s t n e t n! ( s ) n () t 1 1

8 Laplace Transform f () t Fs () f '( t ) sf( s) f (0) t f () t dt Fs () 0 s t e f () t Fs ( ) f ( t T ) u( t T ) st e F() s f (0) lim sf( s) lim f( t) t Laplace Transform Operations s lim sf( s) s0 *

9 Laplace Transform Ex) Solve a differential equation shown below. Sol) 2 d y dy 3 2y 24 y(0) 10 and y'(0) 0 2 dt dt s 2 Y( s) 10s 0 3 sy( s) 10 2 Y( s) 24 s 24 10s 30 Ys () 2 2 s( s 3s 2) s 3s s 30 s( s 1)( s 2) ( s 1)( s 2) Fs Y () s s 1 s 2 f ( t) 12 4e t 2e y 2t

10 Laplace Transform Ex) Solve a differential equation shown below. Sol) 2 d y dy 2 5y 20 y(0) 0 and y'(0) 10 2 dt dt s 2 Y( s) sy( s) 05 Y( s) 20 s Ys () 2 2 s( s 2s 5) s 2s 5 4 4s s 2 Ys () s s 2s 5 s 2s 5 s s 2s 5 4 4( s 1) 3(2) Ys () s ( s 1) (2) ( s 1) (2) ( ) 4 4 t t y t e cos2t 3e sin 2t

11 Periodic Signal Representation Time vs Frequency 시간축 주파수축

12 Fourier Series Harmonic Analysis : 주기적인신호는기본주기와이의정수배주기를갖는 sine 파 ( 고조파 :harmonics) 형의합으로나타낼수있다. Fundamental Harmonics

13 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 function. A constant, DC waveform t f = s(t) = Acos(t+) where t : time, : frequency, A : amplitude, : phase angle A - / -A V, I = Acos(t+) An AC, sine waveform t T = 1/f = 2/

14 Harmonics Analysis Figure Harmonic coefficients of the aortic pressure waveform Figure Harmonic reconstruction of the aortic pressure waveform.

15 Effect of Higher Harmonics Original waveform N=1 N=3 Reconstructed waveform N=7 N=19 N=79 abruptly changing points in time

16 Effect of Higher Harmonics

17 Effect of Higher Harmonics

18 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.

19 Example Problem Fourier Series

20 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.

21 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

22 Exponential Fourier Series Euler s formula : Relationship to trigonometry : Proofs : using Talyor series,

23 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

24 Transition from Fourier Series to Fourier Transform Continuous Aperiodic signal s frequency components. Fourier Series Fourier Transform T, 0 =2/T 0, m 0 t t Fourier Series Fourier Transform

25 Aperiodic Signal Representation Time vs Frequency Bandwidth

26 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 c m of the exponential Fourier series approaches X() as T. Aperiodic function = a periodic function that repeats at infinity Example Problem

27 Linearity Properties of the Fourier Transform Time Shifting / Delay Frequency Shifting Convolution theorem

28 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

29 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 Fourier Transform t t Discrete Time Fourier Transform Discrete Fourier Transform

30 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(N 2 ) 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.

31 Biosignal Representation Time vs Frequency biosignals power spectrum

32 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

33 Signal Filtering Filtering : remove unwanted frequency components Low-Pass, High-Pass, Band-Pass, Band-Stop via Hardware and/or Software

34 Signal Filtering using Fourier Transform Selected parts of the frequency spectrum H(f) Low-pass Filter Band-pass Filter

35 Signal Filtering using Fourier Transform Rejection of the selected parts of the frequency spectrum H(f) Notch Filter

36 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

37 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 healthy 30-year-old male driving car and then hiking uphill. 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 44-year-old female with low heart rate variability while suffering from headaches and pounding sensation in her head.

38 Heart Rate Variability (HRV)

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

40 2D Fourier Transform Fourier transform can be generalized to higher dimensions:

41 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.

42 Image Processing using Fourier Transform Smoothing LPF operation;

43 Image Processing using Fourier Transform Sharpening HPF operation;

44 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.

45 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 Sir Godfrey N. Hounsfield, UK Central Research Laboratories, EMI, London, UK The Nobel Prize in Physiology or Medicine 1979 "for the development of computer assisted tomography"

46 X-ray Imaging System differential attenuation of x-rays to produce an image contrast di n Idx di / dx I I e 0 n I x n : atoms per unit volume of the material I : X-ray intensity at x I 0 : incident X-ray intensity :linear attenuation coefficient[np/cm or cm -1 ]

47 X-ray Imaging System Linear Attenuation Coefficient I 0 x I I = I 0 e -x I N-1 N I 1 x x x x x I = I 0 e -( N-1+ N )x i = ln(i 0 /I)/x

48 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

49 Reconstruction Problem Is the problem mathematically solvable? (1) Iterative method (2) Fourier transform method (3) Back projection method 1 w C 2,1 w 2,2 w 2, C 3 c 1,c 2,c, 3,.c 256 C C = w 1,1 w 1,2 w 1, w 65536,,1 w 65536,,

50 Algebraic Reconstruction Technique cross section f g N j q1 q i1 ij fij N f q ij Where q=indicator for the iteration #. f ij (calculated element) N elements per line g j (measured projection)

51 Iterative ray-by-ray reconstruction Object Next Iteration 1 st Iteration

52 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.

53 Radon Transform (Cont.) y y Object f(x,y) y Projection x 0 x p x f x y dy ( ') (, ) ' x' x x =x 1 x 1 p ( x ') R[ f ( x, y)] f ( x, y) ( x cos y sin x ') dxdy f ( x 'cos y 'sin, x 'sin y 'cos ) dy ' where x' cos sin x y' sin cos y or x cos sin x' y sin cos y'

54 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 ( ) [ p ( x ')] 1 p ( x ')exp( i x ') dx ' f ( x 'cos y 'sin, x 'sin y 'cos )exp( ix ') dx' dy ' f ( x, y)exp[ i ( x cos y sin )] dxdy F( cos, sin ) F(, ) F(, ) x y

55 Fourier Transform Method f(x,y) inverse 2-D transform construct 2-D Spectrum F(, ) 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.

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