Biomedical Engineering Image Formation II PD Dr. Frank G. Zöllner Computer Assisted Clinical Medicine Medical Faculty Mannheim Fourier Series - A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (complex exponentials) - The study of Fourier series is a branch of Fourier analysis Example PD Dr. Zöllner I Folie 29 I 9/9/2014 1
Fourier Series Example II http://en.wikipedia.org/wiki/fourier_series PD Dr. Zöllner I Folie 30 I 9/9/2014 Fourier Series f (t) = a 0 2 + a cos(nωt) + n b n sin(nωt) = c n e inω t, with ω = 2π T n =1 n = a 0 = 1 T a n = 2 T b n = 2 T c n = 1 T c +T c c +T c c +T c c +T c f (t)dt f (t) cos(nωt)dt f (t) sin(nωt)dt f (t)e inω t dt Fourier coefficients T frequency of repetition of f(t) PD Dr. Zöllner I Folie 31 I 9/9/2014 2
Fourier Series Example III! How would the Fourier series look like? How many coefficients are necessary to describe the original signal? http://www.thefouriertransform.com/series/sawexample.php PD Dr. Zöllner I Folie 32 I 9/9/2014 Fourier Series! periodic function represented in time domain! alternative representation: frequency domain " same information " different visualisation The Fourier Transform and its Applications, Lecture Notes Prof. Osgood, Stanford PD Dr. Zöllner I Folie 33 I 9/9/2014 3
Fourier Series! periodic function represented in time domain! alternative representation: frequency domain " same information " different visualisation The Fourier Transform and its Applications, Lecture Notes Prof. Osgood, Stanford PD Dr. Zöllner I Folie 34 I 9/9/2014 Fourier Decomposition (FD) S(x,y,t) signal FFT S(x,y,t) time 0.25 Hz ventilation breathing cycle ventilation perfusion 0.95 Hz perfusion heart cycle Deimling et al. ISMRM 2008:2639. Bauman et al. MRM 2009:656-664. PD Dr. Zöllner I Folie 35 I 9/9/2014 4
Peridoc vs aperiodic functions PD Dr. Zöllner I Folie 36 I 9/9/2014 Fourier Transform Fourier transform = decomposition of continuous aperiodic signals into a continuous spectrum + = + PD Dr. Zöllner I Folie 37 I 9/9/2014 5
Fourier Transform Example: Box Function FT PD Dr. Zöllner I Folie 38 I 9/9/2014 Inverse Fourier Transform The Fourier transform of f(x) is defined as This integral, which is a function of k may be written F(k). Transforming F(k) by the same formula, we have* *the second transformation is not exactly the same as the first. In this form, two successive transformations yield the original function. PD Dr. Zöllner I Folie 39 I 9/9/2014 6
Further Examples PD Dr. Zöllner I Folie 40 I 9/9/2014 Example: k-space in MRI FT PD Dr. Zöllner I Folie 41 I 9/9/2014 7
k-space Information standard low-pass high-pass PD Dr. Zöllner I Folie 42 I 9/9/2014 Properties Linearity: PD Dr. Zöllner I Folie 43 I 9/9/2014 8
Properties The shift property: Function and Fourier transform. Delay leads to phase shift in Fourier space. Proof by substitution. PD Dr. Zöllner I Folie 44 I 9/9/2014 Example: MRI image space (abs) k-space (real) k-space (imag) k-space (abs) PD Dr. Zöllner I Folie 45 I 9/9/2014 9
Convolution Theorem The convolution of two functions f(x) and g(x) is: Proof: e.g. PSF substitution: PD Dr. Zöllner I Folie 46 I 9/9/2014 Properties continuous functions continuous functions periodic functions discrete functions discrete functions periodic functions periodic, discrete functions periodic, discrete functions PD Dr. Zöllner I Folie 47 I 9/9/2014 10
Fourier Transform of Differential Equation " Conclusion: - f : amplitude is multiplied by iω - a differential equation becomes an algebraic equation in the Fourier space PD Dr. Zöllner I Folie 48 I 9/9/2014 Fourier Transform of Integral Equation " Conclusion: - Integral: divide amplitude by iω - an integral equation becomes an algebraic equation in the Fourier space. PD Dr. Zöllner I Folie 49 I 9/9/2014 11
Sampling theory! so far: continous image function! in practice: images sampled on regular grid! information loss in between grid points! Sampling theory to determine how to sample this grid to minimize information loss PD Dr. Zöllner I Folie 50 I 9/9/2014 Sampling theory PD Dr. Zöllner I Folie 51 I 9/9/2014 12
Sampling Theory - Example oversampled undersampled PD Dr. Zöllner I Folie 52 I 9/9/2014 Nyquist Frequency! Convolution -> replicatations of G(u,v) in frequency space! Overlap unless maximum frequencies u,v are less than Δu /2,Δv /2! These frequencies are called nyquist frequencies " if x=y there is only one nyquist frequency PD Dr. Zöllner I Folie 53 I 9/9/2014 13
Nyquist Frequency! example of before, one dimensional " value of G(U) -> 0 as u -> " suppose the lower frequency (d) wave is also present! the sampling processs adds both contributions! i.e. the amplitude of the low frequency wave is altered! also known as aliasing! Nyquist frequency given by g(x) = G(u)cos(2πu φ(u))du u lim = 1 2Δx PD Dr. Zöllner I Folie 54 I 9/9/2014 Image Formation Summary! Basic principle to form an image " Valid for all major imageing modalities (MRI, PET, CT, US, optical imaging)! Point spread function describes the quality of the imaging system " resolution " sensitivity " noise! Fourier Transform, tool to analyse the raw (image) signals " CT/ MR image reconstruction " Sampling theorem PD Dr. Zöllner I Folie 55 I 9/9/2014 14