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CUNY BME I5000 - Fourier Transform Review

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9K-SpaceSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 201Lucas Parra, CCNYCity College of New YorkBME I5000: Biomedical ImagingLecture FTFourier Transform ReviewLucas C. Parra, [email protected] Blackboard: http://cityonline.ccny.cuny.edu/2Lucas Parra, CCNYCity College of New YorkThe Fourier Transform (FT) is defined as*The FT is an invertible transformationWe can show this using * Notational convention: Use k for spacial, and ω for temporal frequency.H (k )=FT [h(x )]=∫−∞∞dx h (x)e−i 2 π kx∫−∞∞dk e−i 2 π kx=δ(x)∫−∞∞dk H (k )ei 2 π kx=∫−∞∞∫−∞∞dk dx ' h (x ')e−i 2 π k (x '− x)=h(x )h (x)=FT−1[ H (k )]=∫−∞∞dk H (k )ei 2 π kxFourier Transform (1D)3Lucas Parra, CCNYCity College of New YorkH (k )=R(k )+i I (k )=A(k )ei ϕ(k)Fourier Transform – Frequency DomainH(k) is in general complex valued with R(k) = Re(H(k)), I(k) = Im(H(k)) andAmplitude:Phase:Positive frequencies k>0: counterclockwise rotationNegative frequencies k<0: clockwise rotation A(k )=∣H (k )∣=√R(k )2+I (k )2ϕ(k )=arctan(I (k )R(k ))H (k )=δ (k −k0) , h (x )=ei 2 π k0x4Lucas Parra, CCNYCity College of New YorkFourier Transform - Examples5Lucas Parra, CCNYCity College of New YorkFourier Transform - Examples6Lucas Parra, CCNYCity College of New YorkB(k )=FT [h( x)∗g ( x)]=H (k )G (k )Fourier Transform – Convolution Theoremb (x)=∫−∞∞dx ' h (x ' ) g ( x−x ')=h( x)∗g ( x)Convolution is defined asConvolution Theorem states that Note that with the convolution theorem we can implement convolution as a multiplication in the frequency domain. h(x) H(k) B(k) b(x)g(x) G(k)FTFTFT-1×7Lucas Parra, CCNYCity College of New YorkFT [h (x )∗g ( x)]=∫−∞∞dx h( x)∗g (x )e−i 2 π kx==∫−∞∞dx∫−∞∞dx ' h( x ' ) g (x −x ' )e−i 2 π kx=∫−∞∞dx∫−∞∞dx ' h( x ' ) g (x )e−i 2 π k( x+x ')=∫−∞∞dx ' h(x ' )e−i 2 π kx '∫−∞∞dx g (x )e−i 2 πkx=H (k )G(k )Fourier Transform – Convolution TheoremProof8Lucas Parra, CCNYCity College of New YorkFourier Transform - Inverse FilterWith the Convolution Theorem we can derive the inverse convolution (or inverse filter)ThereforeAnd the inverse filter is given by the inverse FT of H-1(k):b (x)=g ( x)∗h( x)⇔ B (k )=G(k ) H (k )G(k )=B(k )H (k )g ( x)= FT−1[1H (k )]∗b( x)9Lucas Parra, CCNYCity College of New YorkThe Fourier transform in 2D is defined asThe inverse transform is defined correspondingly. Note that the exponent is separable and therefor the Fourier transform can be applied sequentially in each dimension!In multiple dimensions with k = [kx, ky, kz, ...]T, r = [x, y, z, ...]TFourier Transform – 2D and higherH (kx, ky)=∫−∞∞∫−∞∞dx dy h (x , y)e−i 2 π(kxx+kyy)H (kx, ky)=∫−∞∞dy e−i 2 π kyy∫−∞∞dx h( x , y)e−i 2 π kxxH (k)=∫−∞∞...∫−∞∞d r h(r )e−i 2 πk⋅r10Lucas Parra, CCNYCity College of New YorkSource: (C.A. Mistretta)Fourie Transform k-space11Lucas Parra, CCNYCity College of New YorkNote that the in images the phase carries most of the information2D Fourier Transform – Phase and Amplitude12Lucas Parra, CCNYCity College of New YorkNote that the in images the phase carries most of the information2D Fourier Transform – Phase and Amplitude13Lucas Parra, CCNYCity College of New YorkThe 2D convolution with a PSF h(x,y) is The Convolution Theorem applies in higher dimensions as well.Even though h(x,y) may not be separable (h(x,y) ≠ h(x)*h(y)) with the convolution theorem we never have to truly compute a 2D convolution.Fourier Transform – 2D Convolutionb (x , y )=∫−∞∞∫−∞∞dx ' dy ' h(x ' , y ' ) g ( x−x ' , y− y ' )=h (x , y )∗g (x , y)B(kx, ky)=G (kx, ky)H (kx, ky)14Lucas Parra, CCNYCity College of New YorkConsider stationary oscillatory input to a LSI system h(x):The output is the input times the FT of the impulse responseThe oscillation with frequency k has been modified in phase φ and amplitude A Fourier Domain System Responseg ( x)=ei 2 π k xb (x)=h (x)∗g ( x)=∫−∞∞dx ' h( x ' )ei 2 π k (x− x' )=H (k )ei 2 π k x H (k )A=∣H (k )∣ ϕ=arg (H (k ))H (k )= Aei ϕH (k )ei 2 π kxei 2 π k x15Lucas Parra, CCNYCity College of New YorkFT inversion formula tells us that arbitrary input g(x) can be decomposed into sum of oscillations weighted by G(k)The system response to that is given by the convolution theoremFourier Domain System ResponseB(k )=H (k )G (k )...H(k1) H(k2) H(k3) G(k1)G(k2) G(k3)FT FT-1 g(x) b(x) g ( x)=∫−∞∞dk G(k )ei 2 π kx16Lucas Parra, CCNYCity College of New YorkRadon inverse filter kernel, H(k) = | k | (high pass filter)Fourier Domain System Response17Lucas Parra, CCNYCity College of New YorkGaussian low-pass filter kernel, Fourier Domain System ResponseH (k )=exp (−k22 σ2), σ=1.2718Lucas Parra, CCNYCity College of New YorkFourier Domain System Response 2D19Lucas Parra, CCNYCity College of New YorkThe numerical implementation of the FT is discrete in x and k is referred to as Discrete Fourier Transform (DFT). ●A fast algorithm is available (FFT) to compute the DFT in only N log2 N operations instead of N2.●Convolution with long filters is therefore often implemented using the FFT. ●FFT requires N to be a power of 2. Fourier Transform – FFTg [ x ]=1N∑k=0N −1G [k ]ej 2 π kx / NG[ k ]=∑x=0N −1g [ x ]e− j 2 π kx / N20Lucas Parra, CCNYCity College of New YorkTo filter signals with lengths not a power of 2: ●Pad zeros at the end up to next power of 2 ●FFT●multiply each frequency●inverse FFT ●keep only fist L values.Matlab b(x,y) = h(x,y) * g(x,y) using fft (assumes h smaller than g): L = size(g);N = 2.^ceil(log2(L));b = ifft2(fft2(g,N(1),N(2)).*fft2(h,N(1),N(2)));if isreal([g;h]) b = real(b); endb = b(1:L(1),1:L(2));Demonstrate: fftshift, inverse filtering, separability. Assignment FT: Generate image with randomized phase and original amplitude, and image with random amplitude and original phase.Fourier Transform – Filter with


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