Fourier Sampling. Fourier Sampling. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2005 Linear Systems Lecture 3.
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1 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2005 Linear Sstems Lecture 3 Fourier Sampling F Instead of sampling the signal, we sample its Fourier Transform Sample??? F -1 Fourier Sampling (1 / comb( / G S ( = G( 1 comb k x = G( δ( n n= = G(n δ( n n= 1
2 Fourier Sampling -- Inverse Transform Χ = * = 1/ Fourier Sampling -- Inverse Transform g S (x = F 1 [ G S ( ] = F 1 G( 1 comb k x = F 1 [ G( ] F 1 1 comb k x = g(x comb( x = g(x δ(x n n= = g(x 1 = 1 n= g(x n= δ(x n n Nquist Condition FOV (Field of View 1/ To avoid overlap, 1/ > FOV, or equivalentl, <1/FOV 2
3 Aliasing FOV (Field of View 1/ Aliasing occurs when 1/ < FOV Intuitive view of Aliasing FOV = 2/FOV =1/FOV Aliasing Example =1 1/ =1 3
4 2D Comb Function comb(x, = δ(x m, n m= n= = δ(x mδ( n m= n= = comb(xcomb( Scaled 2D Comb Function comb(x /Δx, /Δ = comb(x /Δxcomb( /Δ = ΔxΔ δ(x mδxδ( nδ m= n= Δ Δx * = 1/ k X = 4
5 2D k-space sampling 1 G S (,k = G(,k comb, k = G(,k δ( m,k n m= n= = G(m,n δ( m,k n m= n= 2D k-space sampling [ ] g S (x, = F 1 G S (,k 1 = F 1 G(,k comb, k 1 = F 1 [ G(,k ] F 1 comb, k = g(x, comb( x comb( = g(x δ(x m δ( n m= n= 1 = g(x m= 1 = g(x m= n= n= δ(x m δ( n m, n Nquist Conditions 1/ k Y > FOV Y 1/ k X > FOV X 1/ k Y FOV Y FOV X 1/ k X 5
6 Aliasing Windowing Windowing the data in Fourier space G W (,k = G(,k W,k ( Results in convolution of the object with the inverse transform of the window ( x, = g( x, w(x, g W Resolution 6
7 ( = rect W,k Windowing Example rect k W k k w( x, = F 1 rect x W rect k kx W k = W k sinc( xsinc W k ( g W ( ( x, = g(x, W k sinc( xsinc W k Windowing Example g(x, = δ(x + δ(x 1 [ ]δ( g W ( ( x, = [ δ(x + δ(x 1 ]δ( W k sinc( xsinc W k = W k ([ δ(x + δ(x 1 ] sinc( x sinc( W k = W k ( sinc( x + sinc( (x 1 sinc( W k =1 =1.5 = 2 w E = 1 w(0 Example Effective Width w(xdx w E = 1 1 sinc( xdx = F[ sinc( x ] kx = 0 = 1 = 1 rect = 0 1 w E 1 7
8 Resolution and spatial frequenc With a window of width the highest spatial frequenc is /2. This corresponds to a spatial period of 2/. 1 = Effective Width 2 Sampling and Windowing * = * = X X = G SW Sampling and Windowing Sampling and windowing the data in Fourier space 1 (,k = G(,k comb, k rect, k W k Results in replication and convolution in object space. g SW ( x, = W k g( x, comb( x, sinc( xsinc(w k 8
9 Discrete Fourier Transform Idea: If we sample and window in the Fourier domain, we obtain a finite number of discrete Fourier samples. When we reconstruct the object, we should have the same number of pixels in our object. Also, the windowing process, has band-limited the sampled Fourier transform, so this allows us to sample the replicated object at discrete points. 1D Sampling and Windowing * * = Χ Χ = Discrete Fourier Transform x = * = 9
10 Discrete Fourier Transform FOV N x = W N x = FOV X 1/ = W Note that FOV X = 1 = δ x, our measure for resolution. N x G DFT (,k = G(,k DFT Sampling, windowing, and replication in Fourier space 1 comb, k rect, k W comb k, k W k g DFT Replication, convolving, sampling in object space ( x, = W k g( x, comb( x, sinc( xsinc(w k W k comb( x, DFT FOV F[ g D (x] = g D (xe j 2πm x dx 0 FOV = g D ( n /W x δ(x n /W x e j 2πmxx dx 0 n= = g D ( n /W x δ(x n /W x e j 2πm x dx n= 0 = g D ( n /W x n= 0 = g D [ n] n= 0 e j 2πmn / N e j 2πmn /W x This is what MATLAB computes when ou use fft 10
11 DFT Basis Functions DFT : G[m] = n= 0 g[ n] e j 2πmn / N Basis Functions are therefore : b m [n] = e j 2πmn / N Are these orthonormal?? Inverse DFT : g[n] = 1 N m= 0 G[ m] e j 2πmn / N DFT : G[r,s] = m= 0n= 0 g[ m,n] Basis Functions are therefore : b r,s [m,n] = e j 2π ( rm +sn / N Are these orthonormal?? Inverse DFT : g[m,n] = 1 N 2 2D DFT ( / N j 2π rm+sn e r= 0 s= 0 G[ r,s] e j 2π ( rm +sn / N In general, the number of points along each dimension need not be the same (e.g. N 1 N 2. How does this change the expressions? 11
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