Topics. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2004 Lecture 4 2D Fourier Transforms
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1 Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2004 Lecture 4 2D Fourier Transforms Topics 1. 2D Signal Representations 2. 2D Fourier Transform 3. Transform Pairs 4. FT Properties 1
2 2D Signal a 0 0 b a b = + c d c 0 0 d Image Decomposition 1 0 a b 0 1 a b = + c d c d g[m,n] = aδ[m,n] + bδ[m,n 1] + cδ[m 1,n] + dδ[m 1,n 1] = g[k,l] δ[m k,n l] k= 0 l= 0 k= 0 l= = c k,l b k,l [m,n] 2
3 Orthonormal Basis Functions Discrete b k,l,b k, l = b k,l [m,n] b Continuous m= n= = δ[k k,l l ] b kx,b k x, k y = b kx k, (x, yb k x, = δ(k x k x k y l [m,n] k y (x, y dxdy Are these orthonormal? Example
4 Example Are these orthonormal? 1/2 1/2 1/2 1/2 1/2 1/2-1/2-1/2 1/2-1/2 1/2-1/2 1/2-1/2-1/2 1/2 Discrete Expansion Coefficients The discrete expansion is g[m,n] = c k,l k= l= b k,l [m,n] If the basis functions b k,l [m,n] are orthonormal then c k,l = b k,l,g = = m= n= m= n= b k,l = c b k,l [m,n]g[m,n] [m,n] c k, l b k, l [m,n] k = l = k, l b k,l k = l = m= n= [m,n] b k, l [m,n] = c k, l δ[k k,l l ] k = l = = c k,l 4
5 Continuous Expansion Coefficients The continuous expansion is g(x, y = ( c k x b kx (x,ydk x dk y If the basis functions b kx (x,y are orthonormal then ( = b kx,g = b kx c k x (x,yg(x, ydxdy ( = b kx (x, y c k x, k b y k x, k y (x,yd k x d k y dxdy ( k y k x, k y = c k x, ( ( b kx (x, yb k x, k y (x, ydxdyd k x d k y = c δ(k x k x k y d k x d k y = c k x Separable Basis Functions Discrete b k,l [m,n] = b k [m]b l [n] e.g. δ[m k,n l] = δ[m k]δ[n l] Continuous b kx (x, y = b kx (xb ky (y e.g. δ( x x i,y y i = δ( x x i δ y y i ( 5
6 b 1 [m] = 1 0 Separable Basis Functions b 1 [n] = [1 0] b 2 [n] = [0 1] 1 0 b 1 [m] = b 1 [n] = [1 0] b 2 [m] = 0 1 b 2 [n] = [0 1] 0 0 b 2 [m] = Separable Basis Functions b 1 [n] = [1 1]/ 2 b 2 [n] = [1 1]/ 2 b 1 [m] = 1 1 1/2 1/2 1/2-1/2 2 1 b 1 [m] = 1 1 1/2 1/ /2-1/2 b 1 [n] = [1 1]/ 2 1/2 1/2 b 1 [m] = /2-1/2 b 1 [m] = b 2 [n] = [1 1]/ 2 1/2-1/2-1/2 1/2 b k,l [m,n] = b k [m] b l [n] b k [m] = exp( πmk / 2 b l [n] = exp( πnl / 2 6
7 Example x = Basis Functions Coefficients Sum Object Fourier Basis Functions Recall that for 1D the basis functions are complex exponentials b kx ( x = e j 2πk x x For 2D, we use the separable 2D functions b kx ( x, y = b kx ( xb ky y ( = e j 2πk x x e j 2πk yy = e j 2π (k xx +k y y Are they orthonormal? 7
8 Plane Waves e j 2π (k x x +k y y = cos( 2π(k x x + k y y + j sin( 2π(k x x + k y y 1 k x 2 + k y 2 1/k y 1/k x cos(2πk x x cos(2πk y y cos(2πk x x +2πk y y Plane Waves A θ 1/k y D θ B 1/k x C ΔABC ~ ΔBDC AC BC = AB BD BD = AB BC AC = θ = arctan k y k x 1 1 k x k y 1 k x k y = 1 k x 2 + k y 2 8
9 Fourier Transform 2D Fourier Transform = F[ g( x, y ] = e j 2π k x x +k y y (,g = g(x, y ( dxdy e j 2π k x x +k y y Inverse Fourier Transform g(x, y = ( dk x dk y e j 2π k x x +k y y Separable Functions ( is said to be a separable function if it can be g x, y written as g( x, y = g X ( xg Y ( y The Fourier Transform is then separable as well. = g(x,y e j 2π ( k xx +k y y dxdy = g X ( xe j 2πkxx dx g Y ( y e j 2πkyy dy = G X (k x G Y (k y Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y 9
10 Example g(x, y = Π(xΠ(y = sinc(k x sinc(k y Example (sinc/rect y 1/2-1/2 1/2 x -1/2 Example (sinc/rect 10
11 11
12 12
13 g(x, y = δ(x, y = δ(xδ(y =1 g(x, y = δ(x = δ(k y!!! g(x, y =1+ e j 2πax = δ k x ( + δ(k x + aδ(k y g(x, y =1+ e j 2πay = δ(k x + δ(k x δ(k y a 13
14 g(x, y = cos(2π(ax + by = 1 2 δ(k x aδ(k y b δ(k x + aδ(k y + b = δ(k x + g(x, y =??? δ(k x + cδ(k y + δ(k x δ(k y d δ(k x aδ(k y b δ(k x + aδ(k y + b 14
15 15
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16 Linearity F ag(x, y + bh(x, y Basic Properties [ ] = a + bh(k x Scaling [ ] = 1 ab G k x F g(ax,by a, k x b Shift F[ g(x a,y b ] = e j 2π (k xa +k y b Modulation [ ] = a b j 2π (xa +yb F g(x, ye Modulation Example x = * = 16
17 Convolution/Multiplication Convolution [ ] = H(k x F g(x, y h(x,y Multiplication [ ] = H(k x F g(x, yh(x, y Multiplication in one domain translates into convolution in the other domain. Convolution Example 17
18 F[ g (x, y ] = G ( k x, k y Symmetry If g(x, y is real then g(x, y = g (x, y, so G ( k x, k y = k y k x Parseval s Relations Energy is preserved 2 2 g(x, y dxdy = G(kx dkx dk y So is the inner product g,h = G,H g (x, y h(x, ydxdy = G (k x H(k x dk x dk y 18
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![Topics. Example. Modulation. [ ] = G(k x. ] = 1 2 G ( k % k x 0) G ( k + k x 0) ] = 1 2 j G ( k x](/thumbs/95/123667623.jpg "Topics. Example. Modulation. [ ] = G(k x. ] = 1 2 G ( k % k x 0) G ( k + k x 0) ] = 1 2 j G ( k x")
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