Convolution Theorem Modulation Transfer Function y(t)=g(t)*h(t)
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1 MTF = Fourier Transform of PSF Bioengineering 28A Principles of Biomedical Imaging Fall Quarter 214 CT/Fourier Lecture 4 Bushberg et al 21 Convolution Theorem Modulation Transfer Function y(t=g(t*h(t g(t + = + h(t + + H(f Y(f=G(fH(f G(f f Bushberg et al 21 f 1
2 Convolution/Modulation Theorem [ ] e j 2πk xx F{ g(x h(x } = g(u h(x udu = g(u h(x u e j 2πk xx = g(uh(k x e j 2πk xu du = G(k x H(k x dxdu dx 2D Convolution/Multiplication Convolution [ ] = G(k x H(k x F g(x, y h(x,y Multiplication [ ] = G(k x H(k x F g(x, yh(x, y Convolution in the spatial domain transforms into multiplication in the frequency domain. Dual is modulation F{ g(xh(x } = G( k x H(k x Application of Convolution Thm. $ 1 x x <1 Λ(x = % & otherwise 1 F(Λ(x = 1 x e j 2πkxx dx =?? 1 ( Application of Convolution Thm. Λ(x = Π(x Π(x F(Λ(x = sinc 2 ( k x = * 2
3 Convolution Response of an Imaging System g(x,y G(k x h 1 (x,y h 2 (x,y h 3 (x,y MODULE 1 MODULE 2 MODULE 3 H 1 (k x H 2 (k x H 3 (k x z(x,y Z(k x z(x,y=g(x,y**h 1 (x,y**h 2 (x,y**h 3 (x,y Z(k x =G(k x H 1 (k x H 2 (k x H 3 (k x System MTF = Product of MTFs of Components Useful Approximation FWHM System = FWHM 12 + FWHM 22 + FWHM N 2 FWHM 1 =1mm FWHM 2 = 2mm FWHM system = 5 = 2.24mm Bushberg et al 21 PollEv.com/be28a 3
![Modulation F g(xe j 2πk x [ ] = G(k x δ(k x k = G( k x k F[ g(xcos( 2πk x ] = 1 2 G ( k k x + 1 2 G ( k + k x F[ g(xsin( 2πk x ] = 1 2 j G ( k k](/docs-images/77/74871888/images/4-1.jpg "x 1 2 j G ( k + k x PollEv.com/be28a Amplitude Modulation (e.g.")
4 Modulation F g(xe j 2πk x [ ] = G(k x δ(k x k = G( k x k F[ g(xcos( 2πk x ] = 1 2 G ( k k x G ( k + k x F[ g(xsin( 2πk x ] = 1 2 j G ( k k x 1 2 j G ( k + k x PollEv.com/be28a Amplitude Modulation (e.g. AM Radio g(t 2g(t cos(2πf t Modulation x = 2cos(2πf t G(f -f f G(f-f + G(f+f * = 4
![Summary of Basic Properties Linearity F [ ag(x, y+ bh(x, y ] = ag(k x + bh(k x Scaling F [ g(ax,by ] = 1 ab G!](/docs-images/77/74871888/images/5-1.jpg "k x a, k $ x # & \" b % Duality F{ G(x } = g( k x Shift F [ g(x a, y b ] = G(k x e j2π (k xa+k yb Convolution F [ g(x, y**h(x, y ] = G(k x H(k x Multiplication F [ g(x, yh(x, y ] = G(k x **H(k x")
5 Summary of Basic Properties Linearity F [ ag(x, y+ bh(x, y ] = ag(k x + bh(k x Scaling F [ g(ax,by ] = 1 ab G! k x a, k $ x # & " b % Duality F{ G(x } = g( k x Shift F [ g(x a, y b ] = G(k x e j2π (k xa+k yb Convolution F [ g(x, y**h(x, y ] = G(k x H(k x Multiplication F [ g(x, yh(x, y ] = G(k x **H(k x Modulation F( g(x, ye j2π (xa+yb * + = G(k x a b In-class Exercise µ(x, y = rect(x, ycos( 2π (x + y Sketch the object Calculate and sketch the Fourier Transform of the Object G(ρ,θ = Projection-Slice Theorem g(l,θe j 2πρl dl = f (x,y δ(x cosθ + y sinθ le j 2πρl dx dy dl = f (x,y e j 2πρ x cosθ +y sinθ [ ] u= ρ cosθ,v= ρ sinθ = F 2D f (x,y ( dx dy g(x, y = Π(xΠ(y G(k x = sinc(k x sinc(k y y 1/2 (sinc/rect -1/2 1/2 x -1/2 Prince&Links 26 5
6 Projection-Slice Theorem Projection-Slice Theorem g(l, l l U(k x, = µ(x, ye j 2π (k x x +k y y dxdy [ ] = µ(x, ydy e j 2πk x x dx = g(x, e j 2πk x x dx = g(l, e j 2πkl dl In-Class : µ(x, y = cos2πx l g(l,θ [ ] U(k x = µ(x, ye j 2π (k x x +k y y dxdy = F 2D µ(x, y U(k x = G(k,θ k x = k cosθ k y = k sinθ k = k 2 2 x + k y F G(k,θ = l g(l,θe j 2πkl dl G(ρ,θ = Projection-Slice Theorem g(l,θe j 2πρl dl = f (x,y δ(x cosθ + y sinθ le j 2πρl dx dy dl = f (x,y e j 2πρ x cosθ +y sinθ [ ] u= ρ cosθ,v= ρ sinθ = F 2D f (x,y ( dx dy In-class Exercise µ(x, y = rect(x, ycos( 2π (x + y Sketch this object. What are the projections at theta = and 9 degrees? For what angle is the projection maximized? PollEv.com/be28a Prince&Links 26 6
7 Fourier Reconstruction Polar Version of Inverse FT F Interpolate onto Cartesian grid then take inverse transform µ(x, y = 2π G(k x e j 2π (k xx +k y y dk x dk y = G(k,θe j 2π (xk cosθ +yk sinθ kdkdθ π = G(k,θe j 2πk(x cosθ +y sinθ k dkdθ Note : g(l,θ + π = g( l,θ So G(k,θ + π = G( k,θ µ(x, y = Filtered Backprojection π π G(k,θe j 2π (xk cosθ +yk sinθ k dkdθ = k G(k,θe j 2πkl dkdθ π = g (l,θdθ where l = x cosθ + y sinθ g (l,θ = k G(k,θe j 2πkl dk = g(l,θ F 1 k = g(l,θ q(l [ ] Backproject a filtered projection Fourier Interpretation Density N circumference N 2π k Low frequencies are oversampled. So to compensate for this, multiply the k-space data by k before inverse transforming. Kak and Slaney; 7
8 Ram-Lak Filter Projection Reconstruction Path F k max =1/Δs x F -1 Filtered Projection Back- Project Projection Reconstruction Path * Filtered Projection Back- Project Kak and Slaney 8
9 Prince and Links 25 Prince and Links 25 9
Convolution/Modulation Theorem. 2D Convolution/Multiplication. Application of Convolution Thm. Bioengineering 280A Principles of Biomedical Imaging
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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
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Topics Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2008 CT/Fourier Lecture 3 Modulation Modulation Transfer Function Convolution/Multiplication Revisit Projection-Slice Theorem Filtered
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