Convolution/Modulation Theorem. 2D Convolution/Multiplication. Application of Convolution Thm. Bioengineering 280A Principles of Biomedical Imaging

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1 Bioengineering 8A Principles of Biomedical Imaging Fall Quarter 15 CT/Fourier Lecture 4 Convolution/Modulation Theorem [ ] e j πx F{ g(x h(x } = g(u h(x udu = g(u h(x u e j πx = g(uh( e j πu du = H( dxdu Convolution in the spatial domain transforms into multiplication in the frequency domain. Dual is modulation F{ g(xh(x } = G( H( dx D Convolution/Multiplication Convolution [ ] = H( F g(x, y h(x, y Multiplication [ ] = H( F g(x, yh(x, y Application of Convolution Thm. $ 1 x x <1 Λ(x = % & otherwise 1 F(Λ(x = 1 x e j πkxx dx =?? 1 (

2 Application of Convolution Thm. Convolution Λ(x = Π(x Π(x F(Λ(x = sinc ( -1 1 = * Response of an Imaging System System MTF = Product of MTFs of Components g(x,y h 1 (x,y h (x,y h 3 (x,y MODULE 1 MODULE MODULE 3 H 1 ( H ( H 3 ( z(x,y Z( z(x,y=g(x,y**h 1 (x,y**h (x,y**h 3 (x,y Z( = H 1 ( H ( H 3 ( Bushberg et al 1 PollEv.com/be8a

3 Useful Approximation FWHM System = FWHM 1 + FWHM + FWHM N FWHM 1 =1mm FWHM = mm FWHM system = 5 =.4mm PollEv.com/be8a Modulation s F g(xe j πk x [ ] = δ( k = G( k F[ g(xcos( πk x ] = 1 G ( k + 1 G ( k + Amplitude Modulation (e.g. AM Radio g(t g(t cos(πf t cos(πf t F[ g(xsin( πk x ] = 1 j G ( k 1 j G ( + k -f f G(f G(f-f + G(f+f 3

4 Modulation x = Shift Theorem F{ g(x a } = F{ g(x δ(x a } = G( e jπa * = { ( } F{ g(x a, y b } = F g(x, y δ x a, y b = e jπ (a+k y b Shifting the function doesn't change its spectral content, so the magnitude of the transform is unchanged. Each frequency component is shifted by a. This corresponds to a relative phase shift of - πa /(spatial period = - πa For example, consider exp( jπ x. Shifting this by a yields exp( jπ (x a = exp( jπ xexp( jπa Summary of Basic Properties Linearity F [ ag(x, y+ bh(x, y ] = a + bh( Scaling F [ g(ax,by ] = 1 ab G! a, k $ x # & " b % Duality F{ G(x } = g( Shift F [ g(x a, y b ] = e jπ (a+k y b Convolution F [ g(x, y**h(x, y ] = H( Multiplication F [ g(x, yh(x, y ] = **H( Modulation F( g(x, ye jπ (xa+yb * + = G(k a,k b x y g(l,θ Projection-Slice Theorem µ ( x, y U ( G(k,θ = k y g(l,θe j πkl dl k = k cosθ k y = k sinθ k = + k y U( = G(k,θ Modified from Prince&Links 6 4

5 Projection-Slice Theorem Projection-Slice Theorem G(k,θ = g(l,θe jπkl dl = µ(x, y δ(x cosθ + ysinθ le jπkl dx dy dl = µ(x, y e = µ(x, y = F D µ(x, y ( dx dy jπ cosθ+ysinθ ( dx dy e jπ x+k y y [ ] kx =k cosθ =ksinθ g(l, l U(, = µ(x, ye j π (x +k y y dxdy [ ] = µ(x, ydy e j π x dx = g(x, e j πkxx dx = g(l, e j πkl dl l Suetens g(x, y = Π(xΠ(y = sinc( sinc(k y (sinc/rect g(x, y = Π(xΠ(y = sinc( sinc(k y (sinc/rect y 1/ -1/ 1/ -1/ x Projection at θ = ; g(l, = rect(l F(g(l, = sinc(k = k cosθ = k; k y = k sinθ = = sinc(ksinc( = sinc(k Projection at θ = 9; g(l,9 = rect(l F(g(l,9 = sinc(k = k cosθ = ; k y = k sinθ = k = sinc(sinc(k = sinc(k 5

6 g(x, y = Π(xΠ(y = sinc( sinc(k y (sinc/rect g(x, y = Π(xΠ(y = sinc( sinc(k y (sinc/rect Projection at θ = 45; # l & # g(l,45 = Λ% ( = rect% $ / ' $ F(g(l,45 = sinc # k & % ( $ ' = k cosθ = k ; k y = k sinθ = k # & # & # = sinc% k (sinc% k ( = sinc % k $ ' $ ' $ l & # l & (*rect% ( / ' $ / ' sinc # k & # % ( = sinc % k $ ' $ & ( ' / / & ( ' Projection at <θ < 9 = k cosθ; k y = k sinθ = sinc( k cosθsinc( k sinθ ( ( g(l,θ = F 1 sinc( k cosθsinc( k sinθ = F 1 ( sinc( k cosθ * F 1 sinc( k sinθ = 1 cosθ rect # l & % (* 1 $ cosθ ' sinθ rect # l & % ( $ sinθ ' 1 = cosθ sinθ rect # l & # l & % (*rect% ( $ cosθ ' $ sinθ ' In-class Exercise µ(x, y = rect(x, ycos( π (x + y In-class Exercise µ(x, y = rect(x, ycos( π (x + y Sketch the object Calculate and sketch the Fourier Transform of the Object Sketch this object. What are the projections at theta = and 9 degrees? For what angle is the projection maximized? PollEv.com/be8a 6

7 Fourier Reconstruction Polar Version of Inverse FT F Interpolate onto Cartesian grid then take inverse transform µ(x, y = π e j π ( x +k y y d dk y = G(k,θe j π (xk cosθ +yk sinθ kdkdθ π = G(k,θe j πk(x cosθ +y sinθ k dkdθ Note : g(l,θ + π = g( l,θ So G(k,θ + π = G( k,θ Suetens Suetens µ(x, y = Filtered Backprojection π G(k,θe j π (xk cosθ +yk sinθ k dkdθ = k G(k,θe j πkl dkdθ π π = g (l,θdθ where l = x cosθ + y sinθ g (l,θ = k G(k,θe j πkl dk = g(l,θ F 1 k = g(l,θ q(l [ ] Backproject a filtered projection Fourier Interpretation Density N circumference N π k Low frequencies are oversampled. So to compensate for this, multiply the k-space data by k before inverse transforming. Suetens Kak and Slaney; Suetens 7

8 Ram-Lak Filter Projection Reconstruction Path F k max =1/Δs x F -1 Filtered Projection Back- Project Suetens Suetens Projection Reconstruction Path * Filtered Projection Back- Project Suetens Kak and Slaney 8

9 Prince and Links 5 Prince and Links 5 9

10 Filters Filtered Projections (a (b Which image used the Hanning filter? PollEv.com/be8a 1