1TT Liu, BE280A, UCSD Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006CT/Fourier Lecture 2TT Liu, BE280A, UCSD Fall 2006Topics• Modulation• Modulation Transfer Function• Convolution/Multiplication• Revisit Projection-Slice Theorem• Filtered Backprojection2TT Liu, BE280A, UCSD Fall 2006Modulation! F g(x)ej 2"k0x[ ]= G(kx) #$(kx% k0) = G kx% k0( )F g(x)cos 2"k0x( )[ ]=12G kx% k0( )+12G kx+ k0( )F g(x)sin 2"k0x( )[ ]=12 jG kx% k0( )%12 jG kx+ k0( )TT Liu, BE280A, UCSD Fall 2006ExampleAmplitude Modulation (e.g. AM Radio)g(t)2cos(2πf0t)2g(t) cos(2πf0t)G(f)-f0f0G(f-f0)+ G(f+f0)3TT Liu, BE280A, UCSD Fall 2006Modulation Examplex=*=TT Liu, BE280A, UCSD Fall 2006Bushberg et al 20014TT Liu, BE280A, UCSD Fall 2006Modulation Transfer Function (MTF)orFrequency ResponseBushberg et al 2001TT Liu, BE280A, UCSD Fall 2006Modulation Transfer FunctionBushberg et al 20015TT Liu, BE280A, UCSD Fall 2006Eigenfunctions! z(x) = g(x) " ej 2#kxx= g(u)$%%&ej 2#kx(x$u)du= G(kx)ej 2#kxxThe fundamental nature of the convolution theorem may bebetter understood by observing that the complex exponentialsare eigenfunctions of the convolution operator.g(x) z(x)! ej 2"kxxThe response of a linear shift invariant system to a complexexponential is simply the exponential multiplied by the FT ofthe system’s impulse response.TT Liu, BE280A, UCSD Fall 2006MTF = Fourier Transform of PSFBushberg et al 20016TT Liu, BE280A, UCSD Fall 2006Convolution/Multiplication! h(x) = H(kx"##$)ej 2%kxxdkxNow consider an arbitrary input h(x).h(x) g(x) z(x)Recall that we can express h(x) as the integral of weightedcomplex exponentials.Each of these exponentials is weighted by G(kx) so that theresponse may be written as! z(x) = G(kx)H(kx"##$)ej 2%kxxdkxTT Liu, BE280A, UCSD Fall 2006Convolution/ModulationTheorem! F g(x) " h(x){ }= g(u) " h(x # u)du#$$%[ ]e# j 2&kxx#$$%dx= g(u) h(x # u)#$$%e# j 2&kxx#$$%dxdu= g(u)H(kx)e# j 2&kxu#$$%du= G(kx)H(kx)Convolution in the spatial domain transforms intomultiplication in the frequency domain. Dual ismodulation! F g(x)h(x){ }= G kx( )" H(kx)7TT Liu, BE280A, UCSD Fall 20062D Convolution/Multiplication! ConvolutionF g(x, y) ""h(x, y)[ ]= G(kx,ky)H(kx,ky)MultiplicationF g(x, y)h(x , y)[ ]= G(kx,ky) ""H(kx,ky)TT Liu, BE280A, UCSD Fall 2006Application of Convolution Thm.! "(x) =1# x x <10 otherwise$ % & F("(x)) = 1# x( )#11'e# j 2(kxxdx = ??-1 18TT Liu, BE280A, UCSD Fall 2006Application of Convolution Thm.! "(x) = #(x) $ #(x)F ("(x)) = sinc2kx( )-1 1 *=TT Liu, BE280A, UCSD Fall 2006Convolution Example9TT Liu, BE280A, UCSD Fall 2006Response of an Imaging SystemG(kx,ky)H1(kx,ky) H2(kx,ky) H3(kx,ky)g(x,y)h1(x,y) h2(x,y) h3(x,y)MODULE 1 MODULE 2 MODULE 3z(x,y)Z(kx,ky)Z(kx,ky)=G(kx,ky) H1(kx,ky) H2(kx,ky) H3(kx,ky)z(x,y)=g(x,y)**h1(x,y)**h2(x,y)**h3(x,y)TT Liu, BE280A, UCSD Fall 2006System MTF = Product of MTFs of ComponentsBushberg et al 200110TT Liu, BE280A, UCSD Fall 2006Useful Approximation ! FWHMSystem= FWHM12+ FWHM22+L FWHMN2ExampleFWHM1=1mmFWHM2= 2mmFWHMsystem= 5 = 2.24 mmTT Liu, BE280A, UCSD Fall 200611TT Liu, BE280A, UCSD Fall 2006Projection TheoremSuetens 2002! U(kx,0) =µ(x, y)e" j 2#(kxx + kyy)"$$%"$$%dxdy=µ(x, y)dy"$"$%[ ]"$"$%e" j 2#kxxdx= g(x,0)"$"$%e" j 2#kxxdx= g(l,0)"$"$%e" j 2#kldl! g(l,0)In-Class Example:! µ(x, y) = cos2"xlTT Liu, BE280A, UCSD Fall 2006Projection TheoremSuetens 2002! U(kx,ky) =µ(x, y)e" j 2#(kxx + kyy)"$$%"$$%dxdy= F2Dµ(x, y)[ ]! G(k,") = g(l,")e# j 2$kl#%%&dlF! U(kx,ky) = G(k,")! kx= k cos"ky= k sin"k = kx2+ ky2! g(l,")l12TT Liu, BE280A, UCSD Fall 2006Projection Slice TheoremPrince&Links 2006! G(",#) = g(l,#)e$ j 2%"l$&&'dl= f (x, y)$&&'$&&'((x cos#+ y sin#$ l)e$ j 2%"ldx dy$&&' dl= f (x, y)$&&'$&&'e$ j 2%"x cos#+y sin#( )dx dy= F2Df (x, y)[ ]u="cos#,v="sin#TT Liu, BE280A, UCSD Fall 2006Fourier ReconstructionSuetens 2002FInterpolate onto Cartesiangrid then take inversetransform13TT Liu, BE280A, UCSD Fall 2006Polar Version of Inverse FTSuetens 2002! µ(x, y) = G(kx,ky"##$"##$)ej 2%(kxx +kyy )dkxdky= G(k,&0#$02%$)ej 2%(xk cos&+yk sin&)kdkd&= G(k,&"##$0%$)ej 2%k(x cos&+y sin&)k dkd&! Note :g(l,"+#) = g($l,")SoG(k,"+#) = G($k,")TT Liu, BE280A, UCSD Fall 2006Filtered BackprojectionSuetens 2002! µ(x, y) = G(k,"#$$%0&%)ej 2&(xk cos"+yk sin")k dkd"= k G(k,"#$$%0&%)ej 2&kldkd"= g'(l,")d"0&%! g"(l,#) = k G(k,#$%%&)ej 2'kldk= g(l,#) " F$1k[ ]= g(l,#) " q(l)! where l = x cos"+ y sin"Backproject a filtered projection14TT Liu, BE280A, UCSD Fall 2006Fourier InterpretationKak and Slaney; Suetens 2002! Density "Ncircumference"N2#kLow frequencies areoversampled. So tocompensate for this,multiply the k-space databy |k| before inversetransforming.TT Liu, BE280A, UCSD Fall 2006Ram-Lak FilterSuetens 2002kmax=1/Δs15TT Liu, BE280A, UCSD Fall 2006Reconstruction PathSuetens 2002FxF-1ProjectionFilteredProjectionBack-ProjectTT Liu, BE280A, UCSD Fall 2006Reconstruction PathSuetens 2002ProjectionFilteredProjectionBack-Project*16TT Liu, BE280A, UCSD Fall 2006ExampleKak and SlaneyTT Liu, BE280A, UCSD Fall 2006ExamplePrince and Links 200517TT Liu, BE280A, UCSD Fall 2006ExamplePrince and Links
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