1TT Liu, BE280A, UCSD Fall 2007Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2007CT/Fourier Lecture 3TT Liu, BE280A, UCSD Fall 2007Topics• Modulation Transfer Function• Convolution/Multiplication• Modulation• Revisit Projection-Slice Theorem• Filtered BackprojectionTT Liu, BE280A, UCSD Fall 2007! z(x) = g(x) " ej 2#kxx= g(u)$%%&ej 2#kx(x$ u)du= G(kx)ej 2#kxxg(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 20072TT Liu, BE280A, UCSD Fall 2007MTF = Fourier Transform of PSFBushberg et al 2001TT Liu, BE280A, UCSD Fall 2007Bushberg et al 2001TT Liu, BE280A, UCSD Fall 2007Modulation Transfer Function (MTF)orFrequency ResponseBushberg et al 2001TT Liu, BE280A, UCSD Fall 2007Modulation Transfer FunctionBushberg et al 20013TT Liu, BE280A, UCSD Fall 2007Convolution/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 2007Convolution/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#$$%du= G(kx)H(kx)Convolution in the spatial domain transforms intomultiplication in the frequency domain. Dual ismodulation! F g(x)h( x){ }= G kx( )" H(kx)TT Liu, BE280A, UCSD Fall 20072D Convolution/Multiplication! ConvolutionF g(x, y ) ""h(x, y)[ ]= G(kx,ky)H(kx,ky)MultiplicationF g(x, y )h(x, y)[ ]= G(kx,ky) ""H(kx,ky)TT Liu, BE280A, UCSD Fall 2007Application of Convolution Thm.! "(x) =1# x x <10 otherwise$ % & F("(x)) = 1# x( )#11'e# j 2(kxxdx = ??-1 14TT Liu, BE280A, UCSD Fall 2007Application of Convolution Thm.! "(x) = #(x) $ #(x)F ("(x)) = sinc2kx( )-1 1 *=TT Liu, BE280A, UCSD Fall 2007Convolution ExampleTT Liu, BE280A, UCSD Fall 2007Response of an Imaging SystemG(kx,ky)H1(kx,ky) H2(kx,ky) H3(kx,ky)g(x,y)h1(x,y) h2(x,y) h3(x,y)MODULE 1 MODULE 2 MODULE 3z(x,y)Z(kx,ky)Z(kx,ky)=G(kx,ky) H1(kx,ky) H2(kx,ky) H3(kx,ky)z(x,y)=g(x,y)**h1(x,y)**h2(x,y)**h3(x,y)TT Liu, BE280A, UCSD Fall 2007System MTF = Product of MTFs of ComponentsBushberg et al 20015TT Liu, BE280A, UCSD Fall 2007Useful Approximation ! FWHMSystem= FWHM12+ FWHM22+L FWHMN2ExampleFWHM1=1mmFWHM2= 2mmFWHMsystem= 5 = 2.24 mmTT Liu, BE280A, UCSD Fall 2007ModulationAmplitude Modulation (e.g. AM Radio)g(t)2cos(2πf0t)2g(t) cos(2πf0t)G(f)-f0f0G(f-f0)+ G(f+f0)TT Liu, BE280A, UCSD Fall 2007Modulation! F g(x)ej 2"k0x[ ]= G(kx) #$(kx% k0) = G kx% k0( )F g(x)cos 2"k0x( )[ ]=12G kx% k0( )+12G kx+ k0( )F g(x)sin 2"k0x( )[ ]=12 jG kx% k0( )%12 jG kx+ k0( )TT Liu, BE280A, UCSD Fall 2007Modulation Examplex=*=6TT Liu, BE280A, UCSD Fall 2007Projection TheoremSuetens 2002! U(kx,0) =µ(x, y)e" j 2#(kxx +kyy)"$$%"$$%dxdy=µ(x, y)dy"$"$%[ ]"$"$%e" j 2#kxxdx= g(x,0)"$"$%e" j 2#kxxdx= g(l,0)"$"$%e" j 2#kldl! g(l,0)In-Class Example:! µ(x, y) = cos2"xlTT Liu, BE280A, UCSD Fall 2007Projection TheoremSuetens 2002! U(kx,ky) =µ(x, y)e" j 2#(kxx +kyy)"$$%"$$%dxdy= F2Dµ(x, y)[ ]! G(k,") = g(l,")e# j 2$kl#%%&dlF! U(kx,ky) = G(k,")! kx= k cos"ky= k sin"k = kx2+ ky2! g(l,")lTT Liu, BE280A, UCSD Fall 2007Projection Slice TheoremPrince&Links 2006! G(",#) = g(l,#)e$ j 2%"l$&&'dl= f (x, y)$&&'$&&'((x cos#+ y sin#$ l)e$ j 2%"ldx dy$&&' dl= f (x, y)$&&'$&&'e$ j 2%"x cos#+y sin#( )dx dy= F2Df (x, y)[ ]u="cos#,v="sin#TT Liu, BE280A, UCSD Fall 2007Fourier ReconstructionSuetens 2002FInterpolate onto Cartesiangrid then take inversetransform7TT Liu, BE280A, UCSD Fall 2007Polar Version of Inverse FTSuetens 2002! µ(x, y) = G(kx,ky"##$"##$)ej 2%(kxx +kyy )dkxdky= G(k,&0#$02%$)ej 2%(xk cos&+yk sin&)kdkd&= G(k,&"##$0%$)ej 2%k( x cos&+y sin&)k dkd&! Note :g(l,"+#) = g($l,")SoG(k,"+#) = G ($k,")TT Liu, BE280A, UCSD Fall 2007Filtered BackprojectionSuetens 2002! µ(x, y) = G(k,"#$$%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 projectionTT Liu, BE280A, UCSD Fall 2007Fourier 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 2007Ram-Lak FilterSuetens 2002kmax=1/Δs8TT Liu, BE280A, UCSD Fall 2007Reconstruction PathSuetens 2002FxF-1ProjectionFilteredProjectionBack-ProjectTT Liu, BE280A, UCSD Fall 2007Reconstruction PathSuetens 2002ProjectionFilteredProjectionBack-Project*TT Liu, BE280A, UCSD Fall 2007ExampleKak and SlaneyTT Liu, BE280A, UCSD Fall 2007ExamplePrince and Links 20059TT Liu, BE280A, UCSD Fall 2007ExamplePrince and Links
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