Bioengineering 280A Principles of Biomedical Imaging Fall Quarter 2006 CT/Fourier Lecture 2 Topics Modulation Modulation Transfer Function Convolution/Multiplication Revisit Projection-Slice Theorem Filtered Backprojection 1
Modulation F g(x)e j 2"k 0 [ x ] = G(k x ) #$(k x % k 0 ) = G( k x % k 0 ) F[ g(x)cos( 2"k 0 x) ] = 1 2 G ( k % k x 0) + 1 2 G ( k + k x 0) F[ g(x)sin( 2"k 0 x) ] = 1 2 j G ( k % k x 0) % 1 2 j G ( k + k x 0) Example Amplitude Modulation (e.g. AM Radio) g(t) 2g(t) cos(2πf 0 t) 2cos(2πf 0 t) G(f) -f 0 f 0 G(f-f 0 )+ G(f+f 0 ) 2
Modulation Example x = * = Bushberg et al 2001 3
Modulation Transfer Function (MTF) or Frequency Response Bushberg et al 2001 Modulation Transfer Function Bushberg et al 2001 4
Eigenfunctions The fundamental nature of the convolution theorem may be better understood by observing that the complex exponentials are eigenfunctions of the convolution operator. e j 2"k xx g(x) z(x) z(x) = g(x) "e j 2#k x x & % = g(u) e j 2#k x (x$u) du $% = G(k x )e j 2#k xx The response of a linear shift invariant system to a complex exponential is simply the exponential multiplied by the FT of the system s impulse response. MTF = Fourier Transform of PSF Bushberg et al 2001 5
Convolution/Multiplication Now consider an arbitrary input h(x). h(x) g(x) z(x) Recall that we can express h(x) as the integral of weighted complex exponentials. # h(x) = $ H(k x )e j 2%kxx dk x "# Each of these exponentials is weighted by G(k x ) so that the response may be written as z(x) = # $ G(k x )H(k x )e j 2%kxx dk x "# Convolution/Modulation Theorem { } = g(u) " h(x # u)du F g(x) " h(x) $ $ #$[ %#$ ] e# j 2&k x x % dx $ $ = g(u) % h(x # u) e # j 2&k xx % dxdu #$ #$ $ = g(u)h(k x )e # j 2&k x % u du #$ = G(k x )H(k x ) Convolution in the spatial domain transforms into multiplication in the frequency domain. Dual is modulation F{ g(x)h(x) } = G( k x ) " H(k x ) 6
2D Convolution/Multiplication Convolution [ ] = G(k x,k y )H(k x,k y ) F g(x, y) ""h(x, y) Multiplication [ ] = G(k x,k y ) ""H(k x,k y ) F g(x, y)h(x,y) Application of Convolution Thm. $ 1# x x <1 "(x) = % & 0 otherwise F("(x)) = 1 ' ( 1# x ) e # j 2(kxx dx =?? #1-1 1 7
Application of Convolution Thm. "(x) = #(x) $ #(x) F("(x)) = sinc 2 ( ) k x -1 1 = * Convolution Example 8
Response of an Imaging System g(x,y) G(k x,k y ) h 1 (x,y) h 2 (x,y) h 3 (x,y) MODULE 1 MODULE 2 MODULE 3 H 1 (k x,k y ) H 2 (k x,k y ) H 3 (k x,k y ) z(x,y) Z(k x,k y ) z(x,y)=g(x,y)**h 1 (x,y)**h 2 (x,y)**h 3 (x,y) Z(k x,k y )=G(k x,k y ) H 1 (k x,k y ) H 2 (k x,k y ) H 3 (k x,k y ) System MTF = Product of MTFs of Components Bushberg et al 2001 9
Useful Approximation FWHM System = FWHM 1 2 + FWHM 2 2 +LFWHM N 2 Example FWHM 1 =1mm FWHM 2 = 2mm FWHM system = 5 = 2.24mm 10
Projection Theorem $ $ U(k x,0) = µ(x, y)e " j 2# (k x x +k y % % y) dxdy "$ "$ "$ "$[ %"$ ] = % µ(x, y)dy e " j 2#kxx dx "$ = % g( x,0) e " j 2#k x x dx "$ "$ = % g(l,0) e " j 2#kl dl "$ g(l,0) l In-Class Example: µ(x, y) = cos2"x Suetens 2002 Projection Theorem $ $ U(k x,k y ) = µ(x, y)e " j 2# (k xx +k y y) % % dxdy "$ "$ [ ] = F 2D µ(x, y) U(k x,k y ) = G(k,") k x = k cos" k y = k sin" g(l,") F l k = k x 2 + k y 2 G(k,") = & % #% g(l,")e # j 2$kl dl Suetens 2002 11
G(",#) = Projection Slice Theorem ' & $& g(l,#)e $ j 2%"l dl & & & = ' ' ' f (x,y)((x cos# + y sin# $ l)e $ j 2%"l dx dy dl ' $& & $& $& = f (x, y) e $& ' & $& $ j 2%" x cos# +y sin# [ ] u= " cos#,v= " sin# = F 2D f (x,y) ( ) dx dy Prince&Links 2006 Fourier Reconstruction F Interpolate onto Cartesian grid then take inverse transform Suetens 2002 12
Polar Version of Inverse FT µ(x, y) = # # $ $ G(k x,k y )e j 2% (k xx +k y y ) dk x dk y "# "# 2% # = $ $ G(k,& )e j 2% (xk cos& +yk sin& ) kdkd& 0 0 % # = $ $ G(k,& )e j 2%k(x cos& +y sin& ) k dkd& 0 "# Note : g(l," + #) = g($l,") So G(k," + #) = G($k,") Suetens 2002 µ(x, y) = Filtered Backprojection & $ % % G(k," )e j 2& (xk cos" +yk sin" ) k dkd" 0 #$ & $ = % % k G(k," )e j 2&kl dkd" % 0 & #$ = g ' (l,")d" 0 where l = x cos" + y sin" g " (l,#) = & % $% k G(k,# )e j 2'kl dk = g(l,#) " F $1 k = g(l,#) "q(l) [ ] Backproject a filtered projection Suetens 2002 13
Fourier Interpretation Density " N circumference " Low frequencies are oversampled. So to compensate for this, multiply the k-space data by k before inverse transforming. N 2# k Kak and Slaney; Suetens 2002 Ram-Lak Filter k max =1/Δs Suetens 2002 14
Projection Reconstruction Path F x F -1 Filtered Projection Back- Project Suetens 2002 Projection Reconstruction Path * Filtered Projection Back- Project Suetens 2002 15
Example Kak and Slaney Example Prince and Links 2005 16
Example Prince and Links 2005 17