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 Backprojection Modulation F g(xe j 2"k 0x [ ] = G(k x #$(k x % k 0 = G( k x % k 0 F[ g(xcos( 2"k 0 x ] = 1 2 G ( k % k x 0 + 1 2 G ( k + k x 0 Amplitude Modulation (e.g. AM Radio g(t 2g(t cos(2πf 0 t 2cos(2πf 0 t F[ g(xsin( 2"k 0 x ] = 1 2 j G ( k x % k 0 % 1 2 j G ( k x + k 0 G(f -f 0 f 0 G(f-f 0 + G(f+f 0 1
Modulation x = * = Modulation Transfer Function (MTF or Frequency Response Modulation Transfer Function 2
Eigenfunctions The fundamental nature of the convolution theorem may be better understood by observing that the complex exponentials are eigenfunctions of the convolution operator. MTF = Fourier Transform of PSF e j 2"k x x g(x z(x z(x = g(x " e j 2#k xx % = g(u e j 2#k x (x$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. 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%k x x 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 #$[ #$ ] e# j 2k xx $ $ F{ g(x " h(x } = % % g(u " h(x # udu dx $ $ = g(u % h(x # u e # j 2k xx % dxdu #$ #$ $ = g(uh(k x e # j 2k xu % du #$ = G(k x H(k x 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 3
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 Application of Convolution Thm. $ 1# x x <1 "(x = % 0 otherwise 1 F("(x = ' 1# x e # j 2(kxx dx =?? #1 ( -1 1 Application of Convolution Thm. Convolution "(x = #(x $#(x F("(x = sinc 2 ( k x -1 1 = * 4
Response of an Imaging System System MTF = Product of MTFs of Components 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 Useful Approximation FWHM System = FWHM 1 2 + FWHM 2 2 +LFWHM N 2 FWHM 1 =1mm FWHM 2 = 2mm FWHM system = 5 = 2.24mm 5
Projection Theorem Projection Theorem $ $ U(k x,0 = µ(x, ye " j 2# (k x x +k y % % y dxdy [ % ] = % µ(x, ydy e " j 2#k x x dx $ $ U(k x = µ(x, ye " j 2# (k x x +k y % % y dxdy [ ] = F 2D µ(x, y U(k x = G(k," g(l,0 l = % g(x,0 e " j 2#kxx dx = % g(l,0 e " j 2#kl dl In-Class : µ(x, y = cos2"x g(l," F l k x = k cos" k y = k sin" k = k 2 2 x + k y G(k," = % #% g(l,"e # j 2$kl dl G(",# = Projection Slice Theorem ' $ g(l,#e $ j 2%"l dl = ' ' ' f (x,y((x cos# + y sin# $ le $ 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 Fourier Reconstruction F Interpolate onto Cartesian grid then take inverse transform PrinceLinks 2006 6
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," Filtered Backprojection $ µ(x, y = % % G(k," e j 2 (xk cos" +yk sin" k dkd" 0 #$ $ = % % k G(k," e j 2kl 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 Fourier Interpretation Ram-Lak Filter Density " N circumference " N 2# k Low frequencies are oversampled. So to compensate for this, multiply the k-space data by k before inverse transforming. k max =1/Δs Kak and Slaney; 7
Projection Reconstruction Path Projection Reconstruction Path * Filtered Projection F Back- Project x F -1 Filtered Projection Back- Project Kak and Slaney Prince and Links 2005 8
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