1!Bioengineering 280A"Principles of Biomedical Imaging""Fall Quarter 2010"MRI Lecture 3"!Thomas Liu, BE280A, UCSD, Fall 2008!Sampling in k-space!Aliasing!Aliasing!∆Bz(x)=Gxx!Faster!Slower!2!Intuitive view of Aliasing!! kx= 1/ FOVFOV!! kx= 2 / FOVFourier Sampling!Instead of sampling the signal, we sample its Fourier Transform!???!Sample!F-1!F!Fourier Sampling!! GS(kx) = G(kx)1"kxcombkx"kx# $ % & ' ( = G(kx))(n=*++,kx* n"kx)= G(n"kx))(n=*++,kx* n"kx)kx!(1 /Δkx) comb(kx /Δkx)!Δkx!Fourier Sampling -- Inverse Transform!Χ!=!*!1/Δkx!Δkx!=!3!Fourier Sampling -- Inverse Transform!! gS(x) = F"1GS(kx)[ ]= F"1G(kx)1#kxcombkx#kx$ % & ' ( ) * + , - . / = F"1G(kx)[ ]0 F"11#kxcombkx#kx$ % & ' ( ) * + , - . / = g(x) 0 comb x#kx( )= g(x) 01(x#kxn="223" n)= g(x) 01#kx1(xn="223"n#kx)=1#kxg(xn="223"n#kx)Nyquist Condition!FOV (Field of View)!1/Δkx!To avoid overlap, 1/Δkx> FOV, or equivalently, Δkx<1/FOV!Aliasing!FOV (Field of View)!1/Δkx!Aliasing occurs when 1/Δkx< FOV!Aliasing Example!Δkx=1!1/Δkx=1!4!2D Comb Function!! comb(x, y) ="(x # m, y # n)n=#$$%m=#$$%="(x # m )"(y # n)n=#$$%m=#$$%= comb(x)comb(y)Scaled 2D Comb Function!! comb(x /"x, y /"y) = comb(x /"x)comb(y /"y)= "x"y#(x $ m"x)#(y $ n"y)n=$%%&m=$%%&Δx!Δy!X! =!*!=!1/∆k!2D k-space sampling!! GS(kx,ky) = G (kx,ky)1"kx"kycombkx"kx,ky"ky# $ % % & ' ( ( = G(kx,ky))(n= *++,kx* m"kx,ky* n"ky)m=*++,= G(m"kx,n"ky))(n= *++,kx* m"kx,ky* n"ky)m=*++,5!2D k-space sampling!! gS(x, y) = F"1GS(kx,ky)[ ]= F"1G(kx,ky)1#kx#kycombkx#kx,ky#ky$ % & & ' ( ) ) * + , , - . / / = F"1G(kx,ky)[ ]0 F"11#kx#kycombkx#kx,ky#ky$ % & & ' ( ) ) * + , , - . / / = g(x, y) 00comb x#kx( )comb y#ky( )= g(x) 001(x#kxn="223" m)m="2231(y#ky" n)= g(x) 001#kx#ky1(xn="223"m#kx)1(y "n#ky)m="223=1#kx#kyg(xn="223"m#kx, y "n#ky)m="223Nyquist Conditions!FOVX!FOVY!1/∆kX!1/∆kY!1/∆kY> FOVY!1/∆kX> FOVX!!Windowing!! GWkx,ky( )= G kx,ky( )W kx,ky( )! gWx, y( )= g x, y( )" w(x, y)Windowing the data in Fourier space !Results in convolution of the object with the inverse transform of the window!Resolution!6!Windowing Example!! W kx,ky( )= rectkxWkx" # $ $ % & ' ' rectkyWky" # $ $ % & ' ' ! w x, y( )= F"1rectkxWkx# $ % % & ' ( ( rectkyWky# $ % % & ' ( ( ) * + + , - . . = WkxWkysinc Wkxx( )sinc Wkyy( )! gWx, y( )= g(x, y) ""WkxWkysinc Wkxx( )sinc Wkyy( )Effective Width!! wE=1w(0)w(x)dx"##$wE!! wE=11sinc(Wkxx)dx"##$= F sinc(Wkxx)[ ]kx= 0=1WkxrectkxWkx% & ' ' ( ) * * kx= 0=1WkxExample!! 1Wkx! "1WkxResolution and spatial frequency!! 2Wkx! With a window of width Wkx the highest spatial frequency is Wkx/2.This corresponds to a spatial period of 2/Wkx.! 1Wkx= Effective WidthWindowing Example!! g(x, y) ="(x) +"(x #1)[ ]"(y)! gWx, y( )="(x) +"(x #1)[ ]"(y) $$WkxWkysinc Wkxx( )sinc Wkyy( )= WkxWky"(x) +"(x #1)[ ]$ sinc Wkxx( )( )sinc Wkyy( )= WkxWkysinc Wkxx( )+ sinc Wkx(x #1)( )( )sinc Wkyy( )! Wkx= 1! Wkx= 2! Wkx= 1.57!Sampling and Windowing!X! =!*!=!X!*! =!Sampling and Windowing!! GSWkx,ky( )= G kx,ky( )1"kx"kycombkx"kx,ky"ky# $ % % & ' ( ( rectkxWkx,kyWky# $ % % & ' ( ( ! gSWx, y( )= WkxWkyg x, y( )""comb(#kxx,#kyy) ""sin c(Wkxx)sinc(Wkyy)Sampling and windowing the data in Fourier space !Results in replication and convolution in object space. !Sampling in ky!kx!ky!Gx(t)!Gy(t)!RF!Δky!τy!Gyi!! "ky=#2$Gyi%y! FOVy=1"kySampling in kx!x!Low pass Filter!ADC!x!Low pass Filter!ADC!! cos"0t! sin"0tRF !Signal!One I,Q sample every Δt!M= I+jQ!I!Q!Note: In practice, there are number of ways of implementing this processing. !8!Sampling in kx!kx!ky!! "kx=#2$Gxr"t! FOVx=1"kxGx(t)!t1!ADC!Gxr!Δt!Resolution!! "x=1Wkx=12kx,max =1#2$Gxr%x! Wkx! WkyGx(t)!Gxr!! "x! "y=1Wky=12ky,max =1#2$2Gyp%yGy(t)!τy!! GypExample!! Goal :FOVx= FOVy= 25.6 cm"x="y= 0.1 cm! Readout Gradient :FOVx=1"2#Gxr$tPick $t = 32 µsecGxr=1FOVx"2#$t=125.6cm( )42.57 %106T&1s&1( )32 %10&6s( ) = 2.8675 %10&5T/cm = .28675 G/cm1 Gauss = 1 % 10&4 Teslat1!ADC!Gxr!Δt!Example!! Readout Gradient :"x=1#2$Gxr%x%x=1"x#2$Gxr=10.1cm( )4257 G&1s&1( )0.28675 G/cm( ) = 8.192 ms = Nread'twhere Nread=FOVx"x= 256Gx(t)!Gxr!! "x9!Example!! Phase - Encode Gradient :FOVy=1"2#Gyi$yPick $y= 4.096 msecGyi=1FOVy"2#$y=125.6cm( )42.57 %106T&1s&1( )4.096 %10&3s( ) = 2.2402 %10-7 T/cm = .00224 G/cmτy!Gyi!Example!! Phase - Encode Gradient :"y=1#2$2Gyp%yGyp=1"y2#2$%y=10.1cm( )4257 G&1s&1( )4.096 '10-3s( ) = 0.2868 G/cm =Np2Gyiwhere Np=FOVy"y= 256Gy(t)!τy!! GypSampling!! Wkx! WkyIn practice, an even number (typically power of 2) sample is usually taken in each direction to take advantage of the Fast Fourier Transform (FFT) for reconstruction. !ky!y!FOV/4!1/FOV!4/FOV!FOV!Example!Consider the k-space trajectory shown below. ADC samples are acquired at the points shownwith ! "t = 10 µsec. The desired FOV (both x and y) is 10 cm and the desired resolution (both xand y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Labelthe waveform with the gradient amplitudes required to achieve the desired FOV and resolution.Also, make sure to label the time axis correctly.10!GE Medical Systems 2003!Gibbs Artifact!256x256 image! 256x128 image!Images from http://www.mritutor.org/mritutor/gibbs.htm!*!=!Apodization!Images from http://www.mritutor.org/mritutor/gibbs.htm!*!=!rect(kx)!h(kx )=1/2(1+cos(2πkx)!Hanning Window!sinc(x)!0.5sinc(x)+0.25sinc(x-1)!+0.25sinc(x+1)!Aliasing and Bandwidth!xLPF!ADC!ADC!! cos"0t! sin"0tRF !Signal!I!Q!LPF!x*!x!f!t!x!t!FOV! 2FOV/3!Temporal filtering in the readout direction limits the readout FOV. So there should never be aliasing in the readout direction. !11!Aliasing and Bandwidth!Slower!Faster!x!f!Lowpass filter!in the readout direction to!prevent aliasing.!readout!FOVx!B=γGxrFOVx!GE Medical Systems
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