Sampling in k-space. Aliasing. Aliasing. Bioengineering 280A Principles of Biomedical Imaging. Fall Quarter 2010 MRI Lecture 3. Slower B z (x)=g x x
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1 Sampling in k-space Bioengineering 80A Principles of Biomedical Imaging Fall Quarer 00 MRI Lecure 3 Thomas Liu, BE80A, UCSD, Fall 008 Aliasing Aliasing Slower B z (G Faser
2 Inuiive view of Aliasing FOV Fourier Sampling F k /FOV k /FOV Insead of sampling he signal, we sample is Fourier Transform??? F - Sample Fourier Sampling Fourier Sampling -- Inverse Transform Δk ( /Δk comb(k /Δk k Χ Δk G S (k G(k comb k ( "k "k ' G(k,( k n"k n, G(n"k ( k n"k n /Δk
3 Fourier Sampling -- Inverse Transform g S ( F " [ G S (k ] F " G(k comb k '-, / k k (. F " [ G(k ] 0 F " comb k '-, / k k (. g( 0comb( k g( 0 3(k " n n" g( 0 3( " n k k n" 3 g( " n k k n" Nquis Condiion FOV (Field of View /Δk To avoid overlap, /Δk > FOV, or equivalenl, Δk </FOV Aliasing Aliasing Eample FOV (Field of View Δk /Δk Aliasing occurs when /Δk < FOV /Δk 3
4 D Comb Funcion comb(, "( m, n m n "( m"( n m n comb(comb( Scaled D Comb Funcion comb( /", /" comb( /"comb( /" "" ( m"( n" m n Δ Δ D k-space sampling / k X G S (k,k G(k,k comb k, k "k "k "k "k ( ' G(k,k,,( k m"k,k n"k m n,, G(m"k,n"k ( k m"k,k n"k m n 4
![D k-space sampling Nquis Condiions [ ] g S (, F " G S (k,k F " G(k,k comb k, k '-, k k k k /, (./ F " [ G(k,k ] 0 F " comb k, k '-, k k k k /, (.](/docs-images/76/73408455/images/5-0.jpg "/ g(, 00comb( k comb( k g( 00 3 3(k \" m (k \" n m\" n\" g( 00 3 3( \" m ( \" n k k m\" n\" k k 3 3 g( \" m, \" n k k k k m\" n\" FOV X / k X / k Y FOV Y / k Y > FOV Y / k X > FOV X Windowing Resoluion Windowing")
5 D k-space sampling Nquis Condiions [ ] g S (, F " G S (k,k F " G(k,k comb k, k '-, k k k k /, (./ F " [ G(k,k ] 0 F " comb k, k '-, k k k k /, (./ g(, 00comb( k comb( k g( (k " m (k " n m" n" g( ( " m ( " n k k m" n" k k 3 3 g( " m, " n k k k k m" n" FOV X / k X / k Y FOV Y / k Y > FOV Y / k X > FOV X Windowing Resoluion Windowing he daa in Fourier space G W ( k,k G( k,k W k,k ( Resuls in convoluion of he objec wih he inverse ransform of he window (, g(, " w(, g W 5
6 ( rec W k,k w, Windowing Eample " k ( F " rec g W " ' rec k W k ' k W ( rec k,. k ' W ( k '-. W k sinc( sinc W k ( ( (, g(, "" W k sinc( sinc W k w E w(0 Eample w E " " sinc( d F[ sinc( ] k 0 rec' k ( Effecive Widh w(d k 0 " w E Resoluion and spaial frequenc Wih a window of widh he highes spaial frequenc is /. This corresponds o a spaial period of /. Effecive Widh g(, ["( "( ]"( g W Windowing Eample ( (, ["( "( ]"( W k sinc( sinc W k W k (["( "( ] sinc( sinc( W k W k ( sinc( sinc( ( sinc( W k.5 6
7 Sampling and Windowing G SW Sampling and Windowing Sampling and windowing he daa in Fourier space ( k,k G k,k ( comb k, k "k "k "k "k ( rec k, k ' W ( k ' Resuls in replicaion and convoluion in objec space. X X g SW (, W k g, ( ""comb(k,k ""sinc( sinc(w k Sampling in k Sampling in k RF G ( G ( k Δk RF Signal cos" 0 Low pass Filer ADC I k τ G i "k G i FOV "k sin" 0 Low pass Filer Noe: In pracice, here are number of was of implemening his processing. ADC Q One I,Q sample ever Δ M IjQ 7
8 G r G ( Sampling in k k k " k,ma G r G ( G r " Resoluion W k ADC Δ "k G r " FOV "k " W k k,ma G p G ( G p τ Eample Eample Goal : FOV FOV 5.6 cm " " 0. cm Readou Gradien : FOV " G r Pick 3 µsec G r " FOV ( 5.6cm T s T/cm.8675 G/cm ( ( s G r ADC Δ Readou Gradien : " G r " G ( 0.cm ( 457 G s ( G/cm r 8.9 ms where N read FOV " 56 N read ' G r G ( " Gauss 0 4 Tesla 8
9 Eample Eample Phase - Encode Gradien : FOV " G i Pick msec G i " FOV ( 5.6cm T s ( ( s T/cm.004 G/cm Phase - Encode Gradien : " G p G p " ( 0.cm 457 G s G/cm ( ( '0-3 s G ( G p τ where N p FOV " 56 N p G i τ G i Sampling Eample In pracice, an even number (picall power of sample is usuall aken in each direcion o ake advanage of he Fas Fourier Transform (FFT for reconsrucion. 4/FOV W k Consider he k-space rajecor shown below. ADC samples are acquired a he poins shown wih " 0 µsec. The desired FOV (boh and is 0 cm and he desired resoluion (boh and is.5 cm. Draw he gradien waveforms required o achieve he k-space rajecor. Label he waveform wih he gradien ampliudes required o achieve he desired FOV and resoluion. Also, make sure o label he ime ais correcl. k /FOV FOV FOV/4 9
10 Gibbs Arifac 5656 image 568 image GE Medical Ssems 003 Images from hp:// Apodizaion rec(k h(k /(cos(πk sinc( Hanning Window 0.5sinc(0.5sinc(- 0.5sinc( RF Signal cos" 0 sin" 0 FOV Aliasing and Bandwidh LPF LPF ADC ADC I Q FOV/3 Temporal filering in he readou direcion limis he readou FOV. So here should never be aliasing in he readou direcion. f Images from hp:// 0
11 readou Faser Aliasing and Bandwidh FOV f BγG r FOV Slower Lowpass filer in he readou direcion o preven aliasing. GE Medical Ssems 003
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