surface Image reconstruction: 2D Fourier Transform

Transcription

1 2/1/217 Chapter 2-3 K-space Intro to k-space sampling (chap 3) Frequenc encoding and Discrete sampling (chap 2) Point Spread Function K-space properties K-space sampling principles (chap 3) Basic Contrast mechanism (chap 4) K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms Description of the reconstruction problem k-space k t k G( ) d k k z Need to fill k-space with data points which uniquel describe the imaged object. Think of phase- and frequenc encoding gradients as means of moving (adding new datapoints) in k-space. Limiting discussion to a slice (2D plane), magnetization distribution is given b the 2-dimensional Fourier transform of the spin distribution across the slice M T (t) (r) ep jk rdr slice (r) is obtained from the inverse Fourier transform of M T (t) under the influence of a known gradient configuration (,) 1 2 k M T (,k )ep j k d dk k-space = visualization of the distribution of spatial frequencies in the image. k-space = Fourier transform of the MR image 4 The phase angle of a spin in a slice at a t is given b: t,, tdt Gnt Gt Definition of k: t k i G i tdt Gradient on (in the i) surface The total transverse magnetisation is a function of, k and the position in the slice: M T (, k ) Image reconstruction: (,) 1 2 k 2D Fourier Transform pulsed field gradient along M T (,k )ep j k d dk 5 t K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms 6 1

2 2/1/217 Frequenc and Phase encoding acquisition of a profile Phase encoding: needs multiple echo acquisitions Pulse sequence introduction Field gradients Spatial coding frequenc encoding (gradient on during signal acquisition) Phase encoding (gradient on before signal acquisition, repeated at different amplitudes) 7 FYS-KJM Spatiall encoded Echoes Spin-Echo: SE Use of gradient in a spin-echo eperiment to induce spatiall dependent dephasing and rephasing Gradient Echo: GRE Gradient induced echo Spin Echo: freq. encoding (frequenc encoding) t= t=/2 t= 9 18 dephasing echo FYS-KJM sampling of the signal which contains frequenc information (-ais) 1 Gradient Echo 9 Gradient Echo M T 2 * relaation (FID) t M z z z echo z z T G 2 _ rew dt T T read / 2 T _ rew dt G T _ rewt G read _ r

3 t=/2 2/1/217 Travelling in k-space Gradient Echo 9 k 1 sample k RF G FYS-KJM Gradient Echo 9 k Gradient Echo 9 k repeated N s t s Gradient Echo Spin-Echo k-space travelling 9 k t= t=/2 t= slice selection 9 18 echo 18º rf pulse k read-out phase encode Acquisition of a profile t=

4 2/1/217 acquisition of a profile acquisition of a profile (frequenc encoding) (frequenc encoding) Spatial information in G Spatial information in G 19 2 acquisition of a profile acquisition of a profile (frequenc encoding) (frequenc encoding) Spatial information in G Spatial information in G acquisition of a profile Acquisition of profile (frequenc encoding) (frequenc encoding) Spatial information in G Spatial information in G

5 2/1/217 acquisition of all profiles Repeated acquisition of profiles profiles (frequenc encoding) the field of view: FOV k Spatial information in G FOV depends on the - gradient strengths - sampling of a profile phase phase K K frequenc -127 frequenc Image generation k Signal intensit distribution in the selected slice IMAGE m, 1 2 frequenc encoding k M T 2D FT,k ep i k d dk frequenc frequenc

6 2/1/217 Digital signal sampling Discrete sampling K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms FYS-KJM Discrete sampling PSF() T read / 2 T read / 2 A.e (i..t ) dt T read MR-signal (M T ) U(t) Sampling interval: U(t) = 1 if t [-T read /2, T read /2] and elsewhere Signal sampling is modulated b a Block Function U(t) T read = N.t s In frequenc domain, this translates to: PSF() FFTU(t) U(t)e (i..t ) dt PSF() A i e(i..t ) T read / 2 Tread / 2 PSF() A sin(..t read / 2) sin T read 2 PSF() T read T read 2 periodic function (See eq. 2-24) 33 FYS-KJM PSF() Field of View (FOV) P (,) 1 2 k M T (,k )ep j k d dk (,) 1 2 M T( )ep j d function (point object) image representation of the point object We can calculate the Full Width at Half Height (FWHH) T read, ma T read / 2 T read = N.t s ma 2, min 2 t s FOV 2 2 f s FOV t s FOV (smallest wavelength) wavelength is automaticall calculated b the scanner FOV is entered b the user (see Eq. 2-28) 2 / FYS-KJM FYS-KJM

7 2/1/217 FOV K-space properties Same definition can be done for the FOV 2 2 k, min G, n G, n1 T phase difference at the edge of the FOV Consider a square matri: N 2 = N.N Resolution (): 2 G N t Field of view ():,ma,min s 2 2 FoV k G t s Maimum frequenc in read-out () FoV / 2 ma Min sampling rate (): 1 s / t G FoV / 2 G,n G,n 1 G N G, ma (See Eq. 2-33) T FOV We need to have G N 2 G, ma Field of view ():,ma N FoV G T _ ma Sampling rate (): N G _ ma T FoV FYS-KJM K-space vs image space Back-folding / S(r) F(k) k k Increase 1/t s, N. ρ(,) FT FoV {FT} -1 Object FoV N k t s _ma FoV - ma = -FoV /2 ma = FoV /2 1 (, ) 2 k k M ( k, k ) ep T jk k dk dk N k t k G( ) d k k z 39 Image FoV discarded discarded 4 Back-folding / FFT 1D: Truncation Artefact Object FoV Image FoV Fold-Over artefact

8 2/1/217 FFT 2D: Truncation artefact Truncation artifact Ringing- (or truncation) artifacts in regions with high spatial frequencies (edges) in a phantom. The artifacts are more evident in the right image due to a lower matri (N=112, vs N=256 in the left image) Fold-Over artefact K-space Intro to k-space sampling Frequenc encoding and Discrete sampling Point Spread Function K-space properties K-space sampling Pulse Sequence Basic Contrast mechanisms and TR Basic Contrast 1 9 o Longitudinal and Transverse Relaation: TR

9 2/1/217 1 Longitudinal and Transverse Relaation: at the same moment 9 o magnetisation 1 short T1 9 o 9 o signal 49 5 short T1 long T1 1 9 o 9 o short T2 1 9 o 9 o long T1 repetition 9 o 9 o 9 o TR 9 o 1 long T

10 2/1/217 echo 9 o 9 o o 9 o Short Longer 2 different tissues with different T1/T2 1 9 o 9 o 1 9 o 9 o T1 contrast brain eample T1 contrast 1 9 o White matter 1 9 o 9 o WM Gre matter GM

11 2/1/217 T1 contrast T1 contrast 1 9 o 9 o WM 1 9 o 9 o WM GM GM T1 contrast T1-weighted images TR 4 ms 1 9 o 9 o WM GM 1 ms 3 ms 4 ms Fat bone marrow bright i.v.contrast T1 weighted images: white matter TR short (SE) < 6 ms (can be as low as 1.5ms) gre matter muscle bod fluids gre short < 25 ms bone air black TR < 6 ms < 25 ms

12 2/1/217 T1-w knee PD (*, proton densit) & T2 contrast 1 9 o 9 o * Denoted rho contrast in the compendium PD & T2 contrast PD & T2 contrast 9 o 9 o 9 o 9 o PD contrast long TR, short T2 contrast: increases 1 9 o 1 9 o

13 2/1/217 T2 contrast, long TR, long PD & T2 weighted images TR 25 ms 1 9 o 2 ms 6 ms 1 ms 14 ms Spine / summar T2 weighted images: TR long long > 18 ms > 8 ms T1 weighted T2 weighted Table of relaation s From: Greg J. Stanisz, Magnetic Resonance in Medicine 54: (25) FYS-KJM