# Sampling, Ordering, Interleaving

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1 Sampling, Ordering, Interleaving Sampling patterns and PSFs View ordering Modulation due to transients Temporal modulations Timing: cine, gating, triggering Slice interleaving Sequential, Odd/even, bit-reversed Arbitrary Simultaneous Multislice / Multiband MR 1

2 Sampling & Point-Spread Functions PSF = Fourier transform of sampling pattern k-space: Extent, Density, Windowing PSF: Width, Replication, Ripple (side-lobes) k-space Sampling Point-Spread Function Fourier Transform Extent Spacing Width FOV 2

3 The Discrete sinc function h(x) = k sin( N kx) sin( kx) Function of extent Shows challenge of low N 3

4 Variable Density Sampling 2x undersamling Δk linear with k Minor Aliasing PSF broadens examplee1_vds.m 4

5 Variable Density Sampling: Density Compensated Multiply by Δk No PSF Broadening Higher ringing (center less dominant) need to apodize 5 examplee1_vds.m

6 Question - SNR? What happens to the noise in this sampling/ reconstruction scenario? Higher density at center = averaging Density compensation at outer = amplification σ ~ sqrt(w) 6 Section E1

7 Non-Cartesian Sampling / Gridding Irregularly sampled data Resample to grid to perform DFT gridmat.m k y k y k x k x 7

8 Gridding Steps Jackson 1991 Divide samples by density at location k Want to have uniform signal if we grid 1 s k y Convolve sampled k locations with kernel c(k) Resample at grid points k x FFT Reconstruction De-apodize to undo convolution side effects 8

9 Partial Fourier and PSF Full k-space trajectory is Sf(k), psf is δ(r) Half-k-space trajectory is Sh(k), PSF is sh(r) Sh(k) is real, with even component 0.5 Sf(k) Real{sh(r)} = 0.5 δ(r) Sampling: M(k) Sh(k) m(r)*[0.5 δ(r) + Imag{sh(r)}] If m(r) is real, the image is the real-part of m(r)*sh(r). How can we remove phase when m(r) is complex? 9

10 Partial k-space PSF - Contiguous Odd component is a step function Imaginary PSF is localized Even Odd k k 10

11 Example: Partial k-space PSF - Even/Odd What if we sample even lines on one side an odd on the other? Odd component is a step function modulated by e iπn Imaginary PSF is localized and shifted 11 Section E1

12 Partial k-space PSF - Random Selection Odd component is random 0 or 1 Imaginary PSF is spread out 12

13 Homodyne Reconstruction Sample half k-space plus a little extra Symmetric k-space: low-resolution image phase φ(r) m(r) Use ramp filter to reconstruct m(r)*sh(r) Remove phase: [m(r)*sh(r)] e -iφ(r) If sh(r) is narrow, phase of m(r) is canceled, and realpart leaves m(r)* δ(r) See John Pauly s notes for other recon methods 13

14 Partial Fourier Acquisition/Reconstruction k y F.T. k x k y Phase F.T. k x 14

15 Homodyne (k-space interpretation) From McGibney MRM 1993 H(u) ~ Density Compensation, reduced ringing Assumption Θ(u) is narrow 15

16 k-space Modulation Many sequences acquire multiple lines with transient magnetization: Echo trains: T2 and T2* decay over k-space Magnetization-prepped bssfp, RF-spoiled transients Off-resonance (EPI, Spiral primarily) Temporal signal effects (non-motion): Contrast uptake, inflow, varying B0, 16

17 View Ordering / Grouping Sequential k y k x Centric / Center-out k y k x Interleaved k y Segmented k y k x k x 17 Each color is a different modulation (echo, time, etc)

18 3D Image (ky-kz) View Ordering/Grouping Sequential kz Sequential ky Center-out (φ,kr) k z k z k z k y k y k y Centric (ordered by radius first or azimuth (φ) first) Segment groups by ky, kz, φ, kr Sub-segment groups (ky, kz, φ, kr, randomly) 18

19 Modulation and PSFs Group k-space samples by intensity Reconstruct PSF for each group Multiply by modulation and sum 19

20 Modulation Example 1: FSE Echo Train of 2T2 Peak reduction (area) Decompose Modulation into even / odd parts real{psf} good PSF broadens 20

21 Modulation Example 2: PD FSE Echo Train of 2T2 Peak reduction (area) Symmetric modulation: Real PSF PSF broadens 21

22 Example: Echo-Train + CS + Half-Fourier + Elliptic 2D k-space sampling variation (ky-kz phase encodes) smooth modulation with echo train Random sampling for CS Choose trajectories through regions to minimize change (eddy-current) Random ky-kz SEMAC sampling with partial ky and elliptic Worters 2011

23 Question: Temporal Odd/Even Sampling Sample odd-then-even lines during contrast uptake What will artifact look like if signal change is 2x frame rate? What can we do about it? Parallel image reconstruction (2x frame rate!) k y Aliased flicker True 2x Frames k x 23 Section E1

24 Temporal Sampling (kf and k-t) Temporal SENSE R=3 Ordering k TE3 k f TE2 k x t TE1 24 Section E1

25 Temporal Undersampling: DISCO Temporal Footprint Temporal Resolution 25

26 Temporal Undersampling: PSFs Data Acquisition (ky-kz -space) DISCO, TWIST Cartesian Acquisition with Projection Reconstruction (CAPR) Time-Resolved Imaging of Contrast Kinetics (TRICKS) Saranathan 2012 Madhuranthakam 2006 Korosec 1996 Point-Spread Functions 26

27 Cardiac/Respiratory Acquisition Timing Cine: Exploit periodicity of cardiac/respiratory cycle Sample N ky lines repeatedly, next N lines on next heartbeat. High frame rate and spatial resolution Triggering: Start acquisition based on external trigger (EKG, plethysmograph, respiratory bellows) Gating: Excite continuously, but acquire only after trigger Can combine any/all of these x x x x x x x x x x x x x x x time trigger / gate trigger / gate 27

28 Cine Example If the heart-rate is 72 beats/min, and we want to acquire a cine image with 256x256 matrix, 5ms TR and 40ms temporal resolution a) How many cardiac cycles are needed? 40ms/5ms = 8 views per segment. 256/8 = 32 b) How many TRs can we place per heartbeat if we use 75% of the cardiac cycle? 60s/72 x 75% = 5/8 sec = 0.625s. 625ms / 5ms = 125 TRs c) How many cine frames do we acquire? 625ms/40ms = 15, or 125/8 =15 28 Section E1

29 Slice Interleaving Multislice acquisitions allow volumetric imaging Acquisitions can be sequential or interleaved Interleaving time efficient if there is dead time Different ways to interleave (reduce adjacent-slicesaturation) Sequential: 0, 1, 2, 3, 4, 5, 6, 7 Odd/Even: 0, 2, 4, 6, 1, 3, 5, 7 Bit-reversed: 0, 4, 2, 6, 1, 5, 3, 7 29

30 How Many Slices to Interleave? Usually specify TR, TI, Echo-train-length (ETL), Resolution,... Tells pulse durations (Tseq) and RF power Nmax ~ TR / Tseq Can re-order slices in time slots Additional slices require another acquisition Slice 0 Slice 1 Slice 2 Slice 0 RF TR 30

31 More Flexible Interleaving If Nslices > Nmax, scan is 2x, 3x,... longer Decoupling phase encode number allows flexible interleaving Read-out matrices across then down Slice Number Slice Number Outer Loop Outer Loop Inner Loop Pop/View Number Inner Loop Outer Loop Outer Loop Inner Loop Pop/View Number Inner Loop

32 FLAIR / STIR? Additional dead-time during TI interval Can sometimes interleave other acquisitions Additional constraints on TR, TI, Tseq Inversion Imaging RF TI TR 32

33 Multiband Imaging Imaging multiple slices simultaneously Excitation: Multiply RF(t) by cos(t+φ) Increases SAR Imaging: Hadamard: Excite 1,1 and 1,-1 pattern, add & subtract POMP: Alternate patterns, increase y FOV Parallel Imaging: Use coils to separate slices Blipped sequences: Gz blips induce slice-dependent phase Like 3D k-space with limited excitation Similar to Dixon water/fat: slices are like spectral peaks 33

34 Controlled Aliasing in Parallel Imaging (CAIPI) Breuer F, MRM 2005, D (ky-kz) sampling: kz Hexagonal sampling offsets replicas Reduced aliasing (further apart) 2D Multislice: Alternating phase during excitation or blips Offset replicas allows in-plane coil sensitivities to help separate slices Can think of as 3D k-space z ky y 34

35 Multiband Example Starting with a 1 khz BW excitation, we want to excite two slices that are 5mm thick and 2cm apart (center-to-center) a) How do we modulate the pulse to excite 2 slices? 1kHz 5mm, +/-2kHz 2cm. Modulate with cos(2π 2000 t) b) What is the image if we alternate excitation with cosine and i*sin modulation on successive ky lines, assuming slices are each 1cm away from center? Slices are both shifted 1/4 FOV c) Which FOV should we encode for 2x parallel imaging if we have 2 coils along y? 35 Section E1

36 Lustig, et al. MRM 2007 Compressed Sensing MRI MRI Data Sample less data, randomly Choose compressible image matching data 36 Section E1

37 CS Reconstruction and PSFs ifft Recovery Courtesy M. Lustig, Example inspired by Donoho et. Al,

38 Summary Sampling and PSFs Resolution, FOV, ringing Variable-density and gridding Partial Fourier View ordering and k-space modulation ky-kz and k-t sampling Slice interleaving Simultaneous Multislice Imaging 38 Section E1

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