1. Parallel Imaging Reconstruction
Multiple coils - “parallel imaging”
Reduced acquisition times.
Higher resolution.
Shorter echo train lengths (EPI).
Artefact reduction.
Multiple coils have previously been used for SNR improvements, reduced acquisition times, and the rejection and replacement of isolated, motion-corrupted k-space lines.
This work aims to use a fully sampled k-space and the ability of SMASH-based methods to predict k-space, to detect and correct motion
corrupted data.

2. K-space from multiple coils
views
multiple
receiver coils
coil
sensitivities
k-space
simultaneous or “parallel” acquisition

3. Undersampled k-space gives aliased images
Fourier transform
of undersampled
k-space.
coil 1
FOV/2
coil 2
Dk = 2/FOV
Dk = 1/FOV

4. SENSE reconstruction ra p1 rb p2 coil 1 coil 2 Solve for ra and rb.
Repeat for every pixel pair.

5. Image and k-space domains
object
coil sensitivity
coil view
Image Domain
multiplication x = c s r FT
k-space convolution = R C S
coil k-space
“footprint”
object k-space

6. Generalized SMASH = S C R image domain product k-space convolution
matrix multiplication
gSMASH1 matrix solution
1 Bydder et al. MRM 2002;47:

7. Composition of matrix S
Acquired k-space
coil 1
coil 2
hybrid-space
data column
FTFE S
process column by column

8. Coil convolution matrix C
sensitivities
FTPE
hybrid
space
cyclic permutations of
&

9. gSMASH = S C R coil 1 coil 2 requires matrix inversion missing samples
(can be irregular)
coil 1 =
coil 2 S C R
requires matrix inversion

10. Linear operations Linear algebra.
Fourier transform also a linear operation.
gSMASH ~ SENSE
Original SMASH uses linear combinations of data.

11. SMASH + + + + + - + - PE weighted coil profiles
sum of weighted profiles
Idealised k-space of
summed profiles
0th harmonic
1st harmonic

12. SMASH = R easy matrix inversion data summed with 0th harmonic weights
1st harmonic weights
easy matrix inversion

13. GRAPPA
Linear combination; fit to a small amount of in-scan reference data.
Matrix viewpoint:
C has a diagonal band.
solve for R for each coil.
combine coil images.

14. Linear Algebra techniques available
Least squares sense solutions – robust against noise for overdetermined systems.
Noise regularization possible.
SVD truncation.
Weighted least squares.
Absolute Coil Sensitivities not known.

15. Coil Sensitivities
All methods require information about coil spatial sensitivities.
pre-scan (SMASH, gSMASH, SENSE, …)
extracted from data (mSENSE, GRAPPA, …)

16. Merits of collecting sensitivity data
Pre-scan In data
One-off extra scan.
Large 3D FOV.
Subsequent scans run at max speed-up.
High SNR.
Susceptibility or motion changes.
No extra scans.
Reference and image slice planes aligned.
Lengthens every scan.
Potential wrap problems in oblique scans.

17. PPI reconstruction is weighted by coil normalisation
coil data used
(ratio of two images)
reconstructed object
c load dependent, no absolute measure.
N root-sum-of-squares or body coil image.

18. Handling Difficult Regions
body coil
raw (array/body)
array coil image
thresholded raw
local polynomial fit
filtered threshold
region grow

19. SENSE in difficult regionsra rb p2 p1
coil 1
coil 2

20. Sources of Noise and Artefacts
Incorrect coil data due to:
holes in object (noise over noise).
distortion (susceptibility).
motion of coils relative to object.
manufacturer processing of data.
FOV too small in reference data.
Coils too similar in phase encode (speed-up) direction.
g-factor noise.

21. Tips
Reference data:
avoid aliasing (caution if based on oblique data).
use low resolution (jumps holes, broadens edges).
use high SNR, contrast can differ from main scan.
Number of coils in phase encode direction >> speed-up factor.
Coils should be spatially different.
(Don’t worry about regularisation?)