Combined Compressed Sensing and
Parallel MRI Compared for Uniform and
Random Cartesian Undersampling of K-Space
D. S. Weller1 , J. R. Polimeni2,3 , L. Grady4 ,
L. L. Wald2,3 , E. Adalsteinsson1 , V. K Goyal1
2
1 EECS, Massachusetts Institute of Technology
A. A. Martinos Center, Radiology, Massachusetts General Hospital
3 Radiology, Harvard Medical School
4 Image Analytics and Informatics, Siemens Corporate Research
5-26-2011
2011 IEEE International Conference on Acoustics, Speech, and Signal Processing – BISP-L2.2
Magnetic Resonance Imaging
Martinos Center for Biomedical Imaging (MGH)
• Acquisition time remains an issue: 8 − 10 minutes at 3 T
• Faster acquisitions ⇒ ↓ cost, ↑ comfort, ↑ quality
• SpRING used to recover images from accelerated data
ICASSP 2011 – BISP-L2.2
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Magnetic Resonance Imaging
Martinos Center for Biomedical Imaging (MGH)
• Acquisition time remains an issue: 8 − 10 minutes at 3 T
• Faster acquisitions ⇒ ↓ cost, ↑ comfort, ↑ quality
• SpRING used to recover images from accelerated data
• Undersampling strategies (uniform, random) compared
• Conventional wisdom: random undersampling required
• We show: uniform undersampling may be sufficient
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Accelerated MR Imaging
• Fourier transform (k-space)
sampled along the readout
direction
readout
• One line per coordinate
pair in other two directions
• Too much time to sample
axial slice plane
3-D k-space
F−1
→
Full k-space
ICASSP 2011 – BISP-L2.2
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Accelerated MR Imaging
• Fourier transform (k-space)
sampled along the readout
direction
readout
• One line per coordinate
pair in other two directions
• Too much time to sample
axial slice plane
3-D k-space
F−1
→
• Undersample k-space by a
R = 4 (2 × 2)
ICASSP 2011 – BISP-L2.2
factor of R ⇒ aliasing in
the image domain
3
Reconstructing Accelerated Images
P = 32-channel coil
Parallel Imaging:
• Multiple receivers with
different spatial weightings
[Roemer et al., 1990]
• GRAPPA: kernel from ACS
(small dense k-space
block) yields missing
k-space [Griswold et al., 2002]
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Reconstructing Accelerated Images
Compressed Sensing: [Candès
and Romberg, 2006; Candès et al.,
2006; Donoho, 2006; Lustig et al., 2007]
• Sparsity of MR images
• Fourier transform provides
incoherent sampling
P = 32-channel coil
Parallel Imaging:
• Multiple receivers with
different spatial weightings
[Roemer et al., 1990]
• GRAPPA: kernel from ACS
(small dense k-space
block) yields missing
k-space [Griswold et al., 2002]
ICASSP 2011 – BISP-L2.2
Four-level ‘9-7’ DWT
4
Multi-Channel Sampling Abstraction Model
RΔk
y1[k]
CTFT

I
S1
η1[k]
SP
ηP[k]
RΔk
CTFT

η1 [k]
 .. 
 .  ∼ N (0, Λ)
ηP [k]
yP[k]
• I approximately sparse in transform domain Ψ
• pth coil spatially weights object I with sensitivity Sp
• Samples in k-space spaced R∆k apart
• Noise η iid across frequencies, correlated across coils
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Improving upon Parallel Imaging with SpRING
R = 9 (3 × 3)
R = 16 (4 × 4)
R = 25 (5 × 5)
R = 36 (6 × 6)
GRAPPA results (with 36 × 36 ACS block and 3 × 3 kernel)
• Two forms of error: noise amplification, residual aliasing
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Improving upon Parallel Imaging with SpRING
R = 9 (3 × 3)
R = 16 (4 × 4)
R = 25 (5 × 5)
R = 36 (6 × 6)
GRAPPA results (with 36 × 36 ACS block and 3 × 3 kernel)
• Two forms of error: noise amplification, residual aliasing
• Noise not sparse ⇒ promoting sparsity denoises images
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Improving upon Parallel Imaging with SpRING
R = 9 (3 × 3)
R = 16 (4 × 4)
R = 25 (5 × 5)
R = 36 (6 × 6)
GRAPPA results (with 36 × 36 ACS block and 3 × 3 kernel)
• Two forms of error: noise amplification, residual aliasing
• Noise not sparse ⇒ promoting sparsity denoises images
• CS also can undo incoherent aliasing
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Improving upon Parallel Imaging with SpRING
R = 9 (3 × 3)
R = 16 (4 × 4)
R = 25 (5 × 5)
R = 36 (6 × 6)
GRAPPA results (with 36 × 36 ACS block and 3 × 3 kernel)
• Two forms of error: noise amplification, residual aliasing
• Noise not sparse ⇒ promoting sparsity denoises images
• CS also can undo incoherent aliasing
• SpRING: improve GRAPPA result with CS [Weller et al., 2010]
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Uniform vs. Random Cartesian Subsampling
Uniform
GRAPPA
Random
unaliasing up to P
coherent artifacts
CS
SpRING
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Uniform vs. Random Cartesian Subsampling
GRAPPA
Uniform
Random
unaliasing up to P
coherent artifacts
unaliasing up to P
non-sparse artifacts
CS
SpRING
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Uniform vs. Random Cartesian Subsampling
GRAPPA
CS
Uniform
Random
unaliasing up to P
coherent artifacts
unaliasing up to P
non-sparse artifacts
limited unaliasing
capable of denoising
SpRING
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Uniform vs. Random Cartesian Subsampling
GRAPPA
CS
Uniform
Random
unaliasing up to P
coherent artifacts
unaliasing up to P
non-sparse artifacts
limited unaliasing
capable of denoising
resolves aliasing
also denoising
SpRING
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Uniform vs. Random Cartesian Subsampling
GRAPPA
CS
Uniform
Random
unaliasing up to P
coherent artifacts
unaliasing up to P
non-sparse artifacts
limited unaliasing
capable of denoising
resolves aliasing
also denoising
SpRING
ICASSP 2011 – BISP-L2.2
unaliasing? denoising?
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Uniform vs. Random Cartesian Subsampling
GRAPPA
CS
Uniform
Random
unaliasing up to P
coherent artifacts
unaliasing up to P
non-sparse artifacts
limited unaliasing
capable of denoising
resolves aliasing
also denoising
SpRING
unaliasing? denoising?
• In between: jittered undersampling, poisson-disc
undersampling, tiled undersampling, etc. [Usman and
Batchelor, 2009; Lustig et al., 2009; Lai et al., 2011]
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SpRING: Combining GRAPPA and CS
• Balance fidelity to the GRAPPA solution and joint sparsity
of the solution while preserving the acquired data:
GRAPPA fidelity
z
}|
{
z
}|
{
z }| {
ŷ ∈ arg min kCF−1 (y − G(d))k22 +λ kΨF−1 ykS(α) s.t. d = Ky
y
• y: full-FOV k-space data
• d: acquired (undersampled) k-space data
• G(d): GRAPPA solution (full-FOV k-space estimate)
• C: low-resolution coil combination weights from ACS data
• λ: tuning parameter
• k · kS(α) : joint sparsity penalty function
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SpRING: Combining GRAPPA and CS
• Balance fidelity to the GRAPPA solution and joint sparsity
of the solution while preserving the acquired data:
GRAPPA fidelity
sparsity penalty
z
}|
{
z
}|
{
z }| {
ŷ ∈ arg min kCF−1 (y − G(d))k22 +λ kΨF−1 ykS(α) s.t. d = Ky
y
• y: full-FOV k-space data
• d: acquired (undersampled) k-space data
• G(d): GRAPPA solution (full-FOV k-space estimate)
• C: low-resolution coil combination weights from ACS data
• λ: tuning parameter
• k · kS(α) : joint sparsity penalty function
ICASSP 2011 – BISP-L2.2
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SpRING: Combining GRAPPA and CS
• Balance fidelity to the GRAPPA solution and joint sparsity
of the solution while preserving the acquired data:
GRAPPA fidelity
sparsity penalty
preserve data
z
}|
{
z
}|
{
z }| {
−1
2
−1
ŷ ∈ arg min kCF (y − G(d))k2 +λ kΨF ykS(α) s.t. d = Ky
y
• y: full-FOV k-space data
• d: acquired (undersampled) k-space data
• G(d): GRAPPA solution (full-FOV k-space estimate)
• C: low-resolution coil combination weights from ACS data
• λ: tuning parameter
• k · kS(α) : joint sparsity penalty function
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SpRING: Penalty Functions
• We utilize two choices of kwkS(α) =
P
n sα (wn )
to
approximate the `0 -“norm” in our analysis:
• the convex `1 -norm
sα(wn)
4
L0
L1
2
0
0
ICASSP 2011 – BISP-L2.2
1
2
wn
3
4
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SpRING: Penalty Functions
• We utilize two choices of kwkS(α) =
P
n sα (wn )
to
approximate the `0 -“norm” in our analysis:
• the convex `1 -norm
• the nonconvex Cauchy penalty function
P
kwkS(α) =
n
log(1 + α|wn |2 ) with continuation over α
sα(wn)
4
L0
L1
2
0
Cauchy
0
ICASSP 2011 – BISP-L2.2
1
2
wn
3
4
9
SpRING: Penalty Functions
• We utilize two choices of kwkS(α) =
P
n sα (wn )
to
approximate the `0 -“norm” in our analysis:
• the convex `1 -norm
• the nonconvex Cauchy penalty function
P
kwkS(α) =
n
log(1 + α|wn |2 ) with continuation over α
sα(wn)
4
L0
L1
2
0
Cauchy
0
1
2
wn
3
4
• We enforce joint sparsity by applying the `2 -norm across
the coils and the sparsity penalty function to the result
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Non-Uniform Cartesian Subsampled GRAPPA
• Various strategies exist for performing GRAPPA with
non-uniform subsampling
• iterative methods such as iterative GRAPPA [Lustig and
Pauly, 2007] can handle arbitrary sampling patterns
• direct methods exist for radial [Griswold et al., 2003] and
spiral trajectories [Heidemann et al., 2006]
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Non-Uniform Cartesian Subsampled GRAPPA
• Various strategies exist for performing GRAPPA with
non-uniform subsampling
• iterative methods such as iterative GRAPPA [Lustig and
Pauly, 2007] can handle arbitrary sampling patterns
• direct methods exist for radial [Griswold et al., 2003] and
spiral trajectories [Heidemann et al., 2006]
• We extend GRAPPA to arbitrary Cartesian undersampling
using a direct method with locally-applied kernels
• for each block of k-space, separate GRAPPA kernels are
derived using the ACS data
• G(d) can be incorporated into SpRING without
re-computation at each iteration
• may not be as robust as iterative methods (like L1 SPIR-iT)
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Non-Uniform Cartesian Subsampled GRAPPA
• Divide k-space into Ry × Rz blocks, reconstruct each block
...
...
with kernel defined for local sampling pattern
P coils
ACS
P coils
undersampled k-space (Ry = Rz = 3)
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Non-Uniform Cartesian Subsampled GRAPPA
• Divide k-space into Ry × Rz blocks, reconstruct each block
...
...
with kernel defined for local sampling pattern
P coils
ACS
P coils
undersampled k-space (Ry = Rz = 3)
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Non-Uniform Cartesian Subsampled GRAPPA
• Divide k-space into Ry × Rz blocks, reconstruct each block
...
...
with kernel defined for local sampling pattern
P coils
ACS
P coils
undersampled k-space (Ry = Rz = 3)
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Non-Uniform Cartesian Subsampled GRAPPA
• Divide k-space into Ry × Rz blocks, reconstruct each block
...
...
with kernel defined for local sampling pattern
P coils
ACS
P coils
undersampled k-space (Ry = Rz = 3)
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Shepp-Logan Phantom Reconstruction
• 8-channel Shepp-Logan phantom (128 × 128 slice)
• simulated sensitivities for 8-channel array coil
• noise covariance generated according to [Roemer et al.,
1990]; noise simulated at -30 dB (complex)
• SpRING performed using the `1 -norm and Cauchy penalty
functions and the finite-differences transform
[S1 ,...,SP ]
−→
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Results – Uniform Undersampling
GRAPPA result
SpRING result
`1 penalty
λ = 101.6
RMSE = 0.03123
RMSE = 0.01866
GRAPPA vs. SpRING reconstructions for 8-coil R = 16 (4 × 4)
uniform undersampled Shepp-Logan data with finite differences
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Results – Uniform Undersampling
GRAPPA result
SpRING result
Cauchy penalty
λ = 101.4
α = 107
RMSE = 0.03123
RMSE = 0.003622
GRAPPA vs. SpRING reconstructions for 8-coil R = 16 (4 × 4)
uniform undersampled Shepp-Logan data with finite differences
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Results – Random Undersampling
GRAPPA result
SpRING result
`1 penalty
λ = 102.2
RMSE = 0.06412
RMSE = 0.02058
GRAPPA vs. SpRING reconstructions for 8-coil R = 16 (4 × 4)
random undersampled Shepp-Logan data with finite differences
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Results – Random Undersampling
GRAPPA result
SpRING result
Cauchy penalty
λ = 103
α = 107
RMSE = 0.06412
RMSE = 0.002386
GRAPPA vs. SpRING reconstructions for 8-coil R = 16 (4 × 4)
random undersampled Shepp-Logan data with finite differences
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Reconstructing Real Brain Data
• 32-channel T1 -weighted MPRAGE brain (256 × 256 × 176
sagittal slices; 1.0 mm isotropic) at 3 T
• axial slice extracted, cropped, and 10/32 coils retained (to
exacerbate GRAPPA aliasing error)
• noise covariance measured using noise-only SNR maps
• SpRING performed using the `1 -norm and Cauchy penalty
functions and the four-level ‘9-7’ DWT
sum-ofsquares
−→
10/32 coils
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Results – Uniform Undersampling
GRAPPA result
SpRING result
`1 penalty
λ = 100.8
RMSE = 0.08276
RMSE = 0.06373
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
uniform undersampled brain data with DWT
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Results – Uniform Undersampling
GRAPPA result
SpRING result
Cauchy penalty
λ = 100.6
α = 104
RMSE = 0.08276
RMSE = 0.06364
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
uniform undersampled brain data with DWT
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Results – Random Undersampling
GRAPPA result
SpRING result
`1 penalty
λ = 102.6
RMSE = 0.2162
RMSE = 0.07637
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
random undersampled brain data with DWT
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Results – Random Undersampling
GRAPPA result
SpRING result
Cauchy penalty
λ = 102.2
α = 104
RMSE = 0.2162
RMSE = 0.07977
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
random undersampled brain data with DWT
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Conclusions
• For the very sparse Shepp-Logan phantom, the Cauchy
penalty function is effective at both denoising and undoing
aliasing, even with uniform undersampling
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Conclusions
• For the very sparse Shepp-Logan phantom, the Cauchy
penalty function is effective at both denoising and undoing
aliasing, even with uniform undersampling
• For real data, sparsity combined with uniform
undersampling effectively denoises the image, but aliasing
is mostly unmitigated
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Conclusions
• For the very sparse Shepp-Logan phantom, the Cauchy
penalty function is effective at both denoising and undoing
aliasing, even with uniform undersampling
• For real data, sparsity combined with uniform
undersampling effectively denoises the image, but aliasing
is mostly unmitigated
• Random undersampling does increase the ability of CS to
resolve aliasing in both sparse and real images
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Conclusions
• For the very sparse Shepp-Logan phantom, the Cauchy
penalty function is effective at both denoising and undoing
aliasing, even with uniform undersampling
• For real data, sparsity combined with uniform
undersampling effectively denoises the image, but aliasing
is mostly unmitigated
• Random undersampling does increase the ability of CS to
resolve aliasing in both sparse and real images
• But, is aliasing always significant in real images?
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Conclusions
• For the very sparse Shepp-Logan phantom, the Cauchy
penalty function is effective at both denoising and undoing
aliasing, even with uniform undersampling
• For real data, sparsity combined with uniform
undersampling effectively denoises the image, but aliasing
is mostly unmitigated
• Random undersampling does increase the ability of CS to
resolve aliasing in both sparse and real images
• But, is aliasing always significant in real images?
• Let us revisit the dataset used in the last example...
• Use all 32 channels of data
• Reduce undersampling to a still-aggressive R = 16 (4 × 4)
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Conclusions
GRAPPA result
SpRING result
`1 penalty
λ = 100.4
RMSE = 0.05091
RMSE = 0.03900
GRAPPA vs. SpRING reconstructions for 32-coil R = 16 (4 × 4)
uniform undersampled brain data with DWT
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Conclusions
GRAPPA result
SpRING result
Cauchy penalty
λ=1
α = 104
RMSE = 0.05091
RMSE = 0.03795
GRAPPA vs. SpRING reconstructions for 32-coil R = 16 (4 × 4)
uniform undersampled brain data with DWT
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Conclusions
GRAPPA result
SpRING result
`1 penalty
λ = 100.8
RMSE = 0.09209
RMSE = 0.05179
GRAPPA vs. SpRING reconstructions for 32-coil R = 16 (4 × 4)
poisson-disc undersampled brain data with DWT
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Conclusions
GRAPPA result
SpRING result
Cauchy penalty
λ = 100.4
α = 104
RMSE = 0.09209
RMSE = 0.04955
GRAPPA vs. SpRING reconstructions for 32-coil R = 16 (4 × 4)
poisson-disc undersampled brain data with DWT
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Acknowledgments
• B1 simulator and array coil geometry provided by Prof.
Fa–Hsuan Lin (MGH/Harvard); available online at
http://www.nmr.mgh.harvard.edu/∼fhlin/.
• Funding: NSF CAREER 0643836, NIH R01 EB007942 and
EB006847, NIH NCRR P41 RR014075, and an NSF
Graduate Research Fellowship
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SpRING: the Nullspace Method
• Operate in the nullspace of K: optimize missing data x
T
(y = K x + KT d):
1 0 0 0 0 0
0 1 0 0 0 0
K 0 0 1 0 0 0
T
range of K is nullspace of K
0 0 0 1 0 0 K
0 0 0 0 1 0
0 0 0 0 0 1
• This nullspace method yields an unconstrained problem:
T
x̂ ∈ arg min kCF−1 (K x + KT d − G(d))k22
x
T
+ λkΨF−1 (K x + KT d)kS(α)
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Results – Poisson-Disc Undersampling
GRAPPA result
SpRING result
`1 penalty
λ = 102.2
RMSE = 0.1257
RMSE = 0.07285
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
poisson-disc undersampled brain data with DWT
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Results – Poisson-Disc Undersampling
GRAPPA result
SpRING result
Cauchy penalty
λ = 101.6
α = 104
RMSE = 0.1257
RMSE = 0.07478
GRAPPA vs. SpRING reconstructions for 10-coil R = 25 (5 × 5)
poisson-disc undersampled brain data with DWT
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