Philippe Ciuciu

Online MR Image Reconstruction for Compressed Sensing Acquisition in
T2* Imaging
P. Ciuciu1, 2
Joint work with: L. El Gueddari1,2, E. Chouzenoux3 and J-C. Pesquet3
1CEA/NeuroSpin, Univ. Paris-Saclay, Gif-sur-Yvette, France
2Inria, Parietal team, Univ. Paris-Saclay, France
3Inria, Opis team, CVN, Centrale-Supélec, Univ. Paris-Saclay, France
October, 9 2020
L2S – CNRS – Supélec – Université Paris-Saclay, France
P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 1 / 50

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Outline
1 Context & Motivation
Ultra High Field Imaging
High resolution MRI
Sampling in MRI
2 Multi-channel image reconstruction for non-Cartesian acquisition
State-of the art
Calibrationless reconstruction: Playing with the regularization
Experiments & Results
Summary
3 Online CS-MRI reconstruction
Problem statement: The single channel case
Multi-channel online reconstruction
Summary
4 Conclusion & Outlook

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Outline
High resolution MRI
Sampling in MRI
State-of the art
Summary
Summary

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High resolution imaging: Increasing the SNR
Ultra-high magnetic eld
Signal-to-Noise Ratio (SNR) increases with the eld strength: SNR ∝ B1.65
0
1
Particularly suited for the T∗
2
-weighted imaging contrast
1Pohmann, Speck, and Sche er 2016, Magnetic resonance in medicine.

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Ultra-high magnetic eld
Signal-to-Noise Ratio (SNR) increases with the eld strength: SNR ∝ B1.65
0
Particularly suited for the T∗
2
-weighted imaging contrast1
2017: 7 Tesla MR systems FDA-cleared & CE-marked for clinical practice
1Zwanenburg et al. 2010, European radiology.

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Multi-channel array coils
Parallel imaging: collect multiple k-space data using a multi-receiver coil to boost the SNR 2.
Illustration of multi-receiver coil (phased array).
2Roemer et al. 1990, Magnetic Resonance in Medicine

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Multi-channel array coils
Parallel imaging: collect multiple k-space data using a multi-receiver coil to boost the SNR 2.
Ex-vivo baboon brain acquired at 7T with a single channel and 32-channel receiver coils.
Single channel acquisition. Multi-channel acquisition.
Courtesy Carole Lazarus
2Roemer et al. 1990, Magnetic Resonance in Medicine

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Highest in vivo Human Brain data acquired at 7T
Towards imaging at the meso-scale ( 100µm2 in plane)
Highest in-vivo Human Brain MRI data collected at 7T 3:
• B0 eld strength: 7T
• Multi-channel receiver coil: 32 Rx
• Resolution: 0.12 × 0.12 × 0.6mm3
• Field of View: 20.28 × 20.93 × 1.26 cm3
• Averages: 2
• Total scan time: 50 min
3Stucht et al. 2015, PloS one

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Highest in vivo Human Brain data acquired at 7T
Towards imaging at the meso-scale ( 100µm2 in plane)
Highest in-vivo Human Brain MRI data collected at 7T 3:
• B0 eld strength: 7T
• Multi-channel receiver coil: 32 Rx
• Resolution: 0.12 × 0.12 × 0.6mm3
• Field of View: 20.28 × 20.93 × 1.26 cm3
• Averages: 2
• Total scan time: 50 min
How can we speed-up the acquisition?
3Stucht et al. 2015, PloS one

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Acquisition in MRI
Data collection in MRI is typically performed over a Cartesian grid:

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Compressed Sensing in MRI: Down-sample the k-space
Speeding up the acquisition

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Compressed Sensing: Beyond Nyquist sampling rate
Sparsity
Compressed sensing4,5 (CS) relies on three main assumptions:
1. Sparsity: Image to be reconstructed is represented by only a few non-zero coe cients in an transformed
domain.
4Candès, Romberg, and Tao 2006, IEEE Transactions on information theory.
5Lustig, Donoho, and Pauly 2007, Magnetic Resonance in Medicine.

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Incoherence
1. Sparsity
2. Incoherence (local coherence8: µ(A) =
n
k=1
ak
2
∞
, with A = F Ψ)
Exact recovery with probability 1 − ε if m ≥ Cµ(A)s log(
n
ε
)
8Puy, Vandergheynst, and Wiaux 2011, IEEE Signal Processing Letters; Chau fert et al. 2014, SIAM Journal on Imaging Sciences.
8Lazarus et al. 2019, Magnetic Resonance in Medicine

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Incoherence
1. Sparsity
2. Incoherence (local coherence8: µ(A) =
n
k=1
ak
2
∞
, with A = F Ψ)
Exact recovery with probability 1 − ε if m ≥ Cµ(A)s log(
n
ε
)
Sparkling VDS9 Radial VDS
8Puy, Vandergheynst, and Wiaux 2011, IEEE Signal Processing Letters; Chau fert et al. 2014, SIAM Journal on Imaging Sciences.

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Nonlinear reconstruction: when sparsity meets incoherence
1. Sparsity
2. Local incoherence
3. Nonlinear sparsity-promoting reconstruction:
x = arg min
x∈CN
1
2
FΩS
x − yΩS
2
F
+ λ Ψx 1, (1)
with:
• S: total number of shots acquired (with C samples each)
• ΩS: support of under-sampling scheme in k-space
• FΩS
: under-sampled Fourier operator (e.g. NUFFTa for non-Cartesian
sampling)
• yΩS
: complex-values k-space measurements (S × C N)
• Ψ: sparsifying transform (e.g. wavelet transform (WT))
• λ > 0: Regularization parameter
• x: MR image estimate

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Compressed Sensing: O ine reconstruction
First scan then reconstruct
Two-step scan procedure:
First: Collect data Second: Perform image recon

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Successful application of Compressed Sensing in MRI10
Sparkling VDS Radial VDS

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Successful application of Compressed Sensing in MRI10
Sparkling VDS Radial VDS
Reconstruction takes 1-2 min for a 2D slice and the 3D
scan consists of 208 slices!

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Outline
High resolution MRI
Sampling in MRI
State-of the art
Summary
Summary

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Non-Cartesian multi-channel MR image reconstruction
State of the art method
Methods can be split in two categories:
1. Recovering a full FOV image: Self-Calibrated reconstruction
2. Recovering an image per channel: Calibrationless reconstruction

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Self-Calibrated reconstruction
1. Self-Calibrated reconstruction:
• Solve an inverse problem and recover a single full FOV image:
x = arg min
x∈CN
1
2
L
=1
σ−2 FΩ
S x − y 2
2
+ λ Ψx 1 (2)
• Model the coil sensitivity pro le S for all channels = 1, . . . , L11,12
11Samsonov et al. 2004, Magnetic Resonance in Medicine.
12Uecker et al. 2014, Magnetic Resonance in Medicine.

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Self-Calibrated reconstruction
1. Self-Calibrated reconstruction:
• Solve an inverse problem and recover a single full FOV image:
x = arg min
x∈CN
1
2
L
=1
σ−2 FΩ
S x − y 2
2
+ λ Ψx 1 (2)
• Model the coil sensitivity pro le S for all channels = 1, . . . , L11,12
• Coil sensitivity pro les are subject/scan-speci c → estimated/extracted for each scan
11Samsonov et al. 2004, Magnetic Resonance in Medicine.

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Calibrationless reconstruction
1. Self-calibrated methods, pros & cons:
• good reconstruction performances
• sensitivity pro les need to be set beforehand =⇒ pre-scan lengthen the acquisition
• estimation must be performed during the acquisition =⇒ lengthen the reconstruction

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Calibrationless reconstruction
1. Self-calibrated methods, pros & cons:
• good reconstruction performances
• sensitivity pro les need to be set beforehand =⇒ pre-scan lengthen the acquisition
• estimation must be performed during the acquisition =⇒ lengthen the reconstruction
2. Calibrationless reconstruction
• No longer need coil sensitivity pro les
• Exploit redundancy across channels to impose structured sparsity

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Problem statement
Calibration-less MR image reconstruction problem solved using an analysis formulation:
X = arg min
X∈CN×L
1
2
FΩ
X − Y 2
2
+ R(ΨX). (3)
with:
• Y = [y1, · · · , yL
] with y ∈ CM the th channel-speci c k-space
• X = [x1, · · · , xL
] with x ∈ CN the th channel-speci c reconstructed image.
• FΩ is the forward under-sampling Fourier operator
• Ψ ∈ CNΨ×N linear operator related to a sparse decomposition
• R is a convex regularization term that promotes sparsity with an explicit proximity operator.

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Optimization algorithm
Primal dual optimization
We aim to nd:
X ∈ argmin
X∈CN×L
[f (X) + R(ΨX)] (4)
where:
f is convex, di ferentiable on CN×L and its gradient is β-Lipschitz
R ∈ Γ0
(CNΨ×L) with a closed form proximity operator13, given by:
prox
R
(Z) = argmin
V ∈CNΨ×L
1
2
Z − V 2 + R(V ) (5)
Note: Those are standard assumptions in optimization-based image reconstruction methods.
13Moreau 1962, Comptes Rendus de l’Académie des Sciences de Paris.

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Optimization algorithm
Condat-V˜
u sequence
Using a primal-dual optimization method proposed by Condat14-V˜
u15:
Algorithm 1: Condat-V˜
u algorithm
Data: X0, Z0
Result: XK , ZK
initialize k = 0, τ > 0, κ > 0,;
while k ≤ K do
Xk+1 := Xk − τ ∇f (Xk ) + Ψ∗Zk ;
Wk+1 := Zk + κ Ψ (2Xk+1 − Xk );
Zk+1 := Wk+1 − κ proxg/κ
Wk+1
κ
;
• if
1
τ
− κ|||Ψ|||2 ≥
β
2
then the algorithm weakly converges to the solution of Eq. (4).
• τ and κ hyper-parameters set as follows: τ :=
1
β
, κ :=
β
2|||Ψ|||2
14Condat 2013, Journal of Optimization Theory and Applications.
15V˜
u 2013, Advances in computational mathematics.

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Calibrationless reconstruction: playing with clustering regularization
Clustering using the 2-norm
Clustering using the ∞-norm

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Regularizing through the Group-LASSO
Group-LASSO (GL)16 regularization has been used in multi-task learning in di ferent eld including MRI
reconstruction17.
Ψ =
16Yuan and Lin 2006, Journal of the Royal Statistical Society: Series B (Statistical Methodology).
17Majumdar and Ward 2012, Magnetic Resonance in Medicine.

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reconstruction17.
The group-LASSO penalty is de ned as follows:
RGL
(Z) =
g∈G
Zg 2
• G is the set of groups de ning a partition of Z Z =

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reconstruction17.
RGL
(Z) = Z 2,1
= λ
NΨ
p=1
L
=1
|zp |2
with:
• Z = ΨX
• λ is a positive hyper-parameter
Z =

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reconstruction17.
RGL
(Z) = Z 2,1
= λ
NΨ
p=1
L
=1
|zp |2
Group-LASSO regularization provides tighter recovery guarantees18
Z =
18Chun, Adcock, and Talavage 2016, IEEE Transactions on Medical Imaging

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Joint sparsity regularization
Octagonal Shrinkage and Clustering Algorithm for Regression
Inferring the structure via a pairwise ∞ norm.
OSCAR regularization19:
Rc-OSCAR
(Z) =
NΨ
p=1
λ
L
=1
|zpj | + γ
max{|zp |, |zpk |}
=
NΨ
p=1
λ
L
=1
(γ( − 1) + 1) |zp |↓
where:
Z↓
∈ CNΨ×L the wavelet coe cients sorted in decreasing order, i.e.:
∀p ∈ N, |zp1| ≤ · · · ≤ |zpL|.
λ and γ are some positive hyper-parameters that need to be set
Figure: OSCAR’s unit ball
Note: OSCAR has an explicit proximity operator
19Bondell and Reich 2008, Biometrics.

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Experimental set-up
Sequence parameters:
• Ex-vivo human brain
• 7T Siemens Scanner
• 1Tx/32Rx Nova coil
• GRE Sparkling trajectory
• 0.390 x 0.390 x 1.5 m3 resolution
• Acceleration factor of 20 in time
• Under-sampling factor of 2.5
• Ψ: Daubechies 4 transform
Figure: Sparkling trajectory Figure: Ex-vivo Human brain
Hyper-parameters set using a grid-search procedure to maximize the SSIM score.
Cartesian scan 512×512 was acquired and used for reference.

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Results
Similarity score
iFT 1-ESPIRIT20 p-LORAKS21 group-LASSO22 c-OSCAR23 k-support norm
SSIM 0.884 0.885 0.753 0.897 0.901 0.900
pSNR 28.25 26.48 25.52 28.59 29.77 30.29
NRMSE 0.1911 0.2276 0.2536 0.1859 0.1604 0.1510
21Haldar and Zhuo 2016, Magnetic resonance in medicine.
23El Gueddari et al. 2019, ISBI 2019.

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Results
Magnitude images
Reference 1-ESPIRIT p-LORAKS
group-LASSO c-OSCAR k-support norm

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Interim summary
Summary:
• Investigate two classes of MR image reconstruction methods for non-Cartesian parallel imaging acquisitions
• Impose structured sparsity for calibrationless MR image reconstruction
• Achieve same image quality in the calibrationless setting as in the self-calibrated one at the cost of a larger
memory footprint and computation cost
→ How can we speed-up the reconstruction.

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Outline
High resolution MRI
Sampling in MRI
State-of the art
Summary
Summary

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Speeding-up the reconstruction
Structural MRI acquisition is segmented in time

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Online CS-MRI reconstruction
Single-channel receiver case
Multi-shot acquisition: ∀i ∈ {1, . . . , S}, collect data yi over Γi support with |Γi | = C

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Online CS-MRI reconstruction
Multi-shot acquisition: ∀i ∈ {1, . . . , S}, collect data yi over Γi support with |Γi | = C
Denote ωk
= {Γ(k−1)∗bs +1
, . . . , Γk∗bs
} the bs-dimensional set of shots to be added to the kth mini-batch
de ned as follows: Ωk
= ∪k
i=1
ωi , with k ≤ S/bs
Solve partial image reconstruction:
∀k ∈ {1, . . . , S}, xk = arg min
x∈CN
S
2k
FΩk
x − yΩk
2
F
+ λ Ψx 1, (6)
with:
• FΩk
: undersampled Fourier operator over Ωk
• yΩk
: complex-valued k-space measurements over Ωk
• xk : estimated MR image from incomplete data yΩk
Online time constraints:
nb × Tit
≈ bs × TR.
The problem is convex → nal solution does not depend on the initialization.

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Online optimization algorithm
To solve Pb (6), we adapted the Condat-V˜
u algorithm to the online framework.
Algorithm 1: Online mini-batch reconstruction algorithm for solving Problem (6).
1 initialize k = bs , x0, v0;
2 while k ≤ S − bs do
3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Primal variable

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Dual variable

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Primal gradient step

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Dual proximal step

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Condat-V˜
u sequence

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Warm restart

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3 τk = 1/βk ;
4 κk = βk /(2|||Ψ|||2);
5 for t = 1, 2, . . . , nb do
6 xk
t
= xk
t−1
− τk ∇fΩk
(xk
t−1
) + Ψ∗vk
t−1
;
7 wk
t
= vk
t−1
+ κk Ψ 2xk
t
− xk
t−1
;
8 vk
t
= wk
t
− κk proxg/κk
wk
t
/κk ;
9 (xk+bs
0
, vk+bs
0
) ← (xk
nb
, vk
nb
);
10 k ← k + bs ;
Concatenate batches

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Acquisition parameters for prospective CS
Acquisition parameters
• Field strength: 7 Tesla
• Repetition Time (TR): 550 ms
• Echo Time (TE): 30 ms
• Field of View: 20.4 cm
• Resolution: 0.4 × 0.4 × 3 mm3
• Matrix size: 512 × 512
• Trajectory: Sparkling26
• Number of shots (S): 34
• Acceleration/undersampling factors: 15/2.5
Cartesian Reference
Figure: Ex-vivo baboon brain
Non-Cartesian sampling
Figure: 15-fold accelerated Sparkling
trajectories

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Reconstruction parameters for online image processing
Online processing constraints
nb × Tit
≈ bs × TR.
• Fourier Transform: GPU NUFFT27
• Time per iteration: Tit
78.2 ± 8.9 ms
• Nb. shots/mini-batch: bs ∈ {1, 2, 17, 34}
• Nb. iterations/mini-batch: nb
∈ {5, 11, 93, 200}
Regularization term
• Sparse transform: 2D decimated Symmlet-8 WT
• Hyper-parameter: λ retrospectively tuned
27Lin 2018, Journal of Imaging

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Non-Cartesian online CS-MR image reconstruction
Figure: Evolution of the global cost function (left) and SSIM score (right) over time.

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Figure: Evolution of the global cost function (left) and SSIM score (right) over time.
=⇒ The smaller the mini-batch size, the sooner and better the image solution.

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Evolution of the output xk over the acquisition, mini-batch size of 1.
Figure: Collected shots over time Figure: Evolution of xk over time

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Online reconstruction
Extension to the multi-channel acquisition
Reconstruction method for multi-channel non-Cartesian acquisitions:
1. Self-Calibrated reconstruction
Provides good reconstruction
Requires the coil sensitivity maps estimation
Gradient-Lipschitz constant depends on the sensitivity pro le estimation
2. Calibrationless reconstruction
Slightly better reconstruction
Does not rely on coil sensitivity pro les
Lipschitz constant only depends on the sampling pattern
→ More suited for online reconstruction
27El Gueddari et al. 2020

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Extension to the multi-channel acquisition
Rely on calibrationless reconstruction:
Xk = arg min
X∈CN×L
1
2
FΩk
X − YΩk
2
2
+ R(ΨX). (7)
with sub-band version of OSCAR regularization28:
Z = ΨX, Rb-OSCAR
(Z) =
S
s=1
B
b=1
PbL
=1
λ|zs,b,
| + γ
S
s=1
B
b=1 1≤ < ≤PbL
max{|zs,b,
|, |zs,b,
|}, (8)
λ > 0, γ > 0.
28El Gueddari et al. 2020

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Acquisition parameters for prospective CS
Multi-channel receiver case
Acquisition parameters
• Field strength: 7 Tesla
• Nb of coils (L): 32
• Repetition Time (TR): 550 ms
• Field of View: 20.4 cm
• Resolution: 0.4 × 0.4 × 1.5 mm3
• Trajectory: Sparkling29
• Number of spokes (S): 34
• Acceleration/undersampling factors: 15/2.5
Cartesian reference
Figure: Ex-vivo human brain
Non-Cartesian sampling
Figure: 15-fold accelerated Sparkling
trajectories

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Reconstruction parameters for online image processing
Online processing constraints
nb × Tit
≈ bs × TR.
• Fourier Transform: GPU NUFFT30
• Time per iteration: Tit
4.29 ± 0.111 s
• Nb. shots/mini-batch: bs ∈ {17, 34}
• Nb. iterations/mini-batch: nb
∈ {1, 200}
Regularization term
• Sparse transform: 2D decimated Symmlet-8 WT
• Hyper-parameter: (λ, γ) retrospectively tuned
30Lin 2018, Journal of Imaging

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Results: Prospective non-Cartesian reconstruction
Online multi-channel reconstruction
Figure: Evolution of the global cost function (left) and estimated image obtained by the end of acquisition.

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Pro ling computation time
Which term contributes the most to the global cost function?

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Gradient step Proximity op. step
Linear Operator
Total time per iteration
direct adjoint
S = 34 750 ms ± 32.2 ms 847 ms ± 17.9 ms 998 ms ± 15.8 ms 667 ms ± 16.2 ms 4.29 s ± 111 ms

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• The contribution of the data delity term is predominant during the rst iterations.
• The gradient step only takes a quarter of computing time per iteration.
=⇒ During acquisition: minimizing only the smooth term.

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The online trick for parallel imaging
Algorithm 1: A fast method for online CS+PI calibration-less reconstruction
initialize k = bs , X0, Z0;
while k ≤ S − bs do
for t = 1, 2, . . . , nb do
Xk
t
= Xk
t−1
− ∇fΩk
(Xk
t−1
) /βk ;
Xk+bs
0
← Xk
nb
;
k ← k + bs ;
ZS
0
← ΨXS
0
;
τ ← 1/βS ;
κ = βS /(2|||Ψ|||2);
for t = 1, 2, . . . , 200 do
XS
t
= XS
t−1
− τ ∇fΩS
(XS
t−1
) + Ψ∗ZS
t−1
;
W S
t
= ZS
t−1
+ κΨ 2XS
t
− XS
t−1
;
ZS
t
= W S
t
− κ proxg/κ
W S
t
/κ ;
Adaptive Gradient step
Solving calibrationless problem
via Condat-V˜
u algorithm

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Online calibration-less CS-PI reconstruction
Minimizing only the data- delity term allowed us to reduce the mini-batch size to bs
= 2 shots.
With the online trick

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Comparison with the fully online framework
First approach With the online trick

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Recovered solution at convergence
Reference Calibrationless reconstruction

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Partial summary
• An online reconstruction approach for 2D MR image reconstruction
• A Calibrationless version to make this online framework compatible with multi-channel acquisition and
reconstruction
• Small batches allow to achieve high image quality by the end of acquisition.
• Delivery of good images by the end of acquisitions
• The bigger the computational resources (GPU), the better the image quality!

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Open source software: A fruit of the COSMIC project
Python package for image reconstruction: Python Sparse data Analysis Package (PySAP)
• Free open source Python package
• pysap-mri: dedicated plugin for MRI reconstruction 31
• Unit-tested & Continuous integration
• pysap-tutorials: Examples, Tutorials
• Documentation
31El Gueddari et al. 2020, ISMRM workshop on Data Sampling

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Outline
High resolution MRI
Sampling in MRI
State-of the art
Summary
Summary

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Conclusion & Outlook
Conclusion:
• Propose a provably convergent online reconstruction framework
• Implement a mini-batch formulation that allows to interleave acquisition & recon and favor small batch sizes
• Propose many algorithmic accelerations: GPU NUFFT, warm-restart, online trick, density compensation
• Provide a feedback during the scan
• Share the code and implement online recon on the scanner (Gadgetron)

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Conclusion & Outlook
Conclusion:
• Propose a provably convergent online reconstruction framework
• Implement a mini-batch formulation that allows to interleave acquisition & recon and favor small batch sizes
• Propose many algorithmic accelerations: GPU NUFFT, warm-restart, online trick, density compensation
• Provide a feedback during the scan
• Share the code and implement online recon on the scanner (Gadgetron)
Outlook:
• Faster optimization algorithms (greedy-FISTA32, POGM33) for orthogonal WT
• Further parallelization needed for 3D imaging
32Liang and Schönlieb 2019, Convergence
33Taylor, Hendrickx, and Glineur 2017, SIAM Journal on Optimization

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Bibliography I
Bondell, H.D. and B.J. Reich (2008). “Simultaneous regression shrinkage, variable selection, and supervised
clustering of predictors with OSCAR”. In: Biometrics 64.1, pp. 115–123.
Candès, Emmanuel J., J. Romberg, and T. Tao (2006). “Robust uncertainty principles: exact signal reconstruction
from highly incomplete frequency information”. In: IEEE Transactions on information theory 52.2, pp. 489–509.
Chau fert, Nicolas et al. (2014). “Variable density sampling with continuous trajectories”. In: SIAM Journal on
Imaging Sciences 7.4, pp. 1962–1992.
Chun, I.Y., B. Adcock, and T.M. Talavage (2016). “E cient compressed sensing SENSE pMRI reconstruction with
joint sparsity promotion”. In: IEEE Transactions on Medical Imaging 35.1, pp. 354–368.
Condat, Laurent (2013). “A primal–dual splitting method for convex optimization involving Lipschitzian,
proximable and linear composite terms”. In: Journal of Optimization Theory and Applications 158.2,
pp. 460–479.
El Gueddari, L. et al. (Jan. 2020). Calibration-less parallel imaging compressed sensing reconstruction based on
OSCAR regularization. IEEE Transactions on Medical Imaging. Saclay, France: CEA/NeuroSpin & INRIA
Saclay & CVN Centrale-Supélec.

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Bibliography II
El Gueddari, Loubna et al. (2019). “Calibrationless OSCAR-based image reconstruction in compressed sensing
parallel MRI”. In: ISBI 2019.
El Gueddari, Loubna et al. (2020). “PySAP-MRI: a Python Package for MR Image Reconstruction”. In: ISMRM
workshop on Data Sampling.
Fessler, Je frey A and Bradley P Sutton (2003). “Nonuniform fast Fourier transforms using min-max interpolation”.
In: IEEE transactions on signal processing 51.2, pp. 560–574.
Haldar, Justin P and Jingwei Zhuo (2016). “P-LORAKS: Low-rank modeling of local k-space neighborhoods with
parallel imaging data”. In: Magnetic resonance in medicine 75.4, pp. 1499–1514.
Lazarus, Carole et al. (2019). “SPARKLING: variable-density k-space lling curves for accelerated T2*-weighted
MRI”. In: Magnetic Resonance in Medicine 81.6, pp. 3643–3661.
Liang, Jingwei and Carola-Bibiane Schönlieb (2019). “Improving “Fast Iterative Shrinkage-Thresholding
Algorithm”: Faster, Smarter and Greedier”. In: Convergence 50, p. 12.
Lin, Jyh-Miin (2018). “Python Non-Uniform Fast Fourier Transform (PyNUFFT): An Accelerated Non-Cartesian
MRI Package on a Heterogeneous Platform (CPU/GPU)”. In: Journal of Imaging 4.3, p. 51.

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Bibliography III
Lustig, Miki, D. Donoho, and John M. Pauly (2007). “Sparse MRI: The application of compressed sensing for
rapid MR imaging”. In: Magnetic Resonance in Medicine 58.6, pp. 1182–1195.
Majumdar, Angshul and Rabab Ward (2012). “Calibration-less multi-coil MR image reconstruction”. In: Magnetic
Resonance in Medicine 30.7, pp. 1032–1045.
Moreau, Jean Jacques (1962). “Fonctions convexes duales et points proximaux dans un espace hilbertien”. In:
Comptes Rendus de l’Académie des Sciences de Paris.
Pohmann, Rolf, Oliver Speck, and Klaus Sche er (2016). “Signal-to-noise ratio and MR tissue parameters in
human brain imaging at 3, 7, and 9.4 tesla using current receive coil arrays”. In: Magnetic resonance in medicine
75.2, pp. 801–809.
Puy, Gilles, Pierre Vandergheynst, and Yves Wiaux (2011). “On variable density compressive sampling”. In: IEEE
Signal Processing Letters 18.10, pp. 595–598.
Roemer, P.B. et al. (1990). “The NMR phased array”. In: Magnetic Resonance in Medicine 16.2, pp. 192–225.
Samsonov, Alexei A et al. (2004). “POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic
resonance imaging”. In: Magnetic Resonance in Medicine 52.6, pp. 1397–1406.

View Slide

Bibliography IV
Stucht, Daniel et al. (2015). “Highest resolution in vivo human brain MRI using prospective motion correction”.
In: PloS one 10.7, e0133921.
Taylor, Adrien B, Julien M Hendrickx, and François Glineur (2017). “Exact worst-case performance of rst-order
methods for composite convex optimization”. In: SIAM Journal on Optimization 27.3, pp. 1283–1313.
Uecker, M. et al. (2014). “ESPIRiT– an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets
GRAPPA”. In: Magnetic Resonance in Medicine 71.3, pp. 990–1001.
V˜
u, B.C (2013). “A splitting algorithm for dual monotone inclusions involving cocoercive operators”. In: Advances
in computational mathematics 38.3, pp. 667–681.
Yuan, Ming and Yi Lin (2006). “Model selection and estimation in regression with grouped variables”. In: Journal
of the Royal Statistical Society: Series B (Statistical Methodology) 68.1, pp. 49–67.
Zwanenburg, Jaco JM et al. (2010). “Fluid attenuated inversion recovery (FLAIR) MRI at 7.0 Tesla: comparison
with 1.5 and 3.0 Tesla”. In: European radiology 20.4, pp. 915–922.

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Philippe Ciuciu

Philippe Ciuciu

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