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Philippe Ciuciu

S³ Seminar
October 09, 2020

Philippe Ciuciu

(CEA/NeuroSpin & Inria Saclay Île-de-France Parietal)

https://s3-seminar.github.io/seminars/philippe-ciuciu/

Title — Structured Sparsity Regularization for online MR Image Reconstruction in Accelerated T2* Imaging

Abstract — Reducing acquisition time is a major challenge in high-resolution MRI that has been successfully addressed by Compressed Sensing (CS) theory. While the scan time has been massively accelerated, the complexity of image recovery algorithms has strongly increased, resulting in slower reconstruction processes. In this work we propose an online approach to shorten image reconstruction times in the CS setting. We leverage the segmented acquisition of anatomical MR data in multiple shots to interleave the MR acquisition and image reconstruction steps. This approach is particularly appealing for 2D high-resolution T2*-weighted anatomical imaging. During the scan, acquired shots are stacked together to form mini-batches and image reconstruction may start from incomplete data. We demonstrate the interest and time savings of this online image reconstruction framework for Cartesian and non-Cartesian sampling strategies combined with a single receiver coil. Next, we further extend this formalism to address the more challenging case of multi-receiver phased array acquisition. In this setting, calibrationless image reconstruction leverages structured sparsity regularization to remain compatible with the timing constraints of online image delivery. Our results on ex-vivo 2D T2*-weighted brain images show that high-quality MR images are recovered by the end of acquisition in both acquisition setups.

Biography — Dr. Philippe Ciuciu obtained his PhD in electrical engineering from the University of Paris-Sud in 2000 and his Habilitation degree in 2008. Dr. Ciuciu is now CEA Research Director at NeuroSpin where he has led, since 2018, the Compressed Sensing group in the Inria-CEA Parietal team at NeuroSpin. Dr. Ciuciu’s work has led to more than 200 research outputs including more than 50 peer-reviewed articles in international journals such as SIAM Imaging Sciences, IEEE Trans. on Signal Processing, IEEE Trans. on Medical Imaging, Medical Image Analysis, NeuroImage, Magnetic Resonance in Medicine, etc. He also holds 2 MRI-related patents. His current research interests are in developing accelerated acquisition and image reconstruction techniques, including deep learning techniques, for magnetic resonance imaging (MRI) with applications in clinical and cognitive neuroscience at 3 and 7 Tesla.
As IEEE Senior Member, he has represented the IEEE Signal Processing Society in the International Symposium on Biomedical Imaging for the 2019-2020 period. He has also been appointed to take part to the steering committee of the 2021 ESMRMB conference in Barcelona. Since 2019 he holds a position as Senior Area Editor for the IEEE open Journal of Signal Processing and that of Vice Chair for the Biomedical Image and Signal Analytics (BISA) technical committee of the EURASIP society. In 2020, he has been appointed as Associate Editor to Frontiers in Neuroscience, section Brain imaging methods.

References

[1] El Gueddari L, Ciuciu P, Chouzenoux E, Vignaud A, Pesquet JC. Calibrationless OSCAR-based image reconstruction in compressed sensing parallel MRI. In2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019) Apr 8, 2019 (pp. 1532-1536). IEEE.

[2] El Gueddari L, Chouzenoux E, Vignaud A, Pesquet JC, Ciuciu P. Online MR image reconstruction for compressed sensing acquisition in T2* imaging. InWavelets and Sparsity XVIII Sep 9, 2019 (Vol. 11138, p. 1113819). International Society for Optics and Photonics.

[3] El Gueddari L, Chouzenoux E, Vignaud A, Ciuciu P. Calibration-less parallel imaging compressed sensing reconstruction based on OSCAR regularization. https://hal.inria.fr/hal-02292372/document

S³ Seminar

October 09, 2020
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  1. 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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  2. 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
    Experiments & Results
    Summary
    4 Conclusion & Outlook
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 2 / 50

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  3. 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
    Experiments & Results
    Summary
    4 Conclusion & Outlook
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 3 / 50

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  4. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 4 / 50

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  5. High resolution imaging: Increasing the SNR
    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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 4 / 50

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  6. High resolution imaging: Increasing the SNR
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 5 / 50

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  7. High resolution imaging: Increasing the SNR
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 5 / 50

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  8. 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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 6 / 50

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  9. 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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 6 / 50

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  10. Acquisition in MRI
    Data collection in MRI is typically performed over a Cartesian grid:
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 7 / 50

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  11. Acquisition in MRI
    Data collection in MRI is typically performed over a Cartesian grid:
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 7 / 50

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  12. Acquisition in MRI
    Data collection in MRI is typically performed over a Cartesian grid:
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 7 / 50

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  13. Acquisition in MRI
    Data collection in MRI is typically performed over a Cartesian grid:
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 7 / 50

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  14. Compressed Sensing in MRI: Down-sample the k-space
    Speeding up the acquisition
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 8 / 50

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  15. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 9 / 50

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  16. Compressed Sensing: Beyond Nyquist sampling rate
    Incoherence
    Compressed sensing6,7 (CS) relies on three main assumptions:
    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
    )
    6Candès, Romberg, and Tao 2006, IEEE Transactions on information theory.
    7Lustig, Donoho, and Pauly 2007, Magnetic Resonance in Medicine.
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 10 / 50

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  17. Compressed Sensing: Beyond Nyquist sampling rate
    Incoherence
    Compressed sensing6,7 (CS) relies on three main assumptions:
    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
    6Candès, Romberg, and Tao 2006, IEEE Transactions on information theory.
    7Lustig, Donoho, and Pauly 2007, Magnetic Resonance in Medicine.
    8Puy, Vandergheynst, and Wiaux 2011, IEEE Signal Processing Letters; Chau fert et al. 2014, SIAM Journal on Imaging Sciences.
    9Lazarus et al. 2019, Magnetic Resonance in Medicine
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 10 / 50

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  18. Compressed Sensing: Beyond Nyquist sampling rate
    Incoherence
    Compressed sensing6,7 (CS) relies on three main assumptions:
    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
    6Candès, Romberg, and Tao 2006, IEEE Transactions on information theory.
    7Lustig, Donoho, and Pauly 2007, Magnetic Resonance in Medicine.
    8Puy, Vandergheynst, and Wiaux 2011, IEEE Signal Processing Letters; Chau fert et al. 2014, SIAM Journal on Imaging Sciences.
    9Lazarus et al. 2019, Magnetic Resonance in Medicine
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 10 / 50

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  19. Compressed Sensing: Beyond Nyquist sampling rate
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 11 / 50

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  20. Compressed Sensing: O ine reconstruction
    First scan then reconstruct
    Two-step scan procedure:
    First: Collect data Second: Perform image recon
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 12 / 50

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  21. Compressed Sensing: Beyond Nyquist sampling rate
    Successful application of Compressed Sensing in MRI10
    Sparkling VDS Radial VDS
    9Lazarus et al. 2019, Magnetic Resonance in Medicine
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 13 / 50

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  22. Compressed Sensing: Beyond Nyquist sampling rate
    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!
    9Lazarus et al. 2019, Magnetic Resonance in Medicine
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 13 / 50

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  23. 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
    Experiments & Results
    Summary
    4 Conclusion & Outlook
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 14 / 50

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  24. 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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 15 / 50

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  25. Non-Cartesian multi-channel MR image reconstruction
    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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 16 / 50

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  26. Non-Cartesian multi-channel MR image reconstruction
    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.
    12Uecker et al. 2014, Magnetic Resonance in Medicine.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 16 / 50

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  27. Non-Cartesian multi-channel MR image reconstruction
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 17 / 50

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  28. Non-Cartesian multi-channel MR image reconstruction
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 17 / 50

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  29. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 18 / 50

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  30. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 19 / 50

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  31. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 20 / 50

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  32. Calibrationless reconstruction: playing with clustering regularization
    Clustering using the 2-norm
    Clustering using the ∞-norm
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 21 / 50

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  33. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 22 / 50

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  34. Regularizing through the Group-LASSO
    Group-LASSO (GL)16 regularization has been used in multi-task learning in di ferent eld including MRI
    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 =
    16Yuan and Lin 2006, Journal of the Royal Statistical Society: Series B (Statistical Methodology).
    17Majumdar and Ward 2012, Magnetic Resonance in Medicine.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 22 / 50

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  35. Regularizing through the Group-LASSO
    Group-LASSO (GL)16 regularization has been used in multi-task learning in di ferent eld including MRI
    reconstruction17.
    The group-LASSO penalty is de ned as follows:
    RGL
    (Z) = Z 2,1
    = λ

    p=1
    L
    =1
    |zp |2
    with:
    • Z = ΨX
    • λ is a positive hyper-parameter
    Z =
    16Yuan and Lin 2006, Journal of the Royal Statistical Society: Series B (Statistical Methodology).
    17Majumdar and Ward 2012, Magnetic Resonance in Medicine.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 22 / 50

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  36. Regularizing through the Group-LASSO
    Group-LASSO (GL)16 regularization has been used in multi-task learning in di ferent eld including MRI
    reconstruction17.
    The group-LASSO penalty is de ned as follows:
    RGL
    (Z) = Z 2,1
    = λ

    p=1
    L
    =1
    |zp |2
    Group-LASSO regularization provides tighter recovery guarantees18
    Z =
    16Yuan and Lin 2006, Journal of the Royal Statistical Society: Series B (Statistical Methodology).
    17Majumdar and Ward 2012, Magnetic Resonance in Medicine.
    18Chun, Adcock, and Talavage 2016, IEEE Transactions on Medical Imaging
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 22 / 50

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  37. Joint sparsity regularization
    Octagonal Shrinkage and Clustering Algorithm for Regression
    Inferring the structure via a pairwise ∞ norm.
    OSCAR regularization19:
    Rc-OSCAR
    (Z) =

    p=1
    λ
    L
    =1
    |zpj | + γ
    max{|zp |, |zpk |}
    =

    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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 23 / 50

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  38. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 24 / 50

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  39. 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
    20Uecker et al. 2014, Magnetic Resonance in Medicine.
    21Haldar and Zhuo 2016, Magnetic resonance in medicine.
    22Majumdar and Ward 2012, Magnetic Resonance in Medicine.
    23El Gueddari et al. 2019, ISBI 2019.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 25 / 50

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  40. Results
    Magnitude images
    Reference 1-ESPIRIT p-LORAKS
    group-LASSO c-OSCAR k-support norm
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 26 / 50

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  41. Results
    Magnitude images
    Reference 1-ESPIRIT p-LORAKS
    group-LASSO c-OSCAR k-support norm
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 26 / 50

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  42. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 27 / 50

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  43. 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
    Experiments & Results
    Summary
    4 Conclusion & Outlook
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 28 / 50

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  44. Speeding-up the reconstruction
    Structural MRI acquisition is segmented in time
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 29 / 50

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  45. Speeding-up the reconstruction
    Structural MRI acquisition is segmented in time
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 29 / 50

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  46. Online CS-MRI reconstruction
    Single-channel receiver case
    Multi-shot acquisition: ∀i ∈ {1, . . . , S}, collect data yi over Γi support with |Γi | = C
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 30 / 50

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  47. Online CS-MRI reconstruction
    Single-channel receiver case
    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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 30 / 50

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  48. Online optimization algorithm
    Single-channel receiver case
    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 ;
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  49. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Primal variable
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  50. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Dual variable
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  51. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Primal gradient step
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  52. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Dual proximal step
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  53. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Condat-V˜
    u sequence
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  54. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Warm restart
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  55. Online optimization algorithm
    Single-channel receiver case
    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 ;
    Concatenate batches
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 31 / 50

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  56. Acquisition parameters for prospective CS
    Single-channel receiver case
    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
    26Lazarus et al. 2019, Magnetic Resonance in Medicine
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  57. Reconstruction parameters for online image processing
    Single-channel receiver case
    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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  58. Non-Cartesian online CS-MR image reconstruction
    Single-channel receiver case
    Figure: Evolution of the global cost function (left) and SSIM score (right) over time.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 34 / 50

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  59. Non-Cartesian online CS-MR image reconstruction
    Single-channel receiver case
    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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 34 / 50

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  60. Non-Cartesian online CS-MR image reconstruction
    Single-channel receiver case
    Evolution of the output xk over the acquisition, mini-batch size of 1.
    Figure: Collected shots over time Figure: Evolution of xk over time
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 35 / 50

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  61. 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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 36 / 50

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  62. Online reconstruction
    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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  63. 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
    29Lazarus et al. 2019, Magnetic Resonance in Medicine
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  64. Reconstruction parameters for online image processing
    Multi-channel receiver case
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 38 / 50

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  65. 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 39 / 50

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  66. Pro ling computation time
    Online multi-channel reconstruction
    Which term contributes the most to the global cost function?
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 40 / 50

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  67. Pro ling computation time
    Online multi-channel reconstruction
    Which term contributes the most to the global cost function?
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 40 / 50

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  68. Pro ling computation time
    Online multi-channel reconstruction
    Which term contributes the most to the global cost function?
    • 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.
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 40 / 50

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  69. The online trick for parallel imaging
    Multi-channel receiver case
    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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 41 / 50

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  70. Online calibration-less CS-PI reconstruction
    Multi-channel receiver case
    Minimizing only the data- delity term allowed us to reduce the mini-batch size to bs
    = 2 shots.
    With the online trick
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 42 / 50

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  71. Online calibration-less CS-PI reconstruction
    Multi-channel receiver case
    Comparison with the fully online framework
    First approach With the online trick
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 42 / 50

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  72. Online calibration-less CS-PI reconstruction
    Multi-channel receiver case
    Recovered solution at convergence
    Reference Calibrationless reconstruction
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 42 / 50

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  73. Online reconstruction
    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!
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 43 / 50

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  74. 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
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 44 / 50

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  75. 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
    Experiments & Results
    Summary
    4 Conclusion & Outlook
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 45 / 50

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  76. 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)
    P. Ciuciu (NeuroSpin) Online CS-MR image reconstruction 46 / 50

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  77. 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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