Joint k-q space Compressed Sensing Diffusion MRI in ISMRM 2014

Joint k-q Space Compressed Sensing for Accelerated
Multi-Shell Acquisition and Reconstruction of the
diffusion signal and Ensemble Average Propagator

Transcript

1. Joint k-q Space Compressed Sensing for Accelerated
   Multi-Shell Acquisition and Reconstruction of the
   diffusion signal and Ensemble Average Propagator
   Jian Cheng, Dinggang Shen, Pew-Thian Yap
   May 14, 2014
   

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2. Declaration
   

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3. Outline
   1 Motivation and Objective
   2 6D-CS-dMRI
   3 Experiments
   4 Conclusion
   

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4. Outline
   1 Motivation and Objective
   2 6D-CS-dMRI
   3 Experiments
   4 Conclusion
   

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5. Reconstruction in 3D space
   Reconstruction problem: {S(x, qj )} → P(x, R)
   Fourier transform [Callaghan 1991]
   S(x, q) = S(x, 0) R∈R3
   P(x, R)e−2πiqT RdR
   

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6. Reconstruction in 3D space
   Reconstruction problem: {S(x, qj )} → P(x, R)
   Fourier transform [Callaghan 1991]
   S(x, q) = S(x, 0) R∈R3
   P(x, R)e−2πiqT RdR
   Diffusion Tensor Imaging (DTI) [Basser 1994]
   : assumption is too strong to
   be correct
   Diffusion Spectrum Imaging (DSI) [Wedeen 2000, 2005]
   : assumption is too
   week to be practical
   

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7. Compressed Sensing dMRI in 3D q-space
   Compressed Sensing (CS): reduce the number of samples, keep
   reconstruction quality. [Donoho 2006; Cand`
   es 2006]
   

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8. Compressed Sensing dMRI in 3D q-space
   Compressed Sensing (CS): reduce the number of samples, keep
   reconstruction quality. [Donoho 2006; Cand`
   es 2006]
   CS-dMRI using 3D continuous sparse dictionary
   Simple Harmonic Oscillator based Reconstruction and Estimation
   (SHORE) [¨
   Ozarslan 2009, 2013]
   Spherical Polar Fourier Imaging (SPFI) [Assemlal 2009, Cheng 2010, 2011, 2013],
   Significant reduction of scanning time?
   

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9. Outline
   1 Motivation and Objective
   2 6D-CS-dMRI
   3 Experiments
   4 Conclusion
   

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10. CS-dMRI in joint 6D k-q space (6D-CS-dMRI)
    Reconstruction problem: {S(km , qj )} → P(x , R)
    S(k, q) = x∈R3
    S(x, 0) R∈R3
    P(x, R)e−2πiqT RdRe−2πixT kdx
    Joint sampling in both k-space and q-space. [Mani 2012, 2014; Awate 2013]
    total scanning time reduction: R = RkRq.
    

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11. A naive way of 6D-CS-dMRI
    Two steps: {S(km , qj)} → {S(x , qj )} → P(x, R)
    Pros:
    Fast, easy to be implemented: volume by volume [Lustig 2007],
    voxel by voxel [Cheng 2013]
    

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12. A naive way of 6D-CS-dMRI
    Two steps: {S(km , qj)} → {S(x , qj )} → P(x, R)
    Pros:
    Fast, easy to be implemented: volume by volume [Lustig 2007],
    voxel by voxel [Cheng 2013]
    Cons:
    Not consider the relationship between diffusion signals in the same
    voxel across different DWI volumes.
    Not consider the relationship between diffusion signals in the same
    DWI volume across different voxels.
    

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13. 6D-CS-dMRI using joint optimization
    Reconstruction problem: {S(km , qj )} → P(x , R)
    min
    {ci },{sv}
    Nq
    v=1
    Fp
    sv − sv
    2
    2
    + λ1
    TV(sv
    ) + λ2
    Φsv 1
    + λ3
    Ns
    i=1
    ci 1
    s.t. Mci
    = Re{si} ∀i,
    Nq: number of volumes, v = 1, 2, . . . , Nq
    Ns: number of voxels, i = 1, 2, . . . , Ns
    sv: measurements in k-space for v-th volume
    sv: DWI signal vector in v-th volume
    si: DWI signal vector in i-th voxel
    Fp: partial Fourier transform
    Φ: wavelet basis
    ci: coefficient vector in i-th voxel
    M: basis matrix with basis elements in its columns
    λ1, λ2 , λ3 : regularization parameters
    Re: real part
    

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14. Alternating Direction Method of Multipliers (ADMM)
    Iterative updates in ADMM [Boyd 2011]
    :
    1 update {ci} voxel by voxel:
    c(k+1)
    i
    := arg min
    c
    0.5ρ Mc − (s(k)
    i
    − U(k)
    i
    )
    2
    2
    + λ3
    c 1
    2 update {sv} volume by volume:
    s(k+1)
    v
    := arg min
    s
    Fp
    s − sv
    2
    2
    + λ1
    TV(s) + λ2
    Φs 1
    +0.5ρ sk
    v
    ({ci}) − s + U(k)
    v 2
    3 Lagrangian variable update:
    U(k+1)
    i
    := U(k)
    i
    + Mc(k)
    i
    − s(k)
    i
    Repeat until the change of {ci} is small enough
    

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15. 6D-CS-dMRI using DL-SPF basis
    Reconstruction problem: {S(km, qj)} → S(x, q) → P(x, R)
    DL-SPF basis: [Cheng MICCAI 2013]
    Continuous representation.
    Analytical solutions from DWI signal to the EAP and ODF.
    Sparser than SPF basis and SHORE basis
    

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16. Outline
    1 Motivation and Objective
    2 6D-CS-dMRI
    3 Experiments
    4 Conclusion
    

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17. Evaluation
    Sampling Scheme in Diffusion Spectrum Imaging (DSI)
    I 514 samples in q-space, one b0 image, bmax = 8000 s/mm2, about 1
    hour
    I Nx × Ny × Nz × 514
    11-fold subsampling jointly in k-q space
    I 3-fold random sampling in k-space
    I 3.7-fold random sampling with 138 samples in q-space (3-fold,
    remove antipodal symmetric samples)
    I Nx × Ny × Nz × 138
    

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18. Evaluation
    Correctness and Consistency
    I Synthetic data:
    test correctness by comparing the reconstruction result using 11-fold
    subsampling with the ground truth
    I Real data
    test consistency by comparing the reconstruction result using 11-fold
    subsampling with the reconstruction result using DSI sampling
    Root Mean Square Error (RMSE)
    I In each voxel, RMSE is defined on 514 DWI samples in DSI scheme.
    I In a field, we calculate the mean RMSE.
    

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19. Synthetic Data without noise
    EAP profiles with radius 15µm, 20 × 20 × 1 × 514
    6D-CS-dMRI, no noise Naive, no noise
    Mean RMSE: 3.20% Mean RMSE: 7.97%
    

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20. Synthetic Data with noise
    Add Gaussian noise in measurements in k-space.
    6D-CS-dMRI, SNR=25 Naive, SNR=25
    Mean RMSE: 7.60% Mean RMSE: 12.76%
    

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21. Real data
    No ground truth. No raw data in k space. Retrospective way.
    

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22. Real data
    From HCP project
    6D-CS-dMRI, no noise Naive, no noise
    Mean RMSE: 5.15% Mean RMSE: 7.78%
    

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23. Real data
    Add Gaussian noise in measurements in k-space.
    6D-CS-dMRI, SNR=25 Naive, SNR=25
    Mean RMSE: 8.83% Mean RMSE: 14.06%
    

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24. Outline
    1 Motivation and Objective
    2 6D-CS-dMRI
    3 Experiments
    4 Conclusion
    

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25. Conclusion and Prospective
    Conclusion:
    6D-CS-dMRI performs CS reconstruction of diffusion signal and
    propagator in joint k-q space.
    6D-CS-dMRI has lower RMSE and more robust to noise than its naive
    way.
    6D-CS-dMRI achieves RMSE about 5% by using 11-fold subsampling
    in k-q space.
    Future work:
    Acquire raw complex data in k-space.
    Sampling scheme in q-space: a subset of DSI scheme → multi-shell
    uniform sampling scheme. (poster id: 2558. Wednesday, 4pm-6pm.
    MICCAI 2014)
    Efficient implementation: parallel in different volumes and different
    voxels
    

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26. Thank you!
    

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