Frank Ong presents a k-space diagonal preconditioner to speed up iterative reconstruction for variable density sampled acquisitions. The preconditioner accelerates convergence without compromising SNR, unlike density compensation. It is incorporated into the primal-dual hybrid gradient method for reconstruction, avoiding the inner loops of other k-space preconditioning methods. Experiments demonstrate convergence within 10 iterations using an L2-optimized diagonal preconditioner tailored for each receiver channel.

1. k-space Diagonal Preconditioner:
Speeding Up Iterative Reconstruction For Variable
Density Sampled Acquisitions Without Compromises
Frank Ong, Martin Uecker, and Michael Lustig

2. Speaker Name: Frank Ong
I have the following financial interest or relationship to disclose with regard to the subject
matter of this presentation:
Company Name: GE Healthcare
Type of Relationship: Research support
Declaration of
Financial Interests or Relationships

3. Variable Density Sampling [1]
• Enables auto-calibration parallel imaging
• Robust to motion by averaging in low frequency
Ref: [1] Tsai and Nishimura, MRM 2000

4. Challenge for Iterative Reconstruction
2D UTE radial compressed sensing recon.
• Takes many iterations to converge
• Blurring when not converged
• Goal: accelerate iterative recon.
convergence for variable density sampling

5. Iterative Reconstruction
• For each iteration n, we compute:
Image
Non-Cartesian SENSE
k-space signal
Regularization
Condition number determines convergence
• For variable density sampling, condition number is high, hence slow
convergence

6. Density Compensation in Iterative Recon.
• Empirically shown to accelerate convergence [5]
• Computationally simple and fast
• Sacrifices SNR for convergence [5-6]:
[1] Jackson et al., MRM 1991 [2] Meyer et al., MRM 1992 [3] Hoge et al., MRM 1997 [4] Pipe and Menon, MRM 1999, [5] Pruessmann et al., MRM 2001 [6] Sutton et al, TMI 2003
Density Compensation Operator
• Leverages density compensation in gridding recon. [1-4]

7. Preconditioning
• No SNR penalty and preserves objective function
• Advocated by many works [1-4]
• Theoretically and empirically shown to accelerate iterative convergence
• Need additional FFTs to compensate variable density in k-space
• Requires inner loops to incorporate preconditioner in regularization
Improved condition number
[1] Sutton et al, TMI 2003 [2] Ramani et al., TMI 2011 [3] Weller et al., TMI 2014 [4] Muckley et al., ISMRM 2016 p0521

8. • Need additional FFTs to compensate variable density in k-space
• Requires inner loops to incorporate preconditioner in regularization
Preconditioning
Improved condition number
Overall increases computational complexity
[1] Sutton et al, TMI 2003 [2] Ramani et al., TMI 2011 [3] Weller et al., TMI 2014 [4] Muckley et al., ISMRM 2016 p0521
• No SNR penalty and preserves objective function
• Advocated by many works [1-4]
• Theoretically and empirically shown to accelerate iterative convergence

9. Desired: k-space Preconditioning
Our contribution:
• Present a general k-space preconditioning method without inner loops using PDHG [2-3]
• Derive an L2-optimized k-space diagonal preconditioner
• Demonstrate in experiments convergence in ~10 iterations
[1] Trzasko et al., ISMRM 2014 p1535 [2] Chambolle and Pock, J Math Imaging Vis 2011 [3] Pock and Chambolle, ICCV 2011
Existing work:
• Recently Trzasko et al. [1] shows a k-space preconditioning method, but with inner
loops and off-the-shelf density compensation factor as preconditioner
Want computation efficiency of density compensation and SNR of preconditioning

10. k-space Preconditioning Formulation
[1] Boyd and Vandenberghe, Convex Optimization
Key observations:
• Dual variable lives in k-space
• Can precondition dual problem with density-compensation like operations!
Primal
Dual [1]

11. k-space Preconditioning Formulation
• Primal-dual hybrid gradient [1-2] solves the primal and dual problems together:
• Same per-iteration computational complexity as vanilla iterative recon.
• Can precondition in k-space to accelerate convergence
[1] Chambolle and Pock, J Math Imaging Vis 2011 [2] Pock and Chambolle, ICCV 2011

12. • We consider an L2-optimized diagonal preconditioner, a
commonly used design for preconditioners [1]:
L2-Optimized k-space Diagonal Preconditioner
[1] Chan SIAM J. Sci. And Stat. Comput. 1988

13. • Preconditioner calculation incorporates sensitivity maps
• Different from density compensation, each channel has different preconditioner
• Providing more degrees of freedom
L2-Optimized k-space Diagonal Preconditioner
Preconditioner of 2D variable density spiral trajectory with 8-channel sensitivity maps
Readout sample numbersLow frequency High frequency

14. FISTA [1] Primal-dual Hybrid Gradient [2]
Primal-dual Hybrid Gradient
w/ Proposed Precond.
L1-Wavelet Recon. Of 2D Variable Density Spiral
[1] Beck and Teboulle SIAM J Imaging Sciences 2009 [2] Chambolle and Pock, J Math Imaging Vis 2011

15. FISTA [1] Primal-dual Hybrid Gradient [2]
Primal-dual Hybrid Gradient
w/ Proposed Precond.
L1-Wavelet Recon. Of 2D Variable Density Spiral
[1] Beck and Teboulle SIAM J Imaging Sciences 2009 [2] Chambolle and Pock, J Math Imaging Vis 2011

16. FISTA Primal-dual Hybrid Gradient
Primal-dual Hybrid Gradient
w/ Proposed Precond.
L1-Wavelet Recon. Of 2D UTE Radial

17. FISTA Primal-dual Hybrid Gradient
Primal-dual Hybrid Gradient
w/ Proposed Precond.
L1-Wavelet Recon. Of 2D UTE Radial

18. Conclusion
• Accelerate iterative recon. convergence without:
• SNR penalty of density compensation
• Increased computations of existing preconditioners
• L2-optimized k-space diagonal preconditioner
• Different for each channel
• Experiments shows convergence at ~10 iterations