Alternating direction of multipliers method for parallel MRI reconstruction

• US 8,879,811 B2
• Filed: 02/27/2013
• Issued: 11/04/2014
• Est. Priority Date: 03/28/2012
• Status: Active Grant

• Current Assignee: Siemens Healthcare GMBH (Siemens AG)
• Original Assignee: Siemens AG
• Inventors: Liu, Jun, Lefebvre, Alban, Nadar, Mariappan
• Primary Examiner(s): Bhatnagar, Anand
• Assistant Examiner(s): PARK, SOO JIN
• Application Number:
  US13/778,446
• Publication Number:
  US 20130259343A1
• Time in Patent Office: 615 Days
• Field of Search: None
• US Class Current: 382/128
• CPC Class Codes: G01R 33/5611 Parallel magnetic resonance imaging, e.g. sensitivity encoding [SENSE], simultaneous acquisition of spatial harmonics [SMASH], unaliasing by Fourier encoding of the overlaps using the temporal dimension [UNFOLD], k-t-broad-use linear acquisition speed-up technique [k-t-BLAST], k-t-SENSE structural details of arrays of sub-coils G01R33/3415
• CPC Class Codes: G06T 11/003 Reconstruction from projections, e.g. tomography

Abstract

A method for reconstructing parallel magnetic resonance images includes providing a set of acquired k-space MR image data y, and finding a target MR image x that minimizes ½∥Fv−y∥₂²+λ∥z∥₁ where v=Sx and z=Wx where S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix, W is a redundant Haar wavelet matrix, and λ≧0 is a regularization parameter, by updating

$x^{k+1}={\left({\mu }_{1}I+{\mu }_3{S}^{H}S\right)}^{-1}\left[{\mu }_1{W}^H\left(z^k-b_z^k\right)+{\mu }_3{S}^H\left(v^k-b_v^k\right)\right],z^{k+1}=\mathrm{soft}\left({\mathrm{Wx}}^{k+1}+b_z^k,\frac{1}{{\mu }_1}\right)\mathrm{where}$$\mathrm{soft}\left(x,T\right)=\{\begin{array}{cc}x+T& \mathrm{if}x\le -T,\\ 0& \mathrm{if}x\le T,\\ x-T& \mathrm{if}x\ge T,\end{array}\mathrm{and}{v}^{k+1}={\left(F^{H}F+{\mu }_{3}I\right)}^{-1}\left[F^{H}y+{\mu }_3\left({\mathrm{Sx}}^{k+1}+b_v^k\right)\right],$
where k is an iteration counter, μ₁ and μ₃ are parameters of an augmented Lagrangian function, and b_z and b_v are dual variables of the augmented Lagrangian.

Claims

1. A method for reconstructing parallel magnetic resonance images (MRI), comprising:

providing a set of acquired k-space MR image data y; and

finding a target MR image x that minimizes ½

∥

Fv−

y∥

₂²+λ

∥

z∥

₁ wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF^H=I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W^TW=I, wherein I is an identity matrix, and λ

≧

0 is a regularization parameter, by updating

2. The method of claim 1, wherein v^k+1 can be updated as

3. The method of claim 1, wherein λ

≦

∥

WA^Hy∥

_∞

.

4. The method of claim 1, wherein μ

₁=μ

₃<

1.

5. The method of claim 1, further comprising updating the dual variables as b_z^k+1=b_z^k+(Wx^k+1−

z^k+1) and b_v^k+1=b_v^k+(Sx^k+1−

v^k+1).

6. The method of claim 5, wherein x^k, z^k, and v^k, and dual variables b_z^k and b_v^k are initialized to zero.

7. The method of claim 1, wherein x^k, z^k, and v^k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.

8. A method for reconstructing parallel magnetic resonance images (MRI), comprising:

providing a set of acquired k-space MR image data y; and

finding a target MR image x that optimizes

9. The method of claim 8, further comprising updating the dual variables as

b_z^k+1=b_z^k+(Wx^k+1−

z^k+1),

b_u^k+1=b_u^k+(u^k+1−

Fv^k+1+y), and

b_v^k+1=b_v^k+(Sx^k+1−

v^k+1).

10. The method of claim 8, wherein ε

≦

∥

y∥

₂.

11. The method of claim 8, wherein μ

₁, μ

₂, and μ

₃ are set as μ

₁=μ

₃=0.1ρ and

μ

₂=ρ

wherein

12. The method of claim 9, wherein x^k, z^k, v^k, and u^k, and dual variables b_z^k, b_u^k, and b_v^k are initialized to zero.

13. The method of claim 8, wherein x^k, z^k, v^k and u^k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.

14. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for reconstructing parallel magnetic resonance images (MRI), the method comprising the steps of:

providing a set of acquired k-space MR image data y; and

finding a target MR image x that minimizes ½

∥

Fv−

y∥

₂²+λ

∥

z∥

₁ wherein v=Sx and z=Wx wherein S is a diagonal matrix containing sensitivity maps of coil elements in an MR receiver array, F is an FFT matrix wherein FF^H=I and H denotes a Hermitian adjoint matrix, W is a redundant Haar wavelet matrix satisfying W^TW=I, wherein I is an identity matrix, and λ

≧

0 is a regularization parameter, by updating

15. The computer readable program storage device of claim 14, wherein v^k+1 can be updated as

16. The computer readable program storage device of claim 14, wherein λ

≦

∥

WA^Hy∥

_∞

.

17. The computer readable program storage device of claim 14, wherein μ

₁=μ

₃<

1.

18. The computer readable program storage device of claim 14, the method further comprising updating the dual variables as b_z^k+1=b_z^k+(Wx^k+1−

z^k+1) and b_v^k+1=b_v^k+(Sx^k+1−

v^k+1).

19. The computer readable program storage device of claim 18, wherein x^k, z^k, and v^k, and dual variables b_z^k and b_v^k are initialized to zero.

20. The computer readable program storage device of claim 14, wherein x^k, z^k, and v^k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.

21. A non-transitory program storage device readable by a computer, tangibly embodying a program of instructions executed by the computer to perform the method steps for reconstructing parallel magnetic resonance images (MRI), the method comprising the steps of:

providing a set of acquired k-space MR image data y; and

finding a target MR image x that optimizes

22. The computer readable program storage device of claim 21, the method further comprising updating the dual variables as

b_z^k+1=b_z^k+(Wx^k+1−

z^k+1),

b_u^k+1=b_u^k+(u^k+1−

Fv^k+1+y), and

b_v^k+1=b_v^k+(Sx^k+1−

v^k+1).

23. The computer readable program storage device of claim 21, wherein ε

≦

∥

y∥

₂.

24. The computer readable program storage device of claim 21, wherein μ

₁, μ

₂, and μ

₃ are set as μ

₁=μ

₃=0.1ρ and

μ

₂=ρ

wherein

25. The computer readable program storage device of claim 22, wherein x^k, z^k, v^k, and u^k, and dual variables b_z^k, b_u^k, and b_v^k are initialized to zero.

26. The computer readable program storage device of claim 21, wherein x^k, z^k, v^k and u^k are updated until a relative difference between a current iteration estimate and a previous iteration estimate fall below a predefined threshold.