EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging. Martin Uecker

Transcription

1 EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging Martin Uecker

2 Tentative Syllabus 01: Jan 27 Introduction 02: Feb 03 Parallel Imaging as Inverse Problem 03: Feb 10 Iterative Reconstruction Algorithms : Feb 17 (holiday) 04: Feb 24 Non-Cartesian MRI : Mar 03 (cancelled) 05: Mar 10 GRAPPA/SPIRiT 06: Mar 17 Nonlinear Inverse Reconstruction : Mar 24 (spring recess) 08: Mar 31 SAKE/ESPIRiT 09: Apr 07 Model-based Reconstruction 10: Apr 14 Compressed Sensing 11: Apr 21 Compressed Sensing 12: Apr 28 TBA

3 Parallel MRI Goal: Reduction of measurement time I Subsampling of k-space I Simultaneous acquisition with multiple receive coils I Coil sensitivities provide spatial information I Compensation for missing k-space data

4 Parallel MRI: Undersampling Undersampling Aliasing k phase k read k partition k phase

5 Image Reconstruction as Inverse Problem Forward problem: y = Fx + n x image (and more), F (nonlinear) operator, n noise, y data Regularized solution: Advantages: x = argmin Fx y 2 2 }{{} + αr(x) }{{} data consistency regularization Simple extension to non-cartesian trajectories Modelling of physical effects Prior knowledge via suitable regularization terms

6 Parallel MRI as Inverse Problem Signal from multiple coils (image ρ, sensitivities c j ): s j (t) = d x ρ( x)c j ( x)e i x k(t) Assumption: known sensitivities c j linear relation between image ρ and data y Image reconstruction is a linear inverse problem: V Aρ = y Ra and Rim, Magn Reson Med, 30: (1993) Pruessmann et al., Magn Reson Med, 4: (1999)

7 Autocalibration: Motivation Estimation of the coil sensitivities during the measurement Motivation: Object influences sensitivities (dielectric) Problems with consistency due to movement (e.g. breathing) Example: Movement (R = 2x2) (Sensitivities calibrated in one position and used in the other)

8 Autocalibration: Conventional Methods Method: Complete acquisition of the k-space center Reconstruction of the fully sampled center Division by RSS image (removal of the image content) Postprocessing: smoothing, extrapolation,... k y k x (GRAPPA: Estimation of weights)

9 Autocalibration: Conventional Methods Problems: Artifacts due to erraneous calibration Limitation to small acceleration factors Need for a high number of reference lines SENSE/auto GRAPPA Acceleration factor: 4=2x2, reference lines: 16x16

10 Purpose Problems of previous methods: Calibration only uses part of the data Inaccurate removal of image content (image not known) Goals: Optimal use of all available data Consistent estimation of image and sensitivities New approach necessary

11 Parallel MRI as Nonlinear Inverse Problem New approach: Joint estimation of image and sensitivities from all data Need to solve signal equation for ρ and c j at the same time: s j (t) = d x ρ( x)c j ( x) e i x k(t) V }{{} unknown Ying L, Sheng J. Magn Reson Med, 57: , 2007 Uecker M, Hohage T, Block KT, Frahm J, Magn Reson Med, 60: , 2008

12 Parallel MRI as Nonlinear Inverse Problem Challenges: 1. Inversion of a nonlinear equation F ( x) + n = y 2. Equations are underdetermined d x c j( x) a( x)ρ( x) e i x k(t) a( x) V

13 Newton-Method How does one solve a non-linear equation? Solve: f (x) y = 0 Idea: linearize f (x + x) df (x) x + f (x) Algorithm: until f (x n ) y < ɛ df (x n ) x = y f (x n ) x n+1 x n + x x 0 y

14 Newton-Method How does one solve a non-linear equation? Solve: f (x) y = 0 Idea: linearize f (x + x) df (x) x + f (x) Algorithm: until f (x n ) y < ɛ df (x n ) x = y f (x n ) x n+1 x n + x x 0 y

15 Newton-Method How does one solve a non-linear equation? Solve: f (x) y = 0 Idea: linearize f (x + x) df (x) x + f (x) Algorithm: until f (x n ) y < ɛ df (x n ) x = y f (x n ) x n+1 x n + x x 0 x 1 y

16 Newton-Method How does one solve a non-linear equation? Solve: f (x) y = 0 Idea: linearize f (x + x) df (x) x + f (x) Algorithm: until f (x n ) y < ɛ df (x n ) x = y f (x n ) x n+1 x n + x x 0 x 1 y

17 Newton-Method How does one solve a non-linear equation? Solve: f (x) y = 0 Idea: linearize f (x + x) df (x) x + f (x) Algorithm: until f (x n ) y < ɛ df (x n ) x = y f (x n ) x n+1 x n + x x 0 x 1 x 2 y

18 Newton-Method (Quadratic) convergence: Initial guess close to solution Continuously differentiable Non-zero derivative (Second derivative exists)

19 Newton-Method Nonlinear system of equations: F : R n R m Taylor series: (DF : Jacobian) Fx = y Newton method: y = Fx n+1 = Fx n + DF (x n+1 x n ) + DF (x n+1 x n ) = y Fx n Banach spaces: Fréchet derivative Convergence: Kantorovich theorem

20 Newton for Optimization Minimization of non-linear least-squares problem: φ(x) = 1 2 Fx y 2 2 = 1 2 (Fx y)t (Fx y) Optimality at the solution: φ = 0 DF T (Fx y) = 0 Newton Method: DF T (Fx y) + H (x n+1 x n ) = 0 Hessian: H = DF T DF + terms with higher derivatives (Note: Approximated Hessian is positive semi-definite)

21 Gauss-Newton Method: Landweber Method: DF H DF (x n+1 x n ) = DF H (y Fx n ) x n+1 x n = µdf H (y Fx n ) Levenberg-Marquardt Algorithm: ( ) DF H DF + αi (x n+1 x n ) = DF H (y Fx n ) Linear optimization problem: x n+1 x n = argmin x DF x + Fx n y α x 2 2

22 Iteratively Regularized Gauss-Newton Method (IRGNM) Update rule: x n+1 x n = argmin x DF x + Fx n y α n x + x n 2 2 Similar to Levenberg-Marquardt algorithm Optimization of: Fx y α x 2 2 Iterative reduction of regularization: ( ) 2 n α n = α 0 3

23 Iteratively Regularized Gauss-Newton Method (IRGNM) Solve: F (x) y = 0 F (x + x) DF (x) x + F (x) Algorithm Newton Normal equations Regularization (Number of iterations determines regularization) while F (x n ) y 2 > ɛ Solve (with CG): ( DF H (x n )DF (x n ) + α n I ) x = DF H (x n ) (y F (x n )) + α n (x n x 0 ) x n+1 x n + x ( ) n 2 α n = α 0 3

24 Iteratively Regularized Gauss-Newton Method (IRGNM) Solve: F (x) y = 0 F (x + x) DF (x) x + F (x) Algorithm Newton Normal equations Regularization (Number of iterations determines regularization) while F (x n ) y 2 > ɛ Solve (with CG): ( DF H (x n )DF (x n ) + α n I ) x = DF H (x n ) (y F (x n )) + α n (x n x 0 ) x n+1 x n + x ( ) n 2 α n = α 0 3

25 Iteratively Regularized Gauss-Newton Method (IRGNM) Solve: F (x) y = 0 F (x + x) DF (x) x + F (x) Algorithm Newton Normal equations Regularization (Number of iterations determines regularization) while F (x n ) y 2 > ɛ Solve (with CG): ( DF H (x n )DF (x n ) + α n I ) x = DF H (x n ) (y F (x n )) + α n (x n x 0 ) x n+1 x n + x ( ) n 2 α n = α 0 3

26 Iteratively Regularized Gauss-Newton Method (IRGNM) Solve: F (x) y = 0 F (x + x) DF (x) x + F (x) Algorithm Newton Normal equations Regularization (Number of iterations determines regularization) while F (x n ) y 2 > ɛ Solve (with CG): ( DF H (x n )DF (x n ) + α n I ) x = DF H (x n ) (y F (x n )) + α n (x n x 0 ) x n+1 x n + x ( ) n 2 α n = α 0 3

27 Parallel Imaging Direct application to parallel imaging not possible! I System of equations F (ρ, cj ) = ~y is underdetermined. c If (ρ, cj ) solution, then also (ρ a, aj ): Z F (ρ/a, cj a) = I d~z cj (~z ) ~ a(~z )ρ(~z ) e i k ~z = F (ρ, ci ) a(~z ) Every image is solution with certain sensitivities! A constraint to reasonable solutions is necessary Prior knowledge: Only very smooth sensitivities are possible

28 Adaptation of the Regularization So far: Regularization used to suppress noise New: Regularization suppresses variation in the sensitivities Mathematical realization: Multiplication with weight matrix W Solution of transformed system: F = F W Choice of weight matrix: Image component of W : identity Coil components of W : penalization of high derivatives (e.g. differential operator (I ) l/2 )

29 Nonlinear Inversion The signal equation for unknown image ρ and unknown coil sensitivities c j is a nonlinear equation Fx = y. Forward operator: F : H l ([0, 1] 3, C N ) L 2 ([0, 1] 3, C) L 2 (range( k), C N ) (c j, ρ) y j = d x c j ( x)ρ( x) e i k(t) x Reconstruction: Iteratively regularized Gauss-Newton method (IRGNM) Smoothness penalty for the coil sensitivities: ρ (1 + s k 2 ) l FTc j 2 2

30 Nonlinear Inversion Algorithm: I Initialization: ρ = 1 and cj = 0 I Update rule (IRGNM): (DF (xn )H DF (xn ) + αn I )δx = DF (xn )H (y F (xn )) + αn (x0 xn ) (solved with the conjugate gradient algorithm) Regularization: I αn = q n α0, e.g. q = 1/2 I αn kρk22 + αn k(1 + s ~k 2 )l FTcj k22 (smoothness of the sensitivities) I Implementation: multiplication with a weighting matrix: F 0 = F W Uecker et al., Magn Reson Med 60: (2008)

31 Nonlinear Inversion Example: Siemens Tim Trio 3 T, 12-channel head coil 3D-FLASH, acceleration R = 2 2 iterative reconstruction of image and coil sensitivities Uecker et al., Magn Reson Med 60: (2008)

32 Nonlinear Inversion with Non-Quadratic Regularization Iteratively Regularized Gauss Newton Method (IRGNM) x n+1 x n = argmin δx DF H (x n )δx + F (x n ) y α n R(δx + x n ) Previously: Image regularized with L 2 -norm R(x) = ρ (1 + s k 2 ) l FTc j 2 2 Now: Different regularization terms R(x) = R(ρ) + (1 + s k 2 ) l FTc j 2 2 Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

33 Nonlinear Inversion Quality of the reconstructed images can be improved Acceleration: 3 x 2 L1-Wavelet: Cohen-Daubechies-Feauveau 9/7

34 Nonlinear Inverse Reconstruction with Variational Penalties Experiments: Siemens Tim Trio 3 T, 12-channel head coil 3D FLASH, acceleration: R = 4 (pseudorandom sampling) Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

35 Nonlinear Inversion for Real-Time MRI Advantages: Can be used directly with non-cartesian data No fully-sampled calibration region needed Dynamic update of coil sensitivities Prior knowledge using regularization Disadvantage: High computational demand!

36 Real-Time MRI α Echo RF G z G x G y TR Sequence for fast low-angle shot (FLASH) MRI and interleaved radial k-space scheme. Advantages of radial sampling: Robustness to motion Tolerance to undersampling Continuous updating of image data Self calibration for parallel imaging Zhang et al., J Magn Reson Imaging, 31: , 2010.

37 Gridding Reconstruction 75 spokes, 150 ms Problems: low SNR, streaking artefacts, low temporal resolution

38 Real-Time MRI Reconstruction: Autocalibrated parallel imaging based on nonlinear inversion Algorithm extended to non-cartesian (radial) sampling Further improvements for real-time imaging gridding nonlinear inversion real-time version Figure: short-axis view of a human heart, 15 spokes (30 ms)

39 Image Reconstruction on Graphical Processing Units 30 seconds of a movie: 1500 frames, 1GB data 8x TITAN GPUs Integrated with MRI scanner (TCP/IP) Array compression: 8-12 virtual channels Reconstruction time: 42ms (24 fps) (optimal quality) Uecker et al., Magn Reson Med 63: (2010) Schätz and Uecker, In: Lecture Notes in Computer Science, 7439: (2012)

40 Numerical Implementation Forward operator: F = P k FT M FOV C W W C M FOV FT P k Weighting matrix Multiplication of image and coil sensitivities Restriction to the field of view (FOV) Fourier transformation Sampling operator At the beginning: interpolation onto Cartesian grid: M FOV FT H P H k y Each iteration: convolution with point spread function (PSF) M FOV FT H Pk H }{{ P kft } convolution M FOV Uecker et al., Magn Reson Med 63: (2010)

41 Image Reconstruction on Graphical Processing Units CPU: Preprocessing measurement data whitening and array compression interpolation onto Cartesian grid GPU: Iterative reconstruction iteratively regularized Gauss-Newton method initial guess convolutionbased conjugate gradient algorithm image and sensitivities initialization with previous frame

42 Real-Time MRI Basic nonlinear inverse reconstruction: Short-axis view of the human heart

43 Real-Time MRI β = 0 β = 0.8 Figure: short-axis view of a human heart, 15 spokes (30 ms) Improved regularization: Previous frame as prior knowledge: x βx prev 2 2 Damping factor β to avoid accumulation of errors Enhanced recovery of high frequencies

44 Real-Time MRI unfiltered filtered Figure: short-axis view of a human heart, 15 spokes (30 ms) Median Filter: Applied in the temporal domain Removes streaking artefacts Preserves sharp transitions t

45 Real-Time MRI unfiltered filtered Figure: short-axis view of a human heart, 15 spokes (30 ms) Median Filter: Applied in the temporal domain Removes streaking artefacts Preserves sharp transitions t

46 Real-Time MRI Why does the median filter remove streaking artefacts? Interleaved k-space sampling scheme median Median: invariant to reordering Removes flickering artefacts for static image content Median as L 1 minimization: x { N } x = argmin x x k x 1 x 4 x 5 x 2 x 1 x 3 k=1

47 Real-Time MRI before after Figure: short-axis view of a human heart, 15 spokes (30 ms) Image filter: Edge enhancement Denoising

48 Real-Time MRI Figure: short-axis view of a human heart, 15 spokes (30 ms) Reconstruction steps: 1. Parallel imaging with nonlinear inverse reconstruction 2. Improved regularization 3. Median filter 4. Further image enhancement

49 Real-Time MRI Experiments: Siemens Tim Trio 3 T 32 channel cardiac coil (array compression to 12 virtual channels) RF-spoiled radial FLASH Healthy volunteers Free breathing, no synchronisation to ECG Image reconstruction with eight GTX 580 GPUs (Nvidia)

50 Real-Time MRI: Movies of the Human Heart Acquisition time 50 ms (25 spokes) Spatial resolution 2.0x2.0x8 mm 3

51 Real-Time MRI: Movies of the Human Heart Acquisition time 33 ms (15 spokes) Spatial resolution 1.5x1.5x8 mm 3 Acquisition time 22 ms (11 spokes) Spatial resolution 2.0x2.0x8 mm 3

52 Real-Time MRI: Swallowing and Speaking Acquisition time 50 ms (25 spokes) Spatial resolution 1.7x1.7x10 mm 3 Acquisition time 50 ms (25 spokes) Spatial resolution 1.7x1.7x10 mm 3 Niebergall A, Zhang S, Kunay E, Keydana G, Job M, Uecker M, Frahm J. Magn Reson Med, submitted 2011.

53 Project 2: Non-Cartesian MRI Project: Implement non-cartesian reconstruction Tools: Matlab, reconstruction toolbox, python,... Deadline: Mar 21 Hand in: Working code and plots/results with description. See website for data and instructions.

54 Project 2: Non-Cartesian MRI Step 1: Computation of non-cartesian samples Compute non-uniform DFT matrix Implement nufft and its adjoint 1D and 2D versions Hints: Oversampling 2 Gaussian kernel (not optimal)

55 Project 2: Non-Cartesian MRI Step 2: Implement iterative gridding Reconstruction from non-cartesian data Use Landweber and/or CG Step 3: Implement non-cartesian SENSE Implement non-cartesian SENSE 1. JD O Sullivan. A fast sinc function gridding algorithm for Fourier inversion in computer tomography. IEEE Trans Med Imaging, 4: , JI Jackson, CH Meyer, DG Nishimura, and A Macovski. Selection of a convolution function for Fourier inversion using gridding. IEEE Trans Med Imaging, 3: , KP Pruessmann, M Weiger, P Börnert, and P Boesiger. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med, 46: , 2001.

56 Final Project Project:? Tools: Matlab, reconstruction toolbox, python,... Deadline: Apr 28 Hand in: Working code and plots/results with description.