nonlinear reconstruction methods for magnetic resonance...
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Nonlinear Reconstruction Methodsfor Magnetic Resonance Imaging
Martin Uecker
Biomedizinische NMR Forschungs GmbHam Max-Planck-Institut fur biophysikalische Chemie, Gottingen
Overview
I Image reconstruction as (nonlinear) inverse problem
I Autocalibrated parallel imaging
I Undersampled radial MRI with total-variation penalty
I Model-based reconstruction for fast spin-echo acquisitions
I Real-time imaging
Direct Image Reconstruction
I Assumption: Signal is Fourier transform of the image:
s(t) =
∫d~x ρ(~x)e−i
~k(t)~x
I Image reconstruction with an FFT algorithm
kx
ky
~k(t ′) = γ∫ t′
0 dt ~G (t)
⇒
k-space
⇒iDFT
image
Requirements:I Short readout (signal equation holds for small t only)I Sampling on a Cartesian grid⇒ Line-by-line scanning
Image Reconstruction as Inverse Problem
Forward problem:
y = Fx + n
x image (and more), F (nonlinear) operator, n noise, y data
Regularized solution:
x? = argmin ‖Fx − y‖22︸ ︷︷ ︸
data consistency
+ αR(x)︸ ︷︷ ︸regularization
Advantages:
I Simple extension to non-Cartesian trajectories
I Modelling of physical effects
I Prior knowledge via suitable regularization terms
Extension to Non-Cartesian Trajectories
Cartesian radial spiral
Practical implementation issues:
I Imperfect gradient waveforms (e.g. delays)
I Efficient implementation of the reconstruction algorithm
Extension to Non-Cartesian Trajectories
Cartesian radial spiral
Practical implementation issues:
I Imperfect gradient waveforms (e.g. delays)
I Efficient implementation of the reconstruction algorithm
Modelling of Physical Effects
Examples:
I Coil sensitivities (parallel imaging)
sj(t) =
∫d~x cj(~x)ρ(~x)e−i
~k(t)~x
I T2 Relaxation
sj(t) =
∫d~x cj(~x)e−R(~x)tρ(~x)e−i
~k(t)~x
I Field inhomogeneities
sj(t) =
∫d~x cj(~x)e−i∆B0(~x)tρ(~x)e−i
~k(t)~x
I Diffusion, flow, motion, ...
⇒ Nonlinear equations with additional unknowns (cj ,R,∆B, ...)
Regularization
I Introduces additional information about the solution
I In case of ill-conditioning: needed for stabilization
Common choices:
I Tikhonov (small norm)
R(x) = ‖W (x − xR)‖22 (often: W = I and xR = 0)
I Total variation (piece-wise constant images)
R(x) =
∫d~r
√|∂1x(~r)|2 + |∂2x(~r)|2
I L1 regularization (sparsity)
R(x) = ‖W (x − xR)‖1
Parallel MRI as Inverse Problem
I Signal from multiple coils (image ρ, sensitivities cj):
sj(t) =
∫Vd~x ρ(~x)cj(~x)e−i~x ·
~k(t)
I Assumption: known sensitivities cj⇒ linear relation between image ρ and data y
I Image reconstruction is a linear inverse problem:
Aρ = y
Ra and Rim, Magn Reson Med 30:142–145 (1993)
Pruessmann et al., Magn Reson Med 4:952–962 (1999)
Auto-calibrated Parallel MRI
Estimation of the coil sensitivities during the measurement
I Object influences sensitivities (dielectric)
I Problems with consistency due to movement (e.g. breathing)
Complete acquisition of the k-space center
I Reconstruction of the fully sampled center
I Removal of the image content
I Postprocessing: smoothing, extrapolation, ...
kx
ky
⇒ ⇒
Nonlinear Inversion
The signal equation for unknown image ρ and unknown coilsensitivities cj is a nonlinear equation Fx = y .
Forward operator:
F : H l([0, 1]3 ,CN)× L2([0, 1]3 ,C)→ L2(range(~k),CN)
(cj , ρ) 7→ yj =
∫d~x cj(~x)ρ(~x) e−i
~k(t)·~x
Reconstruction:I Iteratively regularized Gauss-Newton method (IRGNM)I Smoothness penalty for the coil sensitivities:
‖ρ‖22 + ‖(1 + s|~k |2)lFTcj‖2
2
Advantages:I Better estimation of the coil sensitivities
⇒ Improved image quality
Uecker et al., Magn Reson Med 60:674–682, 2008
Nonlinear Inversion
Algorithm:
I Initialization: ρ = 1 and cj = 0
I Update rule (IRGNM):
(DF (xn)HDF (xn) + αnI )δx = DF (xn)H(y − F (xn)) + αn(x0 − xn)
(solved with the conjugate gradient algorithm)
Regularization:
I αn = qnα0, e.g. q = 1/2
I αn‖ρ‖22 + αn‖(1 + s|~k|2)lFTcj‖2
2
(smoothness of the sensitivities)
I Implementation: multiplication with aweighting matrix: F ′ = F ◦W
Nonlinear Inversion
Experiment:
I Siemens Tim Trio 3 T, 12-channel head coil
I 3D-FLASH, acceleration R = 2× 2
Results:
iterative reconstruction of image and coil sensitivities
Undersampled Radial MRI with Total-Variation Constraint
A motivating example:
I Fourier data from 24 spokes (Shepp-Logan phantom)
I With total variation: exact reconstruction!
Candes E, Romberg J, Tao T, Robust uncertainty principles: Exact signal reconstruction from highlyincomplete frequency information. IEEE Transactions on Information Theory, 52:489–509, 2006
Undersampled Radial MRI with Total-Variation Constraint
Forward operator:
A : ρ 7→∫
d~x cj(~x)ρ(~x)e−i~k(t)~x
~k(t)Reconstruction
I Use of a total-variation regularization term:
x? = argmin‖Ax − y‖22 + αTV (~x)
I Total variation (anisotropic version):
TV (x) =
∫d~r |∂1x(~r)|+ |∂2x(~r)|
I Nonlinear conjugate gradient algorithm
Block KT, Uecker M, Frahm J. Magn Reson Med 57:1086-1098, 2007
Undersampled Radial MRI with Total-Variation Constraint
Experiments:
I Siemens Tim Trio 3 T, 12-channel head coil
I Radial spin-echo sequence with 94, 48, 24 spokes
Results:
Image Reconstruction for Fast Spin-echo Acquisitions
90◦ 180◦
TE1
180◦
TE2
180◦
TE3
e−rt
I Repeated acquisition of echo trains
I T2 relaxation during each trainsampling scheme
(R = 3)
Forward operator:
F : (ρ,R) 7→∫
d~x cj(~x)ρ(~x)e−R(~x)te−i~k(t)~x
Reconstruction:
I Minimization of ‖F (ρ,R)− y‖22
I Nonlinear conjugate gradient algorithm
Image Reconstruction for Fast Spin-echo Acquisitions
Experiments:
I Siemens Tim Trio 3 T, 32-channel coil
I 16 echos, ∆TE = 12.2ms,TR = 3000ms, accl. R = 5
Results:
spin density T2 map synthetic images
Sumpf T, Uecker M, Boretius S, Frahm J, Model-based Nonlinear Inverse Reconstruction for T2 Mappingof Highly Undersampled Spin-Echo MRI Data, submitted 2010.
For radial trajectories: Block KT, Uecker M, Frahm J, IEEE Trans Med Imag, 28, 2009
Real-time MRI
Echo
RF
α
Gz
Gx
Gy
TR
Sequence for fast low-angle shot (FLASH) MRI and interleaved radial k-space scheme.
Advantages of radial sampling:
I Robustness to motion
I Tolerance to undersampling
I Continuous updating of image data
I Self calibration for parallel imaging
Zhang et al., J Magn Reson Imaging, 31:101-109, 2010.
Real-time MRI
Objective:I Robust real-time imaging with high temporal resolution
Reconstruction:I Autocalibrated parallel imaging based on nonlinear inversionI Algorithm extended to non-Cartesian (radial) samplingI Further improvements for real-time imaging
gridding nonlinear inversion real-time version
Figure: short-axis view of a human heart, 15 spokes (30 ms)
Real-time MRI
β = 0 β = 0.8
Figure: short-axis view of a human heart, 15 spokes (30 ms)
Improved regularization:
I Previous frame as prior knowledge: ‖x − βxprev‖22
I Damping factor β to avoid accumulation of errors
⇒ Enhanced recovery of high frequencies
Real-time MRI
unfiltered filtered
Figure: short-axis view of a human heart, 15 spokes (30 ms)
Median Filter:I Applied in the temporal domainI Removes streaking artefactsI Preserves sharp transitions t
1
0
0 5 10 15 20 25
Real-time MRI
unfiltered filtered
Figure: short-axis view of a human heart, 15 spokes (30 ms)
Median Filter:I Applied in the temporal domainI Removes streaking artefactsI Preserves sharp transitions t
1
0
0 5 10 15 20 25
Real-time MRI
Why does the median filter remove streaking artefacts?
I Interleaved k-space sampling scheme
median
I Median: invariant to reordering
⇒ Removes flickering artefacts for static image content
Median as L1 minimization:
x3x1
x?
x2x5x4
x? = argminx
{N∑
k=1
‖xk − x‖1
}
Real-time MRI
before after
Figure: short-axis view of a human heart, 15 spokes (30 ms)
Image filter:
I Edge enhancement
I Denoising
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
Real-time MRI
Experiments:
I Siemens Tim Trio 2.9 T
I 32 channel cardiac coil(array compression to 12 virtual channels)
I RF-spoiled radial FLASH
I Healthy volunteers
I Free breathing, no synchronisation to ECG
I Image reconstruction with four Tesla C1060 GPUs (Nvidia)
Real-time MRI: Movies of the Human Heart
Acquisition time 50 ms (25 spokes)Spatial resolution 2.0x2.0x8 mm3
Real-time MRI: Movies of the Human Heart
Short-axis view 2-Chamber view
Acquisition time 18 ms (9 spokes)Spatial resolution 2.0x2.0x8 mm3
Acquisition time 22 ms (11 spokes)Spatial resolution 2.0x2.0x8 mm3
Real-time MRI: Movies of the Human Heart
Vessels + coronary artery 2-Chamber view
Acquisition time 30 ms (15 spokes)Spatial resolution 1.5x1.5x8 mm3
Acquisition time 30 ms (15 spokes)Spatial resolution 1.5x1.5x8 mm3
Real-time MRI: Speaking
Acquisition time 50 ms (25 spokes)Spatial resolution 1.7x1.7x10 mm3
Real-time MRI: Flow Dynamics
Acquisition time 30 ms (21 spokes)Spatial resolution 1.5x1.5x5 mm3
Summary
Image reconstruction as inverse problem:
I Simple extension to non-Cartesian acquisitions
I Modelling of physical effects
I Prior knowledge with regularization
Examples:
I Autocalibrated parallel imaging
I Undersampled radial MRI with total-variation penalty
I Model-based reconstruction for fast spin-echo acquisitions
I Real-time imaging