ee290t: advanced reconstruction methods for magnetic...
TRANSCRIPT
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EE290T: Advanced Reconstruction Methods for MagneticResonance Imaging
Martin Uecker
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Tentative Syllabus
I 01: Jan 27 Introduction
I 02: Feb 03 Parallel Imaging as Inverse Problem
I 03: Feb 10 Iterative Reconstruction Algorithms
I –: Feb 17 (holiday)
I 04: Feb 24 Non-Cartesian MRI
I –: Mar 03 (cancelled)
I 05: Mar 10 GRAPPA/SPIRiT
I 06: Mar 17 Nonlinear Inverse Reconstruction
I –: Mar 24 (spring recess)
I 08: Mar 31 SAKE/ESPIRiT
I 09: Apr 07 Model-based Reconstruction
I 10: Apr 14 Compressed Sensing
I 11: Apr 21 Compressed Sensing
I 12: Apr 28 Final Project: Presentations
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Outline
I Review of last lecture
I Compressed Sensing (and Parallel Imaging)
I IEEE Eta Kappa Nu - Survey
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Nyquist-Shannon Sampling Theorem
Theorem 1: If a function f (t) contains no frequencies higher thanW cps, it is completely determined by giving its ordinates at aseries of points spaced 1/2W seconds apart.1
I Band-limited function
I Regular sampling
I Linear sinc-interpolation
1. CE Shannon. Communication in the presence of noise. Proc Institute of Radio Engineers; 37:10–21 (1949)
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A Puzzling Numerical Experiment1
Exact recovery of Shepp-Logan phantom from incomplete radialFourier samples:
(Figure: Block et al. 2007)
1. EJ Candes, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From HighlyIncomplete Frequency Information. IEEE Trans Inform Theory; 52:489–509 (2006)
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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Sparsity
Definition:
I vector x ∈ Rn
I k-sparse: at most k non-zero entries
Example:
Notation: Number of non-zero entries ‖x‖0 (this is not a norm)
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Sparsity
x ∈ R2
(0
1.3
) (1.51.3
)
Set of sparse vectors is a (non-convex) union of subspaces
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Denoising
I Sparse vector
I Densoising by hard-thresholding
I Densoising by soft-thresholding (shrinkage)
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Denoising
I Sparse vector and random noise
I Densoising by hard-thresholding
I Densoising by soft-thresholding (shrinkage)
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Denoising
I Sparse vector and random noise
I Densoising by hard-thresholding
I Densoising by soft-thresholding (shrinkage)
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Denoising
I Sparse vector and random noise
I Densoising by hard-thresholding
I Densoising by soft-thresholding (shrinkage)
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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Regular Under-Sampling: Point-Spread-Function
I Regular under-sampling in Fourier domain
I Coherent aliasing in the time domain
Point-Spread-Function
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Coherent Aliasing
I Regular under-sampling in Fourier domain
I Coherent aliasing in the time domain
1 12 2 33
Signal
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Random Sampling: Point-Spread-Function
I Random sampling in Fourier domain
I Incoherent aliasing in time domain
noise-like artifacts
Point-Spread-Function
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Incoherent Aliasing
I Random sampling in Fourier domain
I Incoherent aliasing in time domain
Signal
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Linear Measurements
Ax = y x ∈ Rn, y ∈ Rm
Measurements: m >= n
Reconstruction: x = A†y
Matrix A should be nearly orthogonal.Example: Fourier Matrix
A AHA
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Incoherent Linear Measurements
Ax = y x ∈ Rn and k-sparse, y ∈ Rm
Measurements: k log(n) <= m <= n
Reconstruction: ?
Matrix A should be nearly orthogonal (restricted isometry property)Example: Fourier Matrix with some rows removed A = PF
A AHA
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Restricted Isometry Property
A n × p matrix and 1 ≤ s ≤ p
s-restricted isometry property: There is a constant δs such thatfor every s-sparse vector y :
(1− δs)‖y‖22 ≤ ‖Ay‖2
2 ≤ (1 + δs)‖y‖22
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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Compressed Sensing
Ingredients:
I Sparsity
I Incoherence
I Non-linear reconstruction
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)
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L1-Norm and Sparsity
(01
)
Set of vectors with ‖x‖0 ≤ 1 not convex! ⇒ L1 instead of L0
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L1-Norm and Sparsity
(01
)
Set of vectors with ‖x‖0 ≤ 1 not convex! ⇒ L1 instead of L0
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Linear Reconstruction
L2-regularization:
argminx‖Ax − y‖22 + α‖Wx‖2
2
Explicit Solution: (AHA + αWHW
)−1AHy
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Nonlinear Reconstruction
L1-regularization:
argminx‖Ax − y‖22 + α‖Wx‖1
In general: no explicit solution!
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L2-Norm vs L1-Norm
−1
0
1
−1 0 1
‖x‖2 = 1
‖x‖1 = 1
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L1-Norm and Sparsity
Minimize ‖x‖pp subject to Ax = y
Ax = y
‖x‖22
Ax = y
‖x‖11
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Inverse Problem with L1-Regularization
Minimize ‖x‖1 subject to ‖Ax − y‖2 ≤ ε
‖Ax − y‖2 = ε
‖x‖11
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Linear Reconstruction
x = argminz |z − y |2 + λ|z |2
x =1
1 + λy
0
0−1
1
−11
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Soft-Thresholding
x = argminz |z − y |2 + λ|z |
ηλ(x) =
x − λ x > λ0 |x | ≤ λx + λ x < −λ
0
0−1
1
−11
−λ
λ
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Joint Thresholding
Shrink magnitude but keep phase/direction
I complex values:
ηλ(x) =
ηλ(|x |) x
|x | x 6= 0
0 x = 0
I vectors:
ηλ(x) =
ηλ(‖x‖2) x
||x ||2 x 6= 0
0 x = 0
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Joint Thresholding
Shrink magnitude but keep phase/direction
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Iterative Soft-Thresholding (IST)
Landweber1:
xn+1 = xn + µAH(y − Axn)
Iterative Soft-Thresholding2:
zn = xn + µAH(y − Axn)
xn+1 = ηλ(zn)
1. L Landweber. An iteration formula for Fredholm integral equations of the first kind. Amer J Math; 73:615–624(1951) 2. I Daubechies, M Defrise, C De Mol. An iterative thresholding algorithm for linear inverse problems witha sparsity constraint. Comm Pure Appl Math; 57:1413–1457 (2004)
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Nonlinear Reconstruction: Iterative Soft-Thresholding
1. Data consistency: zn = xn + µAH(y − Axn)
2. Soft-thresholding: xn+1 = ηλ(zn)
iteration 0
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Nonlinear Reconstruction: Iterative Soft-Thresholding
1. Data consistency: zn = xn + µAH(y − Axn)
2. Soft-thresholding: xn+1 = ηλ(zn)
iteration 1
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Nonlinear Reconstruction: Iterative Soft-Thresholding
1. Data consistency: zn = xn + µAH(y − Axn)
2. Soft-thresholding: xn+1 = ηλ(zn)
iteration 2
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Nonlinear Reconstruction: Iterative Soft-Thresholding
1. Data consistency: zn = xn + µAH(y − Axn)
2. Soft-thresholding: xn+1 = ηλ(zn)
iteration 9
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Algorithms
I FOCUSS1
I Iterative Soft-Thresholding (IST)2
I Fast iterative Soft-Thresholding Algorithm3
I Split Bregman4
I Nonlinear Conjugate Gradients
I ...
1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc 45:600–616 (1997) 2. I Daubechies, M Defrise, C De Mol. An iterativethresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math; 57:1413–1457(2004) 3. A Beck, M Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAMJ Imaging Sci; 2:183–202 (2009) 4. T Goldstein and S Osher. The Split Bregman Method for L1-RegularizedProblems. SIAM J Imaging Sci; 2:323–343 (2009)
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Statistical Model
Linear measurements contaminated by noise:
y = Ax + n
Gaussian white noise:
p(n) = N (0, σ2) with N (µ, σ2) =1
σ√
2πe−
(x−µ)2
2σ2
Probability of an outcome (measurement) given the image x :
p(y |A, x , λ) = N (Ax , σ2)
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Bayesian Prior
I L2-Regularization: Ridge Regression
N (µ, σ2) =1
σ√
2πe−
(x−µ)2
2σ2 Gaussian prior
I L1-Regularization: LASSO
p(x |µ, b) =1
2be−|x−µ|
b Laplacian prior
I L2 and L1: Elastic net1
1. H Zou, T Hastie. Regularization and variable selection via the elastic net. J R Statist Soc B. 67:301–320 (2005)
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Compressed Sensing in Magnetic Resonance Imaging
I Sparsity in medical imaging
I Incoherent sampling
I Iterative reconstruction
I Combination with parallel imaging
M Lustig, D. Donoho, JM Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging. MagnReson Med; 58:1182–1195 (2007)
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Sparsity Transform
I Medical images are usually not sparse
I Need to apply sparsity transform
Sparsity:
I Wavelet Transform
I Total Variation
I Temporal Constraints
I Prior images
I Adapted dictionaries
I Low rank
I . . .
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Wavelet Transform
I Orthonormal basis (or almost)
I Multi-scale transform
I Localized in frequency and space
I Compresses many signals/images into few coefficients
I Efficient computation: O(N)
I But: not shift-invariant (cycle spinning)
L1-regularization term in wavelet domain;
R(x) = ‖Wx‖1
W wavelet transform
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Wavelet Transform
brain image wavelet transform
Signal concentrated in few coefficients!
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Cycle spinning
Problem: Not shift-invariantSolution: Cycle spinning (or random shifting)
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Artifacts
blurring blocky artifacts good quality
Problem: Not shift-invariantSolution: Random shifting (cycle spinning)
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Total Variation
Definition: For a function f ∈ L1(Ω) with Ω an open set Ω ⊂ Rn,the total variation of f is:
TV f = sup
∫dx f divφ : φ ∈ C 1
c (Ω,Rn), ‖φ‖L∞(Ω) ≤ 1
I Denoising1
I Image Reconstruction2
1. LI Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms. Physica D; 60:259–268(1992) 2. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T ImageProcessing; 4:932–946 (1995)
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Total Variation
For differentiable function in one variable:
TV f =
∫dx |f ′(x)|
For differentiable function in many variables:
TV f =
∫dx‖∇f (x)‖2
Sparsity of the partial derivatives!
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Total Variation
f (x)‖∇f (x)‖2 =√
|∂1f (x)|2 + |∂2f (x)|2
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Total Variation: Approximations
Anisotropic Total Variation:
TV f =
∫dx√|∂1f (x)|2 + |∂2f (x)|2
≈∫
dx |∂1f (x)|+ |∂2f (x)|
Finite differences (backward):
∂1f (x1, x2) ≈ f (x1, x2)− f (x1 − h, x2)
|h|
∂2f (x1, x2) ≈ f (x1, x2)− f (x1, x2 − h)
|h|
h ≈ 1 Pixel
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Total Variation: Staircase Artifacts
I Staircase artifacts (sparse differences)
I Solution: use of higher-order derivatives1
I Total Generalized Variation2,3
1. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T Image Processing;4:932–946 (1995) 2. K Bredies, K Kunisch, T Pock. Total generalized variation. SIAM J Imaging Sci; 3:492–526(2010) 3. F Knoll, K Bredies, T Pock, R Stollberger. Second order total generalized variation (TGV) for MRI.Magn Reson Med; 65:480–491 (2011)
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Total Variation in Time Domain
f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
t
TVt f ≈∫
dx∑l
|fl(x)− fl−1(x)|
HFL Chandarana, T Block, AB Rosenkrantz, R Lim, D Chu, DK Sodickson, R Otazo. Free-breathing dynamiccontrast-enhanced MRI of the liver with radial golden-angle sampling scheme and advanced compressed-sensingreconstruction. Proc. 20th ISMRM (2012)
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GRASP: Compressed-Sensing Reconstruction
Otazo et al, MRM 2010: 64
Narrow data window Few spokes Flickering streak artifacts
Prior knowledge: Contrast uptake occurs “smoothly” and “continuously”
CS approach: Find solution that
CG SENSE-type reconstruction with temporal Total Variation (TV) constraint
has lowest flickering
matches data in all windows
Ground truth Ground truth13 spokes 13 spokes
Dynamic image series Temporal differences
Courtesy of Tobias Block, NYU
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Example: GRASP Liver Imaging
Chandarana et al, ISMRM 2012: 5529
Free-breathing scan over 5 min
Contrast injection after 20 s
Retrospective selection of temporal resolution
Example: 13 spokes 2 s resolution
Enables free-breathing liver perfusion imaging
Here: 384 x 384 x 30 matrix
Spatial resolution 1.0 x 1.0 x 3.0 mm3
Temporal resolution 1.5 s
Top: GriddingBottom: GRASP
Courtesy of Tobias Block, NYU
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Difference to Prior Image
Sparse difference to reference image g
R(x) = ‖x − x0‖1
Prior image: x0 composite image, previous frame, ...
1. GH Chen, J Tang, S Leng. Prior image constrained compressed sensing (PICCS): a method to accuratelyreconstruct dynamic CT images from highly undersampled projection data sets. Med Phys; 35:660–663 (2008) 2.A Fischer, F Breuer, M Blaimer, N Seiberlich, PM Jakob. Accelerated dynamic imaging by reconstructing sparsedifferences using compressed sensing. Proc 16th ISMRM (2008)
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Dictionary Learning
=
Patch Reconstruction
Dictionary Learning Example-based
...
=a +b +cCourtesy of Patrick Virtue, UC Berkeley
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Spatio-temporal Dictionaries
Dictionary based reconstruction of dynamic complex MRI data. Jose Caballero, Anthony Price, Daniel Rueckert,and Joseph V. Hajnal. ISMRM 13
Courtesy of Jose Caballero, Imperial College London
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Low-rank Approximation
Data matrix M ∈ Ct×s , e.g. time × space
Singular-Value-Decomposition: M = UΣVH
Low rank: rank M < K
Decomposition into K (temporal and spatial) basis functions:
M =∑K
ukσkvHk
1. Z Bo, JP Haldar. C Brinegarm ZP Liang. Low rank matrix recovery for real-time cardiac MRI. ISBI; 996-999(2010) 2. JP Haldar, ZP Liang. Spatiotemporal imaging with partially separable functions: A matrix recoveryapproach. ISBI 716–719 (2010) 3. SG Lingala, H Yue, E DiBella, M Jacob. Accelerated Dynamic MRI ExploitingSparsity and Low-Rank Structure: k-t SLR. IEEE Trans Med Imag; 30:1042–1054 (2011) 4. R Otazo, E Candes,DK Sodickson. Low-rank and sparse matrix decomposition for accelerated DCE-MRI with background and contrastseparation. ISMRM Workshop on Data Sampling and Image Reconstruction. Sedona (2013)
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Low-rank + Sparse Reconstruction of Cardiac Cine
I 6-fold acceleration (ky-t random undersampling)
I Temporal resolution: 40 ms
I Spatial resolution: 1.3x1.3x3 mm3
I Std. CS with temporal FFT
CS L+S L S
Courtesy of Ricardo Otazo, NYU
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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
1. DK Sodickson, WJ Manning. Simultaneous acquisition of spatial harmonics (SMASH): Fast imaging withradiofrequency coil arrays. Magn Reson Med; 38:591–603 (1997) 2. KP Pruessmann, M Weiger, MB Scheidegger,P Boesiger. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med; 42:952–962 (1999) 3. MA Griswold, PMJakob, RM Heidemann, M Nittka, V Jellus, J Wang, B Kiefer, A Haase. Generalized autocalibrating partiallyparallel acquisitions (GRAPPA). Magn Reson Med; 47:1202–10 (2002)
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Parallel MRI: Undersampling
Undersampling Aliasing
kread
kphase
kphase
kpartition
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Parallel MRI as Inverse Problem
I Signal from multiple coils (image x , sensitivities cj):
sj(t) =
∫Vd~r x(~r)cj(~r)e−i~r ·
~k(t)
I Assumption: known sensitivities cj⇒ linear relation between image x and data y
I Image reconstruction is a linear inverse problem:
Ax = y
1. JB Ra and CY Rim, Magn Reson Med 30:142–145 (1993) 2. KP Pruessmann, M Weiger, MB Scheidegger, PBoesiger. Magn Reson Med 4:952–962 (1999)
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Parallel MRI: Regularization
I General problem: bad condition
I Noise amplification during image reconstruction
I L2 regularization (Tikhonov):
argminx‖Ax − y‖22 + α‖x‖2
2 ⇔ (AHA + αI )x = AHy
I Influence of the regularization parameter α:
small medium large
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Parallel MRI: Nonlinear Regularization
I Good noise suppression
I Edge-preserving
⇒ Sparsity, nonlinear regularization
argminx‖Ax − y‖22 + αR(x)
Regularization: R(x) = TV (x), R(x) = ‖Wx‖1, . . .
1. JV Velikina. VAMPIRE: variation minimizing parallel imaging reconstruction. Proc. 13th ISMRM; 2424 (2005)2. G Landi, EL Piccolomini. A total variation regularization strategy in dynamic MRI, Optimization Methods andSoftware; 20:545–558 (2005) 2. B Liu, L Ying, M Steckner, J Xie, J Sheng. Regularized SENSE reconstructionusing iteratively refined total variation method. ISBI; 121-123 (2007) 3. A Raj, G Singh, R Zabih, B Kressler, YWang, N Schuff, M Weiner. Bayesian parallel imaging with edge-preserving priors. Magn Reson Med; 57:8–21(2007) 4. M Uecker, KT Block, J Frahm. Nonlinear Inversion with L1-Wavelet Regularization - Application toAutocalibrated Parallel Imaging. ISMRM 1479 (2008) 5. . . .
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Nonlinear Inversion with Non-Quadratic Regularization
Iteratively Regularized Gauss Newton Method (IRGNM)
xn+1 − xn = argminδx‖DFH(xn)δx + F (xn)− y‖22 + αnR(δx + xn)
Previously: Image regularized with L2-norm
R(x) = ‖ρ‖22 + ‖(1 + s|~k|2)lFTcj‖2
2
Now: Different regularization terms
R(x) = R(ρ) + ‖(1 + s|~k |2)lFTcj‖22
Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).
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Nonlinear Inversion
I Quality of the reconstructed images can be improved
I Acceleration: 3 x 2
I L1-Wavelet: Cohen-Daubechies-Feauveau 9/7
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Compressed Sensing and Parallel Imaging
I Parallel imaging
I Sparsity, nonlinear regularization
I Incoherent sampling
1. KT Block, M Uecker, J Frahm. Undersampled radial MRI with multiple coils. Iterative image reconstructionusing a total variation constraint. Magn Reson Med; 57:1086–1098 (2007) 2. C Zhao, T Lang, J Ji. CompressedSensing Parallel Imaging. Proc. 16th ISMRM; 1478 (2008) 3. B Wu, RP Millane, R Watts, P Bones. Applyingcompressed sensing in parallel MRI. Proc. 16th ISMRM; 1480 (2008) 4. KF King. Combining compressed sensingand parallel imaging. Proc. 16th ISMRM; 1488 (2008) 5. B Liu, FM Sebert, YM Zou, L Ying. SparseSENSE:randomly-sampled parallel imaging using compressed sensing. Proc. 16th ISMRM; 3154 (2008) 6. B Liu, K King,M Steckner, J Xie, J Sheng, L Ying. Regularized sensitivity encoding (SENSE) reconstruction using bregmaniterations. Magn Reson Med 61:145–152 (2009) 7. . . .
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Sampling Schemes
uniform random Poisson-disc
Poisson-disc sampling:
I Minimum distance to exploit parallel imaging
I Incoherence for compressed sensing
M Murphy, K Keutzer, SS Vasanawala, M Lustig. Clinically feasible reconstruction time for L1-SPIRiT parallelimaging and compressed sensing MRI. Proc 18th ISMRM; 4854 (2010)
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Variable-Density Sampling
variable-density Poisson-disc radial
Advantages:
I Auto-calibration for parallel imaging
I Graceful degradation
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Compressed Sensing and Parallel Imaging
Regularized SENSE:
argminx‖PFCx − y‖22 + R(x)
P projection onto samples, F Fourier transform, C coil sensitivities,
R(x) regularization
Simple (but slow): IST with R(x) = ‖Wx‖1
zn = xn + µCHFHP(y − PFCxn)
xn+1 = W−1ηλ(Wzn)
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Compressed Sensing and Parallel Imaging
Linear reconstruction:
I R(x) = 0 ⇒ No regularization
I R(x) = ‖x − x0‖22 ⇒ Tikhonov
Nonlinear reconstruction:
I R(x) = TV (x) ⇒ Total Variation
I R(x) = ‖Wx‖1 ⇒ L1-Wavelet
I R(x) = ‖x − x0‖1 ⇒ Prior Image
I · · ·
Incoherent Sampling ⇒ Compressed Sensing
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Example: Undersampled Radial with Total Variation
KT Block, M Uecker, J Frahm. Undersampled Radial MRI with Multiple Coils. Iterative Image ReconstructionUsing a Total Variation Constraint. Magn Reson Med 57:1086-1098 (2007)
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Nonlinear Inverse Reconstruction with Variational Penalties
Experiments:I Siemens Tim Trio 3 T, 12-channel head coilI 3D FLASH, acceleration: R = 4 (pseudorandom sampling)
Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).
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`1-SPIRiT
I Robust auto-calibrating parallel MRI
I Calibration of coil-by-coil operator G in k-space
Optimization problem:
arg minx
α‖Px − y‖22︸ ︷︷ ︸
data consistency
+ β‖(G − Id)x‖22︸ ︷︷ ︸
calibration consistency
+ γR(x)︸ ︷︷ ︸regularization
x estimated k-space, y data, P Projection onto samples, G SPIRiT
operator, R(x) regularization
M Lustig, JM Pauly. SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space. MagnReson Med; 64:457–471 (2010)
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`1-SPIRiT
“POCS-type” algorithm (α = β =∞):
yn = y + (Id − P)xn data consistency
zn = Gyn calibration consistency
xn+1 = W−1ηλ(Wzn) soft-thresholding
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`1-ESPIRiT
ESPIRiT:
I Flexibility and efficiency of SENSE
I Robustness of GRAPPA/SPIRiT
Algorithm:
I Calibration of coil-by-coil operator in k-space
I Sensitivity maps from eigendecomposition
I Extended (“soft”) SENSE reconstruction
M Uecker, P Lai, MJ Murphy, P Virtue, M Elad, JM Pauly, SS Vasanawala, M Lustig. ESPIRiT - An EigenvalueApproach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med. Epub (2013)
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ESPIRiT: Reconstruction with Multiple Maps
Relaxed (“soft”) SENSE using multiple maps.
I Simultaneous reconstruction of multiple images mj
I Data consistency:
N∑i=1
‖yi − PFM∑j=1
S ji m
j‖22︸ ︷︷ ︸
“soft” SENSE
+ Q(m1, · · · ,mM)
mj images, S j multiplication with maps, F Fourier transform,
P sampling operator, y data, Q regularization
I Image combination (e.g. root of sum of squares)
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Example: `1-ESPIRiT
M Uecker, P Lai, MJ Murphy, P Virtue, M Elad, JM Pauly, SS Vasanawala, M Lustig. ESPIRiT - An EigenvalueApproach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med. Epub (2013)
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Conclusion
Compressed Sensing: Theory
I Sparsity
I Incoherent sampling
I Nonlinear reconstruction
Application to Magnetic Resonance Imaging
I Sparsity transform (adapted to application)
I Incoherent sampling schemes
I Efficient iterative reconstruction
I Combination with parallel imaging
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Software
http://www.eecs.berkeley.edu/~mlustig/Software.html
http://www.eecs.berkeley.edu/~uecker/toolbox.html
http://web.eecs.umich.edu/~fessler/code/index.html
http://www.imt.tugraz.at/index.php/research/
agile-gpu-image-reconstruction-library
http://gadgetron.sourceforge.net/
http://codeare.org/
http://impact.crhc.illinois.edu/mri.aspx
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Acknowledgements
I Michael Lustig, University of California, Berkeley
I Tobias Block, New York University
I Ricardo Otazo, New York University
I Patrick Virtue, University of California, Berkeley
I Jose Caballero, Imperial College London
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Final Projects
I Presentation: ≈ 10 minutes, 28th April
I Report