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TRANSCRIPT
Introduction
Topics:
I Image Reconstruction as Inverse Problem
I Parallel Imaging
I Non-Cartestian MRI
I Subspace Methods
I Model-based Reconstruction
I Compressed Sensing
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 05: Mar 03 Nonlinear Inverse Reconstruction
I 06: Mar 10 Reconstruction in k-space
I 07: Mar 17 Reconstruction in k-space
I –: Mar 24 (spring recess)
I 08: Mar 31 Subspace methods
I 09: Apr 07 Model-based Reconstruction
I 10: Apr 14 Compressed Sensing
I 11: Apr 21 Compressed Sensing
I 12: Apr 28 TBA
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 05: Mar 03 Nonlinear Inverse Reconstruction
I 06: Mar 10 Reconstruction in k-space
I 07: Mar 17 Reconstruction in k-space
I –: Mar 24 (spring recess)
I 08: Mar 31 Subspace methods
I 09: Apr 07 Model-based Reconstruction
I 10: Apr 14 Compressed Sensing
I 11: Apr 21 Compressed Sensing
I 12: Apr 28 TBA
Phased Array
Signal is Fourier transform of magnetization image mweighted by coil sensitivities cj :
sj(t) =
∫d~x ρ(~x)cj(~x)e−i2π~k(t)~x
Images of a human brain from an eight channel array:
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)
Parallel Imaging as Inverse Problem
Model: Signal from multiple coils (image ρ, sensitivities cj):
sj(t) =
∫Ωd~x ρ(~x)cj(~x)e−i2π~x ·~k(t) + nj(t)
Assumptions:
I Image is square-integrable function ρ ∈ L2(Ω,C)
I Additive multi-variate Gaussian white noise n
Problem: Find best approximate/regularized solution in L2(Ω,C).
Ω
Discretization of Linear Inverse Problems
Continuous integral operator F : f 7→ g with kernel K :
g(t) =
∫ b
ads K (t, s)f (s)
Discrete system of linear equations:
y = Ax
Considerations:
I Discretization error
I Efficient computation
I Implicit regularization
continuous discreteoperator F Aunknown f x
data g y
Discretization for Parallel Imaging
Discrete Fourier basis:
f (x , y) ≈N∑
l=−N
N∑k=−N
al ,kei2π
(kx
FOVy+ ly
FOVy
)
I Efficient computation (FFT)
I Approximates R(FH) extremely well (for smooth cj)
I Voxels: Dirichlet kernel DN( xFOVx
)DN( yFOVy
)
Ω
FOVx
FOVy
SENSE: Discretization
Weak voxel condition: Images from discretized subspace shouldbe recovered exactly (from noiseless data).
C¯
ommon choice:
f (x , y) ≈∑r ,s
δ(x − rFOVx
Nx)δ(y − s
FOVy
Ny)
I Efficient computation using FFT algorithm
I Periodic sampling (⇒ decoupling)
Problem: Periodically extended k-space.⇒ Error at the k-space boundary!
SENSE: Decoupling
periodic sampling
m2m1
c2ρc1ρ
c2ρc1ρ
x
y
x
y1
y2
spin density ρ, sensitivities ciSystem of decoupled equations:
m1(x, y)· · ·
mn(x, y)
=
c1(x, y1) c1(x, y2)· · · · · ·
cn(x, y1) cn(x, y2)
· ( ρ(x, y1)
ρ(x, y2)
)
Discretization: Summary
I Continuous reconstruction (from finite data) is ill-posed!
I Discretization error
I Implicit regularization(discretized problem might be well-conditioned)
Attention: Be carefull when simulating data! Same discretizationfor simulation and reconstruction⇒ misleading results (inverse crime)
Complex Gaussian Distribution
Random variable Z = X + iYProper complex Gaussian:
Z ∼ CN (µ, σ2) p(Z ) =1
σπe−|Z−µ|2
σ2
mean: µ = E [Z ]variance: σ2 = E [(Z − µ)(Z − µ)?]proper: pseudo-varianceE [(Z − µ)(Z − µ)] = 0 R
I
Multi-Variate Complex Gaussian Distribution
Random vector Z = X + iY
Multi-variate proper complex Gaussian distribution:
Z ∼ CN (µ,Σ)
mean µ = E [Z ]covariance Σ = Cov [Z ,Z ] = E [(Z − µ)(Z − µ)H ]pseudo-covariance E [(Z − µ)(Z − µ)T ] = 0
Multi-Variate Complex Gaussian Distribution
Linear reconstruction: Z = FZ
Z ∼ CN (Fµ,FΣFH)
Full covariance matrix for all pixels: FΣFH
⇒ Not (always) practical(2D size ∝ 109, 3D size: ∝ 1014)
Geometry Factor
Quantity of interest: noise variance of pixel values:
σ(xi ) =√
(FΣFH)ii
Spatially dependent noise:
σund .(x) = g(x)√Rσfull(x)
Acceleration: R, Geometry factor: g
Practical estimation: Monte-Carlo method
σ2(x) = N−1∑j
|Fnj |2
Gaussian white noise nj .
Noise Amplification in Parallel Imaging
R = 1 R = 3 g-factor map
Local noise amplification dependent on:I Sampling patternI Coil sensitivities
Pruessmann et al. Magn Reson Med. 42:952–962 (1999)
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
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. . . .
Parallel MRI: Nonlinear Regularization
I L2, L1-wavelet and TV-regularization
I 3D-FLASH, 12-channel head coil
I 2D-reduction factor of 12 (phantom) and 6 (brain)
Correlated Noise - Whitening
Cholesky decomposition:
Σ = LLH
Lower triangular matrix L:Transform with W = L−1 to uncorrelated and equalized noise:
CN (0,Σ) ⇒ CN (0,WΣWH) = CN (0, I )
Reconstruction problem: Ax = y ⇒ WAx = Wy Normalequations:
AHWHWAx = AHWHWy ⇔ argminx ‖WAx −Wy‖2
In MRI: Noise correlations between receive channels.whitening ⇒ uncorrelated virtual channels: Wy
Parallel MRI: Iterative Algorithms
Signal equation:
si (~k) =
∫Vd~x ρ(~x) ci (~x)e−i2π~x ·~k︸ ︷︷ ︸
encoding functions
Discretization:
A = PkFC
A has size 2562 × (8× 2562)⇒ Iterative methods builtfrom matrix-vector productsAx , AHy
sensitivities
Landweber Iteration
Gradient descent:
φ(x) =1
2‖Ax − y‖2
2
∇φ(x) = AH(Ax − y)
Iteration rule:
xn+1 = xn − µAH(Axn − y) with µ‖AHA‖ ≤ 1
= (I − µAHA)xn − AHy
Explicit formula for x0 = 0:
xn =n−1∑j=0
(I − µAHA)jµAHy
Landweber Iteration
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
iteration 1iteration 2
iteration 10iteration 50
reconstruction of right singular-vectors (µ = 0.9)
xn =n−1∑j=0
(I − µAHA)jµAHAx =∑l
(1− (1− µσ2
l ))n
VlVHl x
geometric series:∑n−1
j=0 x j = 1−xn
1−x SVD: A =∑
j σjUjVHj
Projection Onto Convex Sets (POCS)
Iterative projection onto convex sets:
yn+1 = PBPAyn
A
B
Problems: slow, not stable, A ∩ B = ∅
Projection Onto Convex Sets (POCS)
Iterative projection onto convex sets:
yn+1 = PBPAyn
A
B
Problems: slow, not stable, A ∩ B = ∅
POCSENSE
Iteration:
yn+1 = PCPy yn
(multi-coil k-space y)
Projection onto sensitivities:
PC = FCCHF−1
(normalized sensitivities)
Projection onto data y :
Py y = My + (1−M)y
AA Samsonov, EG Kholmovski, DL Parker, and CR Johnson. POCSENSE: POCS-based reconstruction forsensitivity encoded magnetic resonance imaging. Magn Reson Med, 52:1397–1406, 2004.
POCSENSE
Fully-sampled data should be consistent with sensitivities:
PCyfull = yfull ⇒ F(CCH − I
)F−1yfull = 0
But is not:
F−1yfull C(CCH − I
)F−1yfull
Note: Can be used to validate coil sensitivities.
POCS and Landweber
For parallel MRI: POCS corresponds to Landweber for µ = 1 andnormalized sensitivities.
yn+1 = PCPyyn
= FC CHF−1((I − Pk)yn + y0)︸ ︷︷ ︸xn
Rewrite:
xn+1 = CHF−1((I − Pk)FCxn + y0) with yn = FCxn= CHCxn − CHF−1PkFCxn + CHF−1Pky0 with Pky0 = y0
= xn − AH(Axn − y0) with CHC = I , A = PkFC
Krylov Subspace Methods
Krylov subspace:
Kn = spani=0···n T nb
⇒ Repeated application of T .
Landweber, Arnoldi, Lanczos, Conjugate gradients, ...
Conjugate Gradients
T symmetric (or Hermitian)
Initialization:
r0 = b − Tx0
d0 = r0
Iteration:
qi ⇐ Tdi
α =|ri |2
R rHi qi
xi+1 ⇐ xi + αdi
ri+1 ⇐ ri − αqi
β =|ri+1|2
|ri |2
di+1 ⇐ ri+1 + βdi
Conjugate Gradients
T symmetric (or Hermitian)
Initialization:
r0 = b − Tx0
d0 = r0
Iteration:
qi ⇐ Tdi
xi+1 ⇐ xi + αdi
ri+1 ⇐ ri − αqi
di+1 ⇐ ri+1 + βdi
Conjugate Gradients
T symmetric (or Hermitian)
Initialization:
r0 = b − Tx0
d0 = r0
Iteration:
qi ⇐ Tdi
α =|ri |2
R rHi qi
xi+1 ⇐ xi + αdi
ri+1 ⇐ ri − αqi
β =|ri+1|2
|ri |2
di+1 ⇐ ri+1 + βdi
Conjugate Gradients
T symmetric or HermitianDirections are conjugate basis:
pHi Tpj = δij
(Wikipedia)
Coefficients can be directly computed:
b = Tx = T∑i
αipi =∑i
αiTpi
pHj b =∑i
αipHj Tpi = αjp
Hj Tpj
⇒αj =pHj b
pHj Tpj
Conjugate Gradients on Normal Equations
Normal equations:
AHAx = AHy ⇔ x = argminx ‖Ax − y‖22
Tikhonov:(AHA + αI
)x = AHy ⇔ x = argminx ‖Ax − y‖2
2 + α‖x‖22
CGNE:
Tx = b
with:
T = AHA
b = AHy
Implementation
FFT IFFTP
FFT IFFTP
distribute image
point-wise multiplication with sensitivities
multi-dimensional FFT
sampling mask or projection onto data
inverse FFT
point-wise mult. with conjugate sensitivities
sum channels
repeat
data flow
Conjugate Gradient Algorithm vs Landweber
Size: 41400× 41400 (complex)
0.0001
0.001
0.01
0.1
1
0 20 40 60 80 100
LandweberCG
Nonlinear Methods
I Non-linear Conjugate Gradient
I Landweber xn+1 = xn + µDFH(y − Fx)
I Iteratively Regularized Gauss-Newton Method
I ...
Project 1: Iterative SENSE
Project: Implement and study Cartesian iterative SENSE
I Tools: Matlab, reconstruction toolbox, python, ...
I Deadline: Feb 24
I Hand in: Working code and plots/results with description.
I See website for data and instructions.
Project 1: Iterative SENSE
Step 1: Implement Model
A = PFSAH = SHF−1PH
Hints:
I Use unitary and centered (fftshift) FFT ‖Fx‖2 = ‖x‖2
I Implement undersampling as a mask, store data with zero
I Check 〈x ,Ay〉 =⟨AHx , y
⟩for random vectors x , y
Project 1: Iterative SENSE
Step 2: Implement Reconstruction
Landweber (gradient descent)1
xn+1 = xn + αAH(y − Axn)
Conjugate gradient algorithm2
Step 3: Experiments
I Noise, errors, and convergence speed
I Different sampling
I Regularization
1. L Landweber. An iteration formula for Fredholm integral equations of the first kind. Amer J Math; 73:615–624(1951) 2. MR Hestenes and E Stiefel. Methods of conjugate gradients for solving linear systems. J. Ref. N.B.S.49:409–436 (1952)
Software Toolbox
I Rapid prototyping(similar to Matlab, octave, ...)
I Reproducible research(i.e. scripts to reproduce experiments)
I Robustness and clinically feasible runtime(C/C++, OpenMP, GPU programming)
Programming library
I Consistent API based on multi-dimensional arrays
I FFT and wavelet transform
I Generic iterative algorithms(conjugate gradients, IST, IRGNM, . . . )
I Transparent GPU acceleration of most functions
Command-line tools
I Simple file format
I Interoperability with Matlab
I Basic operations: fft, crop, resize, slice, . . .
I Sensitivity calibration and image reconstruction
Software
I Available for Linux and Mac OS X (64 bit)http://www.eecs.berkeley.edu/~uecker/toolbox.html
I Requirements: FFTW, GSL, LAPACK (CUDA, ACML)
Ubuntu:sudo apt-get install libfftw3-devsudo apt-get install libgsl0-devsudo apt-get install liblapack-dev
Mac OS X:sudo port install fftw-3-singlesudo port install gslsudo port install gcc47
Data Files
Data files store multi-dimensional arrays.
example.hdr ⇐ Text header
example.cfl ⇐ Data: complex single-precision floats
Text header:
# Dimensions
1 230 180 8 2 1 1 1 1 1 1 1 1 1 1 1
Matlab functions:
data = readcfl(’example’);
writecfl(’example’, data)
C Functions (using memory-mapped IO)
Rapid Prototyping
Data processing using command line tools:
# resize 0 320 tmp in
# fft -i 7 out tmp
Load result into Matlab/Octave:
〉 data = squeeze(readcfl(’out’));
〉 imshow3(abs(data), []);
C Programming Example
#i n c l u d e <complex . h>
#i n c l u d e ”num/ f f t . h”#i n c l u d e ” misc /mmio . h”
i n t main ( )
i n t N = 1 6 ;l o n g dims [N ] ;complex f l o a t ∗ i n = l o a d c f l (” i n ” , N, dims ) ;complex f l o a t ∗ out = c r e a t e c f l (” out ” , N, dims ) ;
f f t c (N, dims , 1 + 2 + 4 , out , i n ) ;
Reconstruction Algorithms
I Iterative SENSE1
I Nonlinear inversion2
I ESPIRiT calibration and reconstruction3
I Regularization: L2 and L1-wavelet
1. Pruessmann KP et al. Advances in sensitivity encoding with arbitrary k-space trajectories. MRM46:638-651 (2001)
2. Uecker M et al. Image Reconstruction by Regularized Nonlinear Inversion - Joint Estimation of CoilSensitivities and Image Content. MRM 60:674-682 (2008)
3. Uecker M, Lai P, et al. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSEmeets GRAPPA. MRM EPub (2013)