EE290T: Advanced Reconstruction Methods for Magnetic ...ee290t/sp14/lecture03.pdf · Parallel MRI Goal: Reduction of measurement time I Subsampling of k-space I Simultaneous acquisition

EE290T: Advanced Reconstruction Methods for MagneticResonance Imaging

Martin Uecker

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 –: Mar 24 (spring recess)

I 08: Mar 31 Subspace methods

I 09: Apr 07 Model-based Reconstruction

I 10: Apr 14 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 –: Mar 24 (spring recess)

I 08: Mar 31 Subspace methods

I 09: Apr 07 Model-based Reconstruction



I 12: Apr 28 TBA

Today

I Review of last lecture

I Noise Propagation

I Iterative Reconstruction Algorithms

I Software

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:

Channel Combination

RSS MVUE MMSE

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 MRI: Undersampling

Undersampling Aliasing

kread

kphase

kphase

kpartition

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

Discretization

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)

Today


I Noise Propagation


I Software

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

Noise vs. Bias

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

Today


I Noise Propagation


I Software

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


Initialization:

r0 = b − Tx0

d0 = r0

Iteration:

qi ⇐ Tdi

xi+1 ⇐ xi + αdi



Conjugate Gradients


Initialization:

r0 = b − Tx0

d0 = r0

Iteration:

qi ⇐ Tdi

α =|ri |2

R rHi qi

xi+1 ⇐ xi + αdi


β =|ri+1|2

|ri |2


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

conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

1.1.414

1.7322.


conjugate gradients

Landweber

0

1.1.414

1.7322.


conjugate gradients

Landweber

0

11.

1.4141.732

2.


conjugate gradients

Landweber

0

1

2

1.1.414

1.7322.


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.


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


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)

Today


I Noise Propagation


I Software

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

http://www.eecs.berkeley.edu/~uecker/toolbox.html

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)

Documents

EE290T: Advanced Reconstruction Methods for Magnetic ...ee290t/sp14/lecture03.pdf · Parallel MRI Goal: Reduction of measurement time I Subsampling of k-space I Simultaneous acquisition