Reconstruction Methods for Magnetic Resonance Imagingmuecker1/reco16_lec03.pdf · Introduction...

Reconstruction Methods for Magnetic Resonance 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: Apr 12 Introduction

I 02: Apr 19 Parallel Imaging as Inverse Problem

I 03: Apr 26 –

I 04: May 03 Iterative Reconstruction Algorithms

I 05: May 10 Non-Cartesian MRI

I 06: May 17 Nonlinear Inverse Reconstruction

I 07: May 24 Reconstruction in k-space

I 08: May 31 Reconstruction in k-space

I 09: Jun 07 Subspace methods

I 10: Jun 14 Subspace methods - ESPIRiT

I 11: Jun 21 Model-based Reconstruction

I 12: Jun 28 Model-based Reconstruction

I 13: Jul 05 Compressed Sensing

I 14: Jul 12 Compressed Sensing

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 = ∅

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

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

Non-linear forward problems:

I Non-linear Conjugate Gradient

I Landweber xn+1 = xn + µDFH(y − Fx)

I Iteratively Regularized Gauss-Newton Method

I ...

Non-quadratic regularization:

I IST, FISTA

I ADMM

I ...

Project 1: Iterative SENSE

Project: Implement and study Cartesian iterative SENSE

I Tools: Matlab, reconstruction toolbox, python, ...

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, FISTA, IRGNM, ADMM, . . . )

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://mrirecon.github.io/bart/

I Requirements: FFTW, LAPACK (CUDA, ACML)

Ubuntu:sudo apt-get install libfftw3-devsudo apt-get install liblapack-devsudo apt-get install libpng-dev

Mac OS X:sudo port install fftw-3-singlesudo port install gcc47sudo port install libpng

http://mrirecon.github.io/bart/

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

Matlab/Octave:

〉 data = squeeze(readcfl(’out’));

〉 data = bart(’fft -i 7’, data);

〉 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)

pics: A Tool for Parallel Imaging Compressed Sensing

> bart pics -RA:B:C :D -R ... [-t trj] kspace sens image

I parallel imaging and compressed sensing

I non-Cartesian k-space trajectories

I multiple regularization terms

I A: different types of regularization:`2, `1, total variation, `1-wavelet, (multi-scale) low-rank

I B: transforms along arbitrary dimensions (space, time, etc.)

I C: joint-thresholding along arbitrary dimensions

I D: regularization parameter

Note: Depending on the algorithm additional parameters (step size,

number of iterations, etc.) must be set for optimal results.

Reconstruction Methods for Magnetic Resonance Imagingmuecker1/reco16_lec03.pdf · Introduction...

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