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 Nonlinear Inverse Reconstruction

I k-space Methods

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

Projects

I Two pre-defined homework projects

I Final project

I Groups: 2-3 people

I 1. Project: iterative SENSE (Feb 03 - Feb 24)

I 2. Project: non-Cartesian MRI (Feb 24 - Mar 17)

I 3. Final Project (Mai 17 - Apr 28)

I Grading: 25%, 25%, 50%

Today

I Introduction to MRI

I Basic image reconstruction

I Inverse problems

Introduction to MRI

I Non-invasive technique to look “inside the body”

I No ionizing radiation or radioactive materials

I High spatial resolution

I Multiple contrasts

I Volumes or arbitrary crossectional slices

I Many applications in radiology/neuroscience/clinical research

Disadvantages

Some disadvantages:

I Expensive

I Long measurement times

I Not every patient is admissible(cardiac pacemaker, implants, tattoo, ...)

MRI Scanner

I MagnetI Radiofrequency coils (and electronics)I Gradient system

I Superconducting magnet

I Field strength: 0.25 T - 9.4 T (human)

MRI Scanner

I MagnetI Radiofrequency coils (and electronics)I Gradient system

I Spin excitation

I Signal detection

MRI Scanner

I MagnetI Radiofrequency coils (and electronics)I Gradient system

I Creates magnetic field gradients on top of the static B0 field

I Switchable gradients in all directions

Protons in the Magnetic Field

energy

magnetic field

∆E = ~γB0

B0

I Protons have spin 1/2(“little magnets”)

I Two energy levels ∆E = 2~γ 12B

0

I Larmor precession ∆E = ~ωLarmor

I Excitation with resonant radiofrequency pulses

I Macroscopic behaviour described by Bloch equations

Bloch Equations

Macroscopic (classical) behaviour of the magnetization correspondsto expectation value 〈~s〉 of spin operator (Pauli matrices).

Dynamics described by differential equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

The Basic NMR-Experiment

x

z

y

Equilibrium

~M

RF-Signal

t

x

y

Bloch equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

The Basic NMR-Experiment

x

z

y

Excitation

~M

RF-Signal

t

x

y

Bloch equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

The Basic NMR-Experiment

x

z

y

Precession

~M

RF-Signal

t

x

y

Bloch equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

The Basic NMR-Experiment

x

z

y

T2-decay

~M

RF-Signal

t

x

y

Bloch equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

The Basic NMR-Experiment

x

z

y

T1-recovery

~M

RF-Signal

t

x

y

Bloch equations:

d

dt〈~s〉 = − e

m0〈~s〉 ×

B1x (t)

B1y (t)B0z

+

−1T2〈sx〉

− 1T2〈sy 〉

s0−〈sz 〉T1

Multiple Contrasts

I Measurement of various physical quantities possible

I Advantage over CT / X-Ray (measurement of tissue density)

Proton density T1-weighted T2-weighted

Receive Chain

ADC 0001011101

Coil Preamplifier Analog-Digital-ConverterSample Matching ComputerAmplifierMixer

Thermal noise:

I Sample noise

I Coil noise

I Preamplifier noise

phased-array coil

Equivalent Circuit

Lcoil

vsignal

R

vnoise

R = RC + RS

I RC loss in coil

I RS loss in sample

=> Johnson-Nyquist noise vn

Johnson-Nyquist Noise

Mean-square noise voltage:

v2n = 4kB T R BW

R resistance, T temperature, kB Boltzmann constant, BW bandwidth

I additive Gaussian white noise

I Brownian motion of electrons in body and coil

I Connection between resistance and noise:Fluctuation-Dissipation-Theorem

Spatial Encoding

Nobel price in medicine 2003for Paul C. Lauterburand Sir Peter Mansfield

Principle:

I Additional field gradients

⇒ Larmor frequency dependenton location

Concepts:

I Slice selection(during excitation)

I Phase/Frequency encoding(before/during readout)

PC Lauterbur, Image formation by induced local interactions: examples employing nuclear magnetic resonance,Nature, 1973

Frequency Encoding

I Additional field gradients B0(x) = B0 + Gx · x⇒ Frequency ω(x) = γB0(x)

B

x

B0

B0(x)

spin density ρ(x)

t

I Signal is Fourier transform of the spin density:

~k(t) = γ

∫ t

0dτ ~G (τ) s(t) =

∫Vd~x ρ(~x)e−i2π

~k(t)·~x

(ignoring relaxation and errors)

Frequency Encoding

I Additional field gradients B0(x) = B0 + Gx · x⇒ Frequency ω(x) = γB0(x)

B

x

B0

B0(x)

spin density ρ(x)

t

I Signal is Fourier transform of the spin density:

~k(t) = γ

∫ t

0dτ ~G (τ) s(t) =

∫Vd~x ρ(~x)e−i2π

~k(t)·~x

(ignoring relaxation and errors)

Gradient System

I z gradient

I y gradient

I x gradient

Gradient System

I z gradient

I y gradient

I x gradient

Gradient System

I z gradient

I y gradient

I x gradient

Gradient System

I z gradient

I y gradient

I x gradient

Gradient System

I z gradient

I y gradient

I x gradient

Imaging Sequence

slice excitation

acquisition

radio

α

slice

read

phase

repetition time TR

FLASH (Fast Low Angle SHot)

Imaging Sequence

readout (in-plane)

acquisition

radio

α

slice

read

phase

repetition time TR

FLASH (Fast Low Angle SHot)

Fourier Encoding

acquisition

radio

α

slice

read

phase

repetition time TR

kread

kphase

Fourier Encoding

acquisition

radio

α

slice

read

phase

repetition time TR

kread

kphase

Image Reconstruction

k-space data

Image Reconstruction

inverse Discrete Fourier Transform

Direct Image Reconstruction

I Assumption: Signal is Fourier transform of the image:

s(t) =

∫d~x ρ(~x)e−i2π~x ·

~k(t)

I Image reconstruction with inverse DFT

kx

ky

~k(t) = γ∫ t

0 dτ ~G (τ)

⇒sampling

k-space

⇒iDFT

image

Requirements

I Short readout (signal equation holds for short time spans only)

I Sampling on a Cartesian grid

⇒ Line-by-line scanning

echo

radio

α

slice

readphase

TR ×N2D MRI sequence

measurement time:TR ≥ 2ms2D: N ≈ 256⇒ seconds3D: N ≈ 256× 256⇒ minutes

Sampling

Sampling

Sampling

Sampling

Poisson Summation Formula

Periodic summation of a signal f from discrete samples of itsFourier transform f̂ :

∞∑n=−∞

f (t + nP) =∞∑

l=−∞

1

Pf̂ (

l

P)e2πi l

Pt

No aliasing ⇒ Nyquist criterium

For MRI: P = FOV ⇒ k-space samples are at k = lFOV

image

k-space

I periodic summation from discrete Fourier samples on grid

I other positions can be recovered by sinc interpolation(Whittaker-Shannon formula)

image

k-space

I periodic summation from discrete Fourier samples on grid

I other positions can be recovered by sinc interpolation(Whittaker-Shannon formula)

image

k-space

I periodic summation from discrete Fourier samples on grid

I other positions can be recovered by sinc interpolation(Whittaker-Shannon formula)

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)

Sampling

Point Spread Function

I Signal:

y = PFx

sampling operator P, Fourier transform F

I Reconstruction with (continuous) inverse Fourier transform:

F−1PHPFx = psf ? x

⇒ Convolution with point spread function (PSF)

(shift-invariance)

I Question: What is the PSF for the grid?

Dirichlet Kernel

Dn(x) =n∑

k=−ne2πikx =

sin(π(2n + 1)x)

sin(πx)

(Dn ? f )(x) =

∫ ∞−∞

dy f (y)Dn(x − y) =n∑

k=−nf̂ (k)e2πikx

FWHM ≈ 1.2

D16

Gibbs Phenomenon

I Truncation of Fourier series

⇒ Ringing at jump discontinuities

Rectangular wave and Fourier approximation

Gibbs Phenomenon

I Truncation of Fourier series

⇒ Ringing at jump discontinuities

Rectangular wave and Fourier approximation

Gibbs Phenomenon

I Truncation of Fourier series

⇒ Ringing at jump discontinuities

Rectangular wave and Fourier approximation

Gibbs Phenomenon

I Truncation of Fourier series

⇒ Ringing at jump discontinuities

Rectangular wave and Fourier approximation

Direct Image Reconstruction

Assumptions:

I Signal is Fourier transform of the image

I Sampling on a Cartesian grid

I Signal from a limited (compact) field of view

I Missing high-frequency samples are small

I Noise is neglectable

Fast Fourier Transform (FFT) Algorithms

Discrete Fourier Transform (DFT):

DFTNk {fn}

N−1n=0 := f̂k =

N−1∑n=0

e i2πNknfn for k = 0, · · · ,N − 1

Matrix multiplication: O(N2)

Fast Fourier Transform Algorithms: O(N logN)

Cooley-Tukey Algorithm

Decomposition:

N = N1N2

k = k2 + k1N2

n = n2N1 + n1

With ξN := e i2πN we can write:

(ξN)kn = (ξN1N2)(k2+k1N2)(n2N1+n1)

= (ξN1N2)N2k1n1(ξN1N2)k2n1(ξN1N2)N1k2n2(ξN1N2)N1N2k1n2

= (ξN1)k1n1(ξN1N2)k2n1(ξN2)k2n2

(use: ξAAB = ξB and ξAA = 1 )

Cooley-Tukey Algorithm

Decomposition of a DFT:

DFTNk {fn}

N−1n=0 =

N−1∑n=0

(ξN)knfn

=

N1−1∑n1=0

(ξN1)k1n1(ξN)k2n1

N2−1∑n2=0

(ξN2)k2n2fn1+n2N1

= DFTN1k1

{(ξN)k2n1DFTN2

k2{fn1+n2N1}

N2−1n2=0

}N1−1

n1=0

I Recursive decomposition to prime-sized DFTs

⇒ Efficient computation for smooth sizes

(efficient algorithms available for all other sizes too)

Inverse Problem

Definition (Hadamard): A problem is called well-posed if

1. there exists a solution to the problem (existence),

2. there is at most one solution to the problem (uniqueness),

3. the solution depends continuously on the data (stability).

(we will later see that all three conditions can be violated)

Moore-Penrose Generalized Inverse

(A linear and bounded)

x is least-squares solution, if

‖Ax − y‖ = inf {‖Ax̂ − y‖ : x̂ ∈ X}

x is best approximate solution, if

‖x‖ = inf {‖x̂‖ : x̂ least-squares solution}

Generalized inverse A† : D(A†) := R(A) + R(A)⊥ 7→ X maps datato the best approximate solution.

Tikhonov Regularization

(inverse not continuous: ‖A−1‖ =∞)

Regularized optimization problem:

xδα = argminx‖Ax − y δ‖22 + α‖x‖2

2

Explicit solution:

xδα =(AHA + αI

)−1AH︸ ︷︷ ︸

A†α

y δ

Generalized inverse:

A† = limα→0

A†α

Tikhonov Regularization: Bias vs Noise

Noise contamination:

y δ = y + δn

= Ax + δn

Reconstruction error:

x − xδα = x − (AHA + αI )−1AHy δ

= x − (AHA + αI )−1AHAx + (AHA + αI )−1AHδn

= α(AHA + αI )−1x︸ ︷︷ ︸approximation error

+ (AHA + αI )−1AHδn︸ ︷︷ ︸data noise error

Morozov’s discrepancy principle:Largest regularization with ‖Axδα − y δ‖2 ≤ τδ

Example

y =

3.4525 3.3209 3.30903.3209 3.3706 3.31883.3090 3.3188 3.2780

10.50.2

+

1.6522e − 021.5197e − 033.3951e − 03

=

5.79235.67175.6274

Ill-conditioning: cond(A) = σmax

σmin≈ 10000

Solution:(α = 0)

x̂ = A−1y =

1.254141.02660−0.58866

Regularized solution:(α = 3× 10−6)

x̂α = A†αy =

1.097060.457510.14632

Tikhonov Regularization using SVD

Singular Value Decomposition (SVD):

A = UΣVH =∑j

σjUjVHj

Regularized inverse:

A†α =(AHA + αI

)−1AH =

∑j

σjσ2j + α

VjUHj

Approximation error:

1−σ2j

σ2j + α

=α

σ2j + α

Data noise error:

σjδ

σ2j + α

noise

approximation

α

Image Reconstruction as Inverse Problem

Forward problem:

y = Ax + n

x image (and more), A forward operator, n noise, y data

Regularized solution:

x? = argminx ‖Ax − 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, eddy currents)

I Efficient implementation of the reconstruction algorithm

Modelling of Physical Effects

Examples:

I Coil sensitivities (parallel imaging)

sj(t) =

∫d~x ρ(~x)cj(~x)e−i2π~x ·

~k(t)

I T2 relaxation

s(t) =

∫d~x ρ(~x)e−R(~x)te−i2π~x ·

~k(t)

I Field inhomogeneities

s(t) =

∫d~x ρ(~x)e−i∆B0(~x)te−i2π~x ·

~k(t)

I Diffusion, flow, motion, ...

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

Statistical Interpretation

Probability of x given y : p(x |y)

Likelihood: Probability of a specific outcome given a parameter:p(y |x)

Maximum-Likelihood-Estimator: The estimate is the parameterx which maximizes the likelihood.

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)

Prior Knowledge

Prior knowledge as a probability distribution for the the image:

p(x) = · · ·

A posterior probability distribution p(x |y) given the data y can becomputed using Bayes’ theorem:

p(x |y) =p(y |x)p(x)

p(y)

A point estimate for the image can then be obtained, for exampleby maximum a posteriori estimation (MAP), or by minimizingthe posterior expected value of a loss function, e.g. a minimummean squared error (MMSE) estimator.

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)

Summary

I Magnetic Resonance Imaging

I Direct image reconstruction

I Inverse problems