EE290T: Advanced 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: 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 Today

I Review

I Non-Cartesian MRI

I Non-uniform FFT

I Project 2 Direct Image Reconstruction

I Assumption: Signal is Fourier transform of the image:

Z ~ s(t) = d~x ρ(~x)e−i2π~x·k(t)

I Image reconstruction with inverse DFT

kx ⇒ ⇒ sampling iDFT

~ R t ~ image k(t) = γ 0 dτ G(τ) k-space Requirements

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

I Sampling on a Cartesian grid ⇒ Line-by-line scanning

α echo radio measurement time: slice TR ≥ 2ms read 2D: N ≈ 256 ⇒ seconds phase 3D: N ≈ 256 × 256 ⇒ minutes TR ×N

2D MRI sequence Poisson Summation Formula

Periodic summation of a signal f from discrete samples of its Fourier transform fˆ: ∞ ∞ X X 1 l 2πi l t f (t + nP) = fˆ( )e P P P n=−∞ l=−∞

No aliasing ⇒ Nyquist criterium

l For MRI: P = FOV ⇒ k-space samples are at k = FOV 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 Other positions can be recovered by sinc interpolation (Whittaker-Shannon formula)

image k-space

I Periodic summation from discrete Fourier samples on grid 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 than W cps, it is completely determined by giving its ordinates at a series 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) Fast Fourier Transform (FFT) Algorithms

Discrete Fourier Transform (DFT):

N−1 N−1 X 2π N ˆ i N kn DFTk {fn}n=0 := fk = e fn for k = 0, ··· , N − 1 n=0

Matrix multiplication: O(N2)

Fast Fourier Transform Algorithms: O(N log N) Cooley-Tukey Algorithm

Decomposition of a DFT:

N−1 N N−1 X kn DFTk {fn}n=0 = (ξN ) fn n=0

N1−1 N2−1 X k1n1 k2n1 X k2n2 = (ξN1 ) (ξN ) (ξN2 ) fn1+n2N1 n1=0 n2=0 n oN1−1 = DFT N1 (ξ )k2n1 DFT N2 {f }N2−1 k1 N k2 n1+n2N1 n2=0 n1=0

I Recursive decomposition to prime-sized DFTs ⇒ Eﬃcient computation for smooth sizes

(eﬃcient algorithms available for all other sizes too) Parallel MRI: Iterative Algorithms

Signal equation: Z ~ −i2π~x·~k si (k) = d~x ρ(~x) ci (~x)e V | {z } encoding functions

Discretization:

A = Pk FC

A has size 2562 × (8 × 2562) ⇒ Iterative methods built from matrix-vector products sensitivities Ax, AH y Landweber Iteration

Gradient descent: 1 φ(x) = kAx − yk2 2 2 ∇φ(x) = AH (Ax − y)

Iteration rule:

H H xn+1 = xn − µA (Axn − y) with µkA Ak ≤ 1 H H = (I − µA A)xn − A y

Explicit formula for x0 = 0:

n−1 X H j H xn = (I − µA A) µA y j=0 Projection Onto Convex Sets (POCS)

Iterative projection onto convex sets:

yn+1 = PB PAyn

Problems: slow, not stable, A ∩ B = ∅ POCSENSE

Iteration:

yˆn+1 = PC Py yˆn

(multi-coil k-spacey ˆ)

Projection onto sensitivities:

H −1 PC = FCC F

(normalized sensitivities)

Projection onto data y:

Py yˆ = My + (1 − M)ˆy

AA Samsonov, EG Kholmovski, DL Parker, and CR Johnson. POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging. Magn Reson Med, 52:1397–1406, 2004. Krylov Subspace Methods

Krylov subspace:

n Kn = spani=0···n {T b}

⇒ Repeated application of T .

Landweber, Arnoldi, Lanczos, Conjugate gradients, ... Conjugate Gradients

T symmetric (or Hermitian)

Initialization:

r0 = b − Tx0

d0 = r0 Iteration:

qi ⇐ Tdi 2 |ri | α = H R ri qi xi+1 ⇐ xi + αdi

ri+1 ⇐ ri − αqi 2 |ri+1| β = 2 |ri |

di+1 ⇐ ri+1 + βdi Conjugate Gradients on Normal Equations

Normal equations:

H H 2 A Ax = A y ⇔ x = argminx kAx − yk2

Tikhonov:

H H 2 2 A A + αI x = A y ⇔ x = argminx kAx − yk2 + αkxk2

CGNE:

Tx = b

with:

T = AH A b = AH y Implementation

repeat

FFT P IFFT

sum channels point-wise mult. with conjugate sensitivities data ﬂow inverse FFT sampling mask or projection onto data

multi-dimensional FFT

point-wise multiplication with sensitivities

distribute image Conjugate Gradient Algorithm vs Landweber

conjugate gradients Landweber

2 1 1. 1.414 1.732 0 2. 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 = PFC AH = C H F −1PH

Hints:

I Use unitary and centered (ﬀtshift) FFT kFxk2 = kxk2 I Implement undersampling as a mask, store data with zero H I Check hx, Ayi = A x, y for random vectors x, y Project 1: Iterative SENSE

Step 2: Implement Reconstruction

Landweber (gradient descent)1

H xn+1 = xn + αA (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 Review

I Non-Cartesian MRI

I Non-uniform FFT

I Project 2 Parallel Imaging as Inverse Problem

Model: Signal from multiple coils (image ρ, sensitivities cj ): Z −i2π~x·~k(t) sj (t) = d~x ρ(~x)cj (~x)e + nj (t) Ω 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 Non-Cartesian Trajectories

Examples:

Cartesian radial spiral Non-Cartesian Trajectories

Practical issues:

I Long readout: blurring, artifacts

I Imperfect gradient waveforms (e.g. delays, eddy currents)

I Eﬃcient implementation of the reconstruction algorithm Spiral

Long readout:

I Eﬃcient! ? I T2 -Relaxation ⇒ blurring I Oﬀ-resonance Spiral

Signal equation: Z itγ∆B (~x) −t/T ?(~x) −2πi~k~x y(t) = dx ρ(~x)e 0 e 2 e Ω Oﬀ-resonance:

itγ∆B0(~x) |t − tTE | ∆γB0 ⇒ e ≈ const

Relaxation:

? ? −t/T2 |t − tTE | T2 ⇒ e ≈ const Spiral

Signal equation: Z itγ∆B (~x) −t/T ?(~x) −2πi~k~x y(t) = dx ρ(~x)e 0 e 2 e Ω Oﬀ-resonance:

itγ∆B0(~x) |t − tTE | ∆γB0 ⇒ e ≈ const

Relaxation:

? ? −t/T2 |t − tTE | T2 ⇒ e ≈ const Radial

π I Not eﬃcient (Nyquist: 2 × spokes vs. Cartesian lines) I Motion robustness

I High tolerance to undersampling

I Continuous update possible Radial - Undersampling Artifacts

Left to right: 96, 48, 24 radial spokes Forward Problem

Signal equation:

Z ~ −2πikl~x yl = d~x ρ(~x)e Ω

Non-Cartesian sample positions: kl Discretization: ρ(~x) pixel values at grid positions ~x

~ X −2πikl~x yl = ρ(~x)e ~x ⇒ non-uniform DFT from grid to non-Cartesian positions. Inverse Non-Uniform Fourier Transform

Forward model:

Z ~ ikl~x yl = d~x ρ(~x)e

Inverse problem:

y = Pk Fρ = Aρ

Best-approximation (least-squares + minimum-norm):

−1 H H x = limα→0A AA + αI y Non-uniform FFT

Fourier transform at arbitray (non-Cartesian) positions kl Z −2πikl x yl = dx f (x)e

DFT at arbitray (non-Cartesian) positions kl

N−1 k n X −2πi l yl = xne N n=0

Fast computation: nuFFT Interpolation to Non-Cartesian Data

I Periodic extension: FFT yields Fourier transform on grid

I Values at k-space positions recovered by sinc interpolation

Whittaker-Shannon formula: ∞ X t − nT sin(πx) f (t) = f (nT ) sinc( ) sinc(x) = T πx n=−∞ Approximation of the Interpolation

Computation:

N/2−1 X t − nT f (t) = f (nT ) kern( ) T n=−N/2

I Sinc interpolation: not practical

I Compactly supported kernels: truncated Gaussian, Kaiser-Bessel Kernel with compact support:

kern(x) = 0 for |x| >= wT

⇒ Sum has only 2 ∗ w terms ⇒ Eﬃcient computation Approximation of the Interpolation

Computation (2D):

N/2−1 M/2−1 X X s − nS t − nT f (s, t) = f (nS, mT ) kern( ) kern( ) S T n=−N/2 m=−M/2 Approximation of the Interpolation

Computation (2D):

N/2−1 M/2−1 X X s − nS t − nT f (nS, mT ) kern( ) kern( ) S T n=−N/2 m=−M/2

Isotropic kernel:   N/2−1 M/2−1 s 2 2 X X s − nS t − nT f (nS, mT ) kern +  S T  n=−N/2 m=−M/2 Approximation of the Interpolation

I Sinc interpolation: not practical

I Finite kernel: Kaiser-Bessel interpolation

I Oversampling to reduce approximation errors

stop pass stop Approximation of the Interpolation

I Sinc interpolation: not practical

I Finite kernel: Kaiser-Bessel interpolation

I Oversampling to reduce approximation errors

stop pass stop Approximation of the Interpolation

I Sinc interpolation: not practical

I Finite kernel: Kaiser-Bessel interpolation

I Oversampling to reduce approximation errors

stop pass stop Approximation of the Interpolation

I Sinc interpolation: not practical

I Finite kernel: Kaiser-Bessel interpolation

I Oversampling to reduce approximation errors

stop pass stop Approximation of the Interpolation

I Sinc interpolation: not practical

I Finite kernel: Kaiser-Bessel interpolation

I Oversampling to reduce approximation errors

stop pass stop Gridding

Best approximation:

−1 A† = lim AH AAH + αI α→0 Gridding: approximate inverse:

G = AH D ≈ A†

Approximations:

I Interpolation with a ﬁnite kernel (as discussed) H −1 I In addition: diagonal approximation to AA Gridding - Density Compensation

I Diagonal matrix D in the Fourier domain

I Usually approximated somehow (Voronoi diagrams)

I Intuition: “density compensation”

I Exact inverse for (continuous) Radon transformation Gridding - Density Compensation

Ram-Lak ﬁlter No density compensation Iterative Gridding

Normal equations: AH Ax = AH y

Iterative solution:

I Conjugate gradient algorithm H H I Repeated application of A A to A y

Properties:

I No density compensation needed

I More accurate than gridding

I Required for parallel imaging

I Problem: reconstruction speed Density Compensation as Preconditioner

Can we have both? Left preconditioning: ¯ Ax = y ⇒ DAx = Dy

Idea: AH D ≈ A−1 ⇒ AH DA ≈ I

Apply CG algorithm to:

AH DAx = AH Dy

⇒ faster convergence (usually) Density Compensation as Preconditioner

Problems: H I Usually A D not a good approximation (especially with undersampling)

I Weighted least-squares solution (⇒ suboptimal SNR):

H H 1/2 A DAx = A Dy ⇔ argminx kD (Ax − y)k2

Preconditioner for normal equations:

LAH Ax = LAH y or AH ARz = AH y Rz = x Iterative Reconstruction

Observation: basic operation is AH A.

Convolution with PSF:

H −1 A A = F Pk F | {z } convolution with PSF Convolution

Deﬁnition: Z (f ? g)(~x) = d~y f (~y)g(~x − ~y) Rn Convolution theorem:

F{a ? b} = F{a}·F{b} ⇒ a ? b = F −1 {F{a}·F{b}} Iterative Reconstruction

Main operation:

ρˆ = AH Aρ

Deﬁnition of adjoint:

hσ, ρˆi = hAσ, Aρi

Then:

X ~ Z ~ ρˆ(y) = e+2πikl~y d~x ρ(~x)e−2πikl~x l Z X = d~x ρ(~x) e2πikl (y−x) l | {z } PSF Iterative Reconstruction: Alternative Implementation

Discretization: pixel values on ﬁnite grid: xr , ys Z X 2πikl (y−x) X X 2πikl (yr −xs ) d~x ρ(~x) e ≈ ρ(~xr ) e l s l

Hermitian Toeplitz matrix: Txy = ax−y   a0 a1 ... ?  a a0 a1   1   ......   . . .   ?   a1 a0 a1  ? a1 a0 ⇒ Two applications of FFT algorithm Convolution

Note: Discretization not necessary (at this point)

Poisson summation formula: ∞ ∞ X X 1 l 2πi l t f (t + nP) = fˆ( )e P P P n=−∞ l=−∞

Periodic convolution: 1. Computation of Fourier coeﬃcients 2. Multiplication with Fourier coeﬃcients of (periodic) kernel 3. Summation of Fourier series 1. Truncated PSF (twice FOV) 2. Periodic convolution 3. Restricted support

Convolution

Aperiodic convolution: 1. Truncated PSF (twice FOV) 2. Periodic convolution 3. Restricted support

Convolution

Aperiodic convolution: 2. Periodic convolution 3. Restricted support

Convolution

Aperiodic convolution:

1. Truncated PSF (twice FOV) 3. Restricted support

Convolution

Aperiodic convolution:

1. Truncated PSF (twice FOV) 2. Periodic convolution Convolution

Aperiodic convolution:

1. Truncated PSF (twice FOV) 2. Periodic convolution 3. Restricted support Iterative gridding

NH Nx = NH y

N nuFFT.

Alternative method

H −1 H R F MTF FRx = N×2y

MTF Multiplication with Fourier transform of truncated PSF R Restriction to FOV Non-Cartesian SENSE

Model: Signal from multiple coils (image ρ, sensitivities cj ): Z −i2π~x·~k(t) sj (t) = d~x ρ(~x)cj (~x)e + nj (t) Ω

Cartesian:

A = PFC

P sampling mask

Non-Cartesian:

A = NFC

N non-uniform DFT Refocusing Reconstruction

Long readout: blurring, artifacts

Signal:

Z ~ s(t) = dx ρ(~x)eitγ∆B0(~x)e−2πik(t)~x Ω

Refocusing:

I Takes ﬁeld map ∆B0 into account I Removes blurring/artifacts due to oﬀ-resonance Project 2: Non-Cartesian MRI

Project: Implement non-Cartesian reconstruction

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

I Deadline: Mar 17

I Hand in: Working code and plots/results with description.

I See website for data and instructions. Project 2: Non-Cartesian MRI

Step 1: Computation of non-Cartesian samples

I Compute non-uniform DFT matrix

I Implement nuFFT and its adjoint

I 1D and 2D versions

Hints:

I Oversampling ×2

I Gaussian kernel (not optimal) Project 2: Non-Cartesian MRI

Step 2: Implement iterative gridding

I Reconstruction from non-Cartesian data

I Use Landweber and/or CG

Step 3: Implement non-Cartesian SENSE

I Implement non-Cartesian SENSE

1. JD O’Sullivan. A fast sinc function gridding algorithm for Fourier inversion in computer tomography. IEEE Trans Med Imaging, 4:200-207, 1985. 2. JI Jackson, CH Meyer, DG Nishimura, and A Macovski. Selection of a convolution function for Fourier inversion using gridding. IEEE Trans Med Imaging, 3:473-478, 1991. 3. KP Pruessmann, M Weiger, P B¨ornert,and P Boesiger. Advances in sensitivity encoding with arbitrary k-space trajectories. Magn Reson Med, 46:638-651, 2001.