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Declaration of Conflict of Interest or
Relationship

David Atkinson
:


I have no conflicts of interest to disclose with regard to the subject matter
of this presentation.


Image Reconstruction: Motion Correction


David Atkinson


D.Atkinson@ucl.ac.uk


Centre for Medical Image Computing,

University College London

with thanks to

Freddy Odille, Mark White, David
Larkman
,

Tim Nielsen, Murat
Askoy
, Johannes Schmidt.

Problem: Slow Phase Encoding


Acquisition slower than physiological motion.


motion artefacts.



Phase encode FOV just large enough to prevent
wrap around.


minimises acquisition time,


Nyquist
: k
-
space varies rapidly making interpolation
difficult.



Motion and K
-
Space

k
-
space acquired in time

image

Fourier Transform

ikx
k
k
x
e
S
s


The sum in the Fourier Transform means that motion at any
time can affect every pixel.

K
-
Space Corrections for Affine Motion

Image Motion

Translation (rigid shift)

Rotation

Expansion

General affine

K
-
Space Effect

Phase ramp

Rotation (same angle)

Contraction

Affine transform

[Guy
Shechter

PhD Thesis]

Rotation Example

Time

Example rotation
mid
-
way through scan.

Ghosting in phase encode direction.

Interpolation, Gridding and Missing Data

FFT requires regularly spaced samples.

Rapid variations of k
-
space make interpolation difficult.


K
-
space missing in some regions.

Prospective Motion Correction

Motion determined during scan & plane updated using
gradients.


Prevents pie
-
slice missing data.


Removes need for interpolation.


Prevents through
-
slice loss of data.


Can instigate re
-
acquisition.


Reduces reliance on post
-
processing.



Introduces relative motion of coil sensitivities, distortions & field
maps.


Difficult to accurately measure tissue motion in 3D.


Gradient update can only compensate for affine motion.

Non
-
Rigid Motion


Most physiological motion is non
-
rigid.


No direct correction in k
-
space or using gradients.


A flexible approach is to solve a matrix equation
based on the forward model of the acquisition and
motion.

Forward Model and Matrix Solution

m
E
ρ

“Encoding”
matrix
with

motion, coil
sensitivities etc

Measured data

Artefact
-
free
Image

2
min
m
E
ρ
ρ

Least squares solution:

Conjugate gradient techniques
such as LSQR.

The Forward Model as Image Operations

motion
-
free

patient

motion

coil

sensitivity

sample

shot

=

Measured

k
-
space for

shot

FFT

k i

Image transformation at current shot

Multiplication of image by coil sensitivity map

Fast Fourier Transform to k
-
space

Selection of acquired k
-
space for current shot

Shots

spin
echo

1 readout = 1 shot

single
-
shot EPI

multi
-
shot

Forward Model as Matrix
-
Vector Operations

motion
-
free

patient

motion

coil

sensitivity

sample

shot

=

Measured

k
-
space for

shot

FFT

k i

ρ
E
m


*

Converting Image Operations to Matrices


The trial motion
-
free image is converted to a
column vector.

n

n

n
2

ρ
motion
-
free

patient image

Expressing Motion Transform as a Matrix


?

=

motion

coil

FFT

sample

Measured

image

k i

=


Matrix acts on pixels, not coordinates.


One pixel rigid shift


shifted diagonal.


Half pixel rigid shift


diagonal band, width depends
on interpolation kernel.


Shuffling (non
-
rigid) motion
-

permutation matrix.

Converting Image Operations to Matrices


Pixel
-
wise image multiplication of coil sensitivities
becomes a diagonal matrix.


FFT can be performed by matrix multiplication.


Sampling is just selection from k
-
space vector.

patient

=

motion

coil

FFT

sample

Measured

image

k i

=

Stack Data From All Shots, Averages and
Coils

ρ
E
m

*

Conjugate Gradient Solution


Efficient: does not require
E

to be computed or
stored.


User must supply functions to return result of
matrix
-
vector products



We know the correspondence between matrix
-
vector multiplications and image operations,
hence we can code the functions.


w
E
Ev
H

and

2
min
m
E
ρ
ρ

The Complex Transpose E
H


Reverse the order of matrix operations and take
Hermitian

transpose.


Sampling matrix is real and diagonal hence unchanged by
complex transpose.


FFT changes to
iFFT
.


Coil sensitivity matrix is diagonal, hence take complex
conjugate of elements.


Motion matrix ...



motion



FFT

H

H

H

H

coil

sample

Complex transpose of motion matrix

Options:


Approximate by the inverse motion transform.


Approximate the inverse transform by negating
displacements.


Compute exactly by assembling the sparse matrix
(if not too large and sparse).


Perform explicitly using for
-
loops and
accumulating the results in an array.

Example Applications of Solving Matrix
Eqn

averaged cine

‘sensors’ from central k
-
space
lines input to coupled solver
for motion model and artefact
-
free image.

multi
-
shot DWI

example phase correction

artefact free image

Summary: Forward Model Method


Efficient Conjugate Gradient solution.


Incorporates physics of acquisition including parallel
imaging.


Copes with missing data or shot rejection.


Interpolates in the (more benign) image domain.


Can include other artefact causes e.g. phase errors in
multi
-
shot DWI, flow artefacts, coil motion, contrast uptake.


Can be combined with prospective acquisition.


Often regularised by terminating iterations.


Requires knowledge of motion.


Alternative Iterative Reconstruction



+

rotate

+
shift

rotate

+
shift

coil

1

coil

2

coil

3

weight

with

the

coil

sensitivities

combine

updates

unfold

-


Fourier
-
transform each
interleave.


Initialize image: I=0

fold

measurement

data

[Nielsen et al. #3048]

motion

compensated

uncorrected

Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


ECG,

respiratory bellows,

optical tracking,

ultrasound (#3961),

spirometer

(#1553),

accelerometer (#1550).

Estimating Motion


External measures.


Explicit navigator measures
.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


pencil beam navigator,

central k
-
space lines,

orbital navigators,

rapid, low resolution images,

FID navigators.

Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


repeated
acq

near k
-
space centre,

PROPELLER,

radial & spiral acquisitions,

spiral projection imaging,

Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency
.


Iterative methods.


Motion models.


Predict and compare k
-
space lines.


Detect and minimise artefact source
to make multiple coil images
consistent.


Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


Find model parameters to minimise
cost function e.g. image entropy, coil
consistency.

Estimating Motion


External measures.


Explicit navigator measures.


Self
-
navigated sequences.



Coil consistency.


Iterative methods.


Motion models.


Link a model to scan
-
time signal.


Solve for motion model and image in
a coupled system (GRICS).

Example combined prospective and
retrospective methods at ISMRM 2010


5 mm
gating

window

2x SENSE


20 mm window

Motion corrected
2xSENSE

20 mm window

34.7 min

9 min

9 min

[Schmidt et al #492]

Retrospective correction: motion model from low res images,

LSQR solution, 6 iterations.

Implicit SENSE allows
undersampling

Example combined prospective and
retrospective methods at ISMRM 2010

[
Aksoy

et al #499]

no correction

prospective

optical tracking

additional entropy
-
based

autofocus

Outlook


Reconstruction times, 3D and memory still
challenging.



Expect intelligent use of prior knowledge:
sparsity
, motion models, atlases etc.



Optimum solution target dependent. Power in
combined acquisition and reconstruction methods.

www.ucl.ac.uk/cmic

Physiological Motion can be useful


Functional information


cardiac wall motion, bowel motility.


Elastography
.


Randomising acquisition for compressed sensing
reconstruction.