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.
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