Transcript motion - UCL Computer Science

Declaration of Relevant Financial Interests
or Relationships
David Atkinson:
I have no relevant financial interest or relationship to disclose with regard
to the subject matter of this presentation.
Image Reconstruction: Motion Correction
David Atkinson [email protected]
Centre for Medical Imaging,
Division of Medicine,
University College London
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.
Any Motion Can Corrupt Entire Image
image
s(x)
Fourier Transform
s( x)   S (k ) eikx
k
k-space acquired in time
The sum over all k in the Fourier Transform means that
motion at any time can affect every pixel.
K-Space Corrections for Affine Motion
Image Motion
K-Space Effect
Translation (rigid shift)
Phase ramp
Rotation
Rotation (same angle)
Expansion
Contraction
General affine
Affine transform
Α
V V '
A( x)  Ax  d
1 T
k'  (A ) k
i 2 (k'd)
e
S (k ) 
det( A)
S ' (k' )
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
Eρ  m
“Encoding” matrix
with motion, coil
sensitivities etc
Measured data
Artefact-free Image
min Eρ  m
ρ
2
Least squares solution:
Conjugate gradient techniques
such as LSQR.
The Forward Model as Image Operations
Measured
k-space for
shot
= sample
shot
FFT
k i
coil
sensitivity motion
motionfree
patient
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
single-shot EPI
multi-shot
spin echo
1 readout = 1 shot
Forward Model as Matrix-Vector Operations
Measured
k-space for
shot
=
sample
shot
m
FFT
k i
coil
sensitivity motion
motionfree
patient

E
*
ρ
Converting Image Operations to Matrices
• The trial motion-free image is converted to a
column vector.
motion-free
patient image
ρ
n
n
n2
Expressing Motion Transform as a Matrix
Measured =
image
=
sample
FFT
k i
coil
motion
?
• 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
Measured =
image
=
sample
FFT
k i
coil
motion
Stack Data From All Shots, Averages and
Coils
m
E
*
ρ
Conjugate Gradient Solution
min Eρ  m
ρ
• Efficient: does not require E to be computed or
stored.
• User must supply functions to return result of
matrix-vector products Ev and E H w
• We know the correspondence between matrixvector multiplications and image operations,
hence we can code the functions.
2
The Complex Transpose EH
H
motion
H
coil
H
FFT
H
sample
• 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 ...
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 artefactfree 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.
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 probes
spirometers
accelerometers
Power deposition
Field Probes
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.
Predict and compare k-space lines.
• Coil consistency.
• Iterative methods.
• Motion models.
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).
Golden Angle Sampling For Motion Detection
and High Resolution Imaging
• Determine motion from registration of images
(each from wide temporal range).
• Use motion in CG high resolution reconstruction.
[Hansen et al #749]
Outlook
• Prospective corrections limited to affine motion.
• 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.