Retrospective Motion Correction
David Atkinson
FRIAS Junior Researcher Conference
Freiburg, December 2013.
Precursor 1:
Don’t forget human factors
• Patient Communication
• Patient Comfort
These can make a difference!
Precursor 2:
Take advantage of Prospective Correction
• Reduces motion artefacts.
• Reduces spin history and undersampling effects.
• But, gradients can only compensate affine motion.
• Sensors may not measure internal anatomy.
Retrospective Motion Correction
Measured k-space
artefact-free image
• Direct k-space corrections for affine motion.
• General Matrix Method – inverts a forward model.
• MRI signal model well understood:
– Measured k-space is Fourier Transform of what the coils
see.
– Motion measurement at image resolution is the problem.
Inferring Motion From Data
Autofocus / Consistency
• Iterative trial corrections
on portions of k-space.
• Cost function
– Image quality, e.g. entropy.
– Consistency, e.g. between
coil-based prediction and
trial correction.
• Requires only raw data.
• Many local minima.
• Large number of degrees
of freedom.
Registration
• Affine or non-rigid on
– Low-res snapshot images.
– Images reconstructed from
partial k-space.
– Respiratory bins.
• May require data
“oversampling”.
• Can build a motion
model.
Motion Models
• Registration deformation fields linked to a
“sensor”.
– Deformations from training scan or data itself
– “sensor” can be physical or processed MR signal.
• Predicts corrections for main scan.
• Can account for 3D motion and hysteresis.
Respiratory motion models: A review
Medical Image Analysis 2013; 17:19-42
McClelland et al.
K-Space Corrections for Affine Motion
Α
V V '
A( x)  Ax  d
k'  (A 1 )T k
i 2 (k'd)
e
S (k ) 
S ' (k')
det(A)
Image Motion
K-Space Effect
Translation (rigid shift)
Phase ramp
Rotation
Rotation (same angle)
Expansion
Contraction
General affine
Affine transform
[Guy Shechter Thesis]
Autofocus / Autocorrection / Blind Correction
Manduca et al, Radiology 2000; 215:904 evaluation in 144 patients
Lin et al, MRI 2006; 24:751 and JMRI 2007; 26:191. 26 clinical 3D wrist exams
Respiratory bin, iterate translation that
sharpens image
One 3D translation for all k-space lines in a bin.
Tenengrad sharpness
Nav: 5mm window
Un corrected
Corrected
Nav: 15mm window
Moghari, MRM 2013; 70:1005
Non-Rigid (Non-Affine) Motion
• Physiological motion is often non-rigid.
• No direct correction in k-space or using gradients.
• General Matrix Method (GMM)...
Forward Model
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.
Batchelor et al, MRM 2005;54:1273
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
Forward Model as Matrix-Vector Operations
Measured
k-space for
shot
=
sample
shot
m
FFT
k i
coil
sensitivity motion
motionfree
patient

E
*
ρ
Stack Data From All Shots, Averages and
Coils
m
E
*
ρ
Image Operations as Matrices and Vectors
• The trial motion-free image is converted to a
column vector.
motion-free
patient image
ρ
n
n
n2
Converting Image Operations to Matrices
patient
Measured
k-space
=
sample
FFT
k i
coil
motion
=
• 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.
Expressing Motion Transform as a Matrix
Measured
k-space
=
=
sample
FFT
k i
coil
motion
?
Motion Matrix Acts on Pixels, Not
Coordinates
Geometrical Transformation Matrix Applied to Coordinates
x’
y’
z’
=
*
x
y
z
Affine matrix A
=
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
*
For interpolation, use
non-integer matrix
entries.
Describes “permutation”
non-affine motion
Expressing Motion Transform as a Matrix
Measured
k-space
=
sample
FFT
k i
coil
motion
=
• For small displacements, matrix has a diagonal
band.
• Can describe motion more general than affine.
Conjugate Gradient Solution
min Eρ  m
ρ
2
Internally uses only matrix-vector products:
H
Ev and E w
E large:
• Use sparse representation, or,
• Supply functions that return result of matrix-vector
products.
– Functions use correspondence between matrix-vector
multiplications and image operations.
The Complex Transpose EH
H
motion
H
coil
E  S FCM
E H  M HCH F HS H
H
FFT
H
sample
• 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.
• MH, Hermitian transpose of motion matrix ... ?
MH Complex transpose of motion matrix
Options:
• Approximate by applying the inverse motion
transform as an image operation.
– approximate transform by negating displacements, or
– use deformation field from reverse registration.
• Find MH from sparse representation of M.
• Build explicitly using for-loops and accumulating
the results in an array.
Summary: General Matrix Method
• Inverse Problem with efficient Conjugate Gradient solution.
• Forward model incorporates physics of acquisition
– parallel imaging,
– phase errors in multi-shot DWI, flow effects, coil motion, contrast
changes, ...
•
•
•
•
Copes with some missing data or shot rejection.
Interpolation is in the (more benign) image domain (fwd).
Advantage of complex averaging.
Requires Motion Estimate.
=
=
?
Motion Estimation
• External Sensors
• Explicit Navigation
• Self-navigation
• Data Consistency / Image Quality
• Motion Model
• Hybrid Schemes
Motion Estimation
• External Sensors
cameras, respiratory belt, ...
• Explicit Navigation
• Self-navigation
• Data Consistency / Image Quality
• Motion Model
• Hybrid Schemes
Motion Estimation
•
•
•
•
Pencil beam navigators (diaphragm)
External Sensors K =0 projections
y
Orbital navigators
Explicit Navigation
Snapshot images, (Philips iScan)
FID readouts
Self-navigation
Fat navigators
Data Consistency / Image
Quality
2D phase
navigators (diffusion)
• Motion Model
• Hybrid Schemes
Motion Estimation
• External Sensors
• Explicit Navigation
• Self-navigation
PROPELLOR
Radial, spiral, golden radial
Undersampled recons e.g. CS
• Data Consistency / Image Quality
• Motion Model
• Hybrid Schemes
Motion Estimation
• External Sensors
• Explicit Navigation
• Self-navigation
Autofocus
Metric Optimised Gating
Coil Consistency
• Data Consistency / Image Quality
• Motion Model
• Hybrid Schemes
Motion Estimation
• External Sensors
• Explicit Navigation
• Self-navigation
• Data Consistency / Image Quality
• Motion Model
• Hybrid Schemes
Typically for respiratory motion
Motion Estimation
• External Sensors
• Explicit Navigation
• Self-navigation
• Data Consistency / Image Quality
• Motion Model Large variety – aim to maximise strengths
• Hybrid Schemes
Motion-induced phase errors in multi-shot
diffusion
multi-shot DWI
example phase
correction pt
artefact free image
Atkinson et al. MRM 56:1135–1139 (2006)
Motion model, “sensor” is aliased images
• Training scan to build model.
• Parameterised by mean vertical displacement.
• Undersampled frames compared with warped
model to obtain current value of model parameter.
• GMM applied
White et al MRM 2009; 62:440-449
Locally affine registrations, GMM
Golden radial acquisition
Respiratory bins
Local affine registrations for deformation fields
Motion in full matrix method compared to warp & add, and, gating.
General matrix method
Warp & add bins
5mm gating
Gastao Cruz, submitted to ISMRM 2014
Locally affine registrations, GMM
Golden radial acquisition
Respiratory bins
Local affine registrations for deformation fields
Motion in full matrix method compared to warp & add, and, gating.
General matrix method
Warp & add bins
5mm gating
Gastao Cruz, submitted to ISMRM 2014
Butterfly Navs + Local Linear 3D Autofocus
Paediatric renal tumour
Butterfly nav in x then y then z.
Translation found for each of N coil
elements.
N2 translational motion corrections,
best (gradient entropy) selected locally.
Locally linear, 3D corrections.
Cheng et al, MRM 2012; 68: 1785
GRICs: Coupled reconstruction of motion
and image
Sensors ky=0 profiles from multiple coil elements.
Linear model links deformation to sensors.
Optic flow on image reconstruction residue links image to
motion model parameters.
averaged cine
GRICs
Odille et al, MRM 2010; 63:1247
Conclusions
• Retrospective correction:
– direct, or
– general matrix method.
• Motion estimation the primary problem.
• Clinical uptake limited:
– Algorithms must be reliable or fast.
– Raw data sizes are huge.
• Techniques optimised to anatomy & sequence.
• Effective solutions are possible.