Course

Report
Image
Registration
Carlton CHU
*
*
*
*
*
Smooth
Realign
Normalise
Segment
DARTEL
With slides by John Ashburner, Chloe Hutton and Jesper Andersson
Overview of SPM Analysis
Motion
Correction
Statistical Parametric Map
Design matrix
fMRI time-series
Smoothing
General Linear Model
Spatial
Normalisation
Anatomical Reference
Parameter Estimates
Contents
* Preliminaries
*
*
*
*
Smooth
Rigid-Body and Affine Transformations
Optimisation and Objective Functions
Transformations and Interpolation
* Intra-Subject Registration
* Inter-Subject Registration
Smooth
Smoothing is done by convolution.
Each voxel after smoothing effectively
becomes the result of applying a weighted
region of interest (ROI).
Before convolution
Convolved with a circle
Convolved with a Gaussian
Image Registration
• Registration - i.e. Optimise the parameters
that describe a spatial transformation
between the source and reference
(template) images
• Transformation - i.e. Re-sample according
to the determined transformation
parameters
2D Affine Transforms
* Translations by tx and ty
* x1 = x0 + tx
* y1 = y0 + ty
* Rotation around the origin by  radians
* x1 = cos() x0 + sin() y0
* y1 = -sin() x0 + cos() y0
* Zooms by sx and sy
* x1 = sx x0
* y1 = sy y0
*Shear
*x1 = x0 + h y0
*y1 = y0
2D Affine Transforms
* Translations by tx and ty
* x1 = 1 x0 + 0 y0 + tx
* y1 = 0 x0 + 1 y0 + ty
* Rotation around the origin by  radians
* x1 = cos() x0 + sin() y0 + 0
* y1 = -sin() x0 + cos() y0 + 0
* Zooms by sx and sy:
* x1 = sx x0 + 0 y0 + 0
* y1 = 0 x0 + sy y0 + 0
*Shear
*x1 = 1 x0 + h y0 + 0
*y1 = 0 x0 + 1 y0 + 0
3D Rigid-body Transformations
* A 3D rigid body transform is defined by:
* 3 translations - in X, Y & Z directions
* 3 rotations - about X, Y & Z axes
* The order of the operations matters
1


0

0

0
0
0
1
0
0
1
0
0
Xtrans 
1

 
Ytrans   0

Ztrans   0
 
1   0
Translations
0
0
cos Φ
 sinΦ
sinΦ
cos Φ
0
0
Pitch
about x axis
0 
 cos Θ

 
0  0

0    sinΘ
 
1   0
0
sinΘ
1
0
0
cos Θ
0
0
Roll
about y axis
0 
 cos Ω

 
0    sinΩ

0  0
 
1   0
sinΩ
0
cos Ω
0
0
1
0
0
Yaw
about z axis
0 
0 

0

1 
Voxel-to-world Transforms
* Affine transform associated with each image
* Maps from voxels (x=1..nx, y=1..ny, z=1..nz) to some world
co-ordinate system. e.g.,
* Scanner co-ordinates - images from DICOM toolbox
* T&T/MNI coordinates - spatially normalised
* Registering image B (source) to image A (target) will
update B’s voxel-to-world mapping
* Mapping from voxels in A to voxels in B is by
* A-to-world using MA, then world-to-B using MB-1
* MB-1 MA
Optimisation
* Optimisation involves finding some “best”
parameters according to an “objective function”,
which is either minimised or maximised
* The “objective function” is often related to a
probability based on some model
Objective
function
Most probable solution
(global optimum)
Local optimum
Local optimum
Value of parameter
Objective Functions
* Intra-modal
* Mean squared difference (minimise)
* Normalised cross correlation (maximise)
* Entropy of difference (minimise)
* Inter-modal (or intra-modal)
*
*
*
*
Mutual information (maximise)
Normalised mutual information (maximise)
Entropy correlation coefficient (maximise)
AIR cost function (minimise)
Transformation
* Images are re-sampled. An example in 2D:
for y0=1..ny0 % loop over rows
for x0=1..nx0 % loop over pixels in row
x1 = tx(x0,y0,q) % transform according to q
y1 = ty(x0,y0,q)
if 1x1 nx1 & 1y1ny1 then % voxel in range
f1(x0,y0) = f0(x1,y1) % assign re-sampled value
end % voxel in range
end % loop over pixels in row
end % loop over rows
* What happens if x1 and y1 are not integers?
Simple Interpolation
* Nearest neighbour
* Take the value of the
closest voxel
* Tri-linear
* Just a weighted
average of the
neighbouring voxels
* f5 = f1 x2 + f2 x1
* f6 = f3 x2 + f4 x1
* f7 = f5 y2 + f6 y1
B-spline Interpolation
A continuous function is represented by
a linear combination of basis functions
B-splines are piecewise polynomials
2D B-spline basis functions
of degrees 0, 1, 2 and 3
Nearest neighbour and
trilinear interpolation are
the same as B-spline
interpolation with degrees
0 and 1.
Contents
* Preliminaries
* Intra-Subject Registration
* Realign
* Mean-squared difference objective function
* Residual artifacts and distortion correction
* Coregister
* Inter-Subject Registration
Mean-squared Difference
* Minimising mean-squared difference works for
intra-modal registration (realignment)
* Simple relationship between intensities in one
image, versus those in the other
* Assumes normally distributed differences
Residual Errors from aligned fMRI
* Re-sampling can introduce interpolation errors
* especially tri-linear interpolation
* Gaps between slices can cause aliasing artefacts
* Slices are not acquired simultaneously
* rapid movements not accounted for by rigid body model
* Image artefacts may not move according to a rigid body model
* image distortion
* image dropout
* Nyquist ghost
* Functions of the estimated motion parameters can be
modelled as confounds in subsequent analyses
Movement by Distortion Interaction
of fMRI
*Subject disrupts B0 field,
rendering it inhomogeneous
=> distortions in phaseencode direction
*Subject moves during EPI
time series
*Distortions vary with
subject orientation
=> shape varies
Movement by distortion interaction
Correcting for distortion changes using
Unwarp
Estimate reference from
mean of all scans.
Estimate new distortion
fields for each image:
Estimate
movement
parameters.
• estimate rate of change
of field with respect to
the current estimate of
movement parameters
in pitch and roll.

Unwarp time
series.
+
B0 
B0 
Andersson et al, 2001
Contents
* Preliminaries
* Intra-Subject Registration
* Realign
* Coregister
* Mutual Information objective function
* Inter-Subject Registration
Inter-modal registration
• Match images from same
subject but different
modalities:
– anatomical localisation of
single subject activations
– achieve more precise
spatial normalisation of
functional image using
anatomical image.
Mutual Information
* Used for between-modality registration
* Derived from joint histograms
* MI=
ab P(a,b) log2 [P(a,b)/( P(a) P(b) )]
* Related to entropy: MI = -H(a,b) + H(a) + H(b)
* Where H(a) = -a P(a) log2P(a) and H(a,b) = -a P(a,b) log2P(a,b)
Contents
* Preliminaries
* Intra-Subject Registration
* Inter-Subject Registration
* Normalise
* Affine Registration
* Nonlinear Registration
* Regularisation
* Segment
Spatial Normalisation - Reasons
* Inter-subject averaging
* Increase sensitivity with more subjects
* Fixed-effects analysis
* Extrapolate findings to the population as a whole
* Mixed-effects analysis
* Standard coordinate system
* e.g., Talairach & Tournoux space
Spatial Normalisation - Procedure
* Minimise mean squared difference from template
image(s)
Affine registration
Non-linear registration
T2
T1
Transm
T1
305
EPI
PD
PET
PD
T2
Template Images
“Canonical” images
PET
A wider range of
contrasts can be
registered to a
linear combination
of template images.
T1
PD
SS
Spatial normalisation can
be weighted so that nonbrain voxels do not
influence the result.
Similar weighting masks
can be used for normalising
lesioned brains.
Spatial Normalisation - Templates
Spatial Normalisation - Affine
* The first part is a 12 parameter
affine transform
*
*
*
*
3 translations
3 rotations
3 zooms
3 shears
* Fits overall shape and size
* Algorithm simultaneously minimises
* Mean-squared difference between template and source image
* Squared distance between parameters and their expected values
(regularisation)
Spatial Normalisation - Non-linear
Deformations consist of a
linear combination of smooth
basis functions
These are the lowest
frequencies of a 3D discrete
cosine transform (DCT)
Algorithm simultaneously minimises
* Mean squared difference between template
and source image
* Squared distance between parameters and
their known expectation
Spatial Normalisation - Overfitting
Without
regularisation,
the non-linear
Template
spatial
image
normalisation
can introduce
unnecessary
Non-linear
warps.
registration
using
regularisation.
(2 = 302.7)
Affine
registration.
(2 = 472.1)
Non-linear
registration
without
regularisation.
(2 = 287.3)
Contents
* Preliminaries
* Intra-Subject Registration
* Inter-Subject Registration
* Normalise
* Segment
* Gaussian mixture model
* Intensity non-uniformity correction
* Deformed tissue probability maps
Segmentation
* Segmentation in SPM5 also estimates a spatial
transformation that can be used for spatially
normalising images.
* It uses a generative model, which involves:
* Mixture of Gaussians (MOG)
* Bias Correction Component
* Warping (Non-linear Registration) Component
Gaussian Probability Density
* If intensities are assumed to be Gaussian of mean mk
and variance s2k, then the probability of a value yi is:
Non-Gaussian Probability Distribution
* A non-Gaussian probability density function can be
modelled by a Mixture of Gaussians (MOG):
Mixing proportion - positive and sums to one
Belonging Probabilities
Belonging
probabilities
are assigned
by normalising
to one.
Mixing Proportions
* The mixing proportion gk represents the prior
probability of a voxel being drawn from class k irrespective of its intensity.
* So:
Non-Gaussian Intensity Distributions
* Multiple Gaussians per tissue class allow non-Gaussian
intensity distributions to be modelled.
* E.g. accounting for partial volume effects
Probability of Whole Dataset
* If the voxels are assumed to be independent, then
the probability of the whole image is the product of
the probabilities of each voxel:
* A maximum-likelihood solution can be found by
minimising the negative log-probability:
Modelling a Bias Field
* A bias field is included, such that the required
scaling at voxel i, parameterised by b, is ri(b).
* Replace the means by mk/ri(b)
* Replace the variances by (sk/ri(b))2
Modelling a Bias Field
* After rearranging...
y
r(b)
y r(b)
Tissue Probability Maps
* Tissue probability maps (TPMs) are used instead
of the proportion of voxels in each Gaussian as
the prior.
ICBM Tissue Probabilistic Atlases. These tissue probability maps are
kindly provided by the International Consortium for Brain Mapping, John C.
Mazziotta and Arthur W. Toga.
“Mixing Proportions”
* Tissue probability maps for
each class are included.
* The probability of obtaining
class k at voxel i, given
weights g is then:
Deforming the Tissue Probability
Maps
* Tissue probability
images are
deformed according
to parameters a.
* The probability of
obtaining class k at
voxel i, given
weights g and
parameters a is
then:
The Extended Model
* By combining the modified P(ci=k|) and P(yi|ci=k,), the
overall objective function (E) becomes:
The Objective Function
Optimisation
* The “best” parameters are those that minimise this
objective function.
* Optimisation involves finding them.
* Begin with starting estimates, and repeatedly change
them so that the objective function decreases each
time.
Steepest Descent
Start
Optimum
Alternate between
optimising different groups
of parameters
Schematic of optimisation
Repeat until convergence…
Hold g, m, s2 and a constant, and minimise E w.r.t. b
- Levenberg-Marquardt strategy, using dE/db and d2E/db2
Hold g, m, s2 and b constant, and minimise E w.r.t. a
- Levenberg-Marquardt strategy, using dE/da and d2E/da2
Hold a and b constant, and minimise E w.r.t. g, m and s2
-Use an Expectation Maximisation (EM) strategy.
end
Levenberg-Marquardt Optimisation
* LM optimisation is used for nonlinear registration (a)
and bias correction (b).
* Requires first and second derivatives of the
objective function (E).
* Parameters a and b are updated by
* Increase l to improve stability (at expense of
decreasing speed of convergence).
Expectation Maximisation is used to
update m, s2 and g
* For iteration (n), alternate between:
* E-step: Estimate belonging probabilities by:
* M-step: Set (n+1) to values that reduce:
Regularisation
* Some bias fields and warps are more probable (a
priori) than others.
* Encoded using Bayes rule (for a maximum a posteriori
solution):
* Prior probability distributions modelled by a
multivariate normal distribution.
*
*
*
*
Mean vector
ma and mb
Covariance matrix Sa and Sb
-log[P(a)] = (a-ma)TSa-1(ama) + const
-log[P(b)] = (b-mb)TSb-1(bmb) + const
Initial Affine Registration
The procedure begins with a
Mutual Information affine
registration of the image with
the tissue probability maps.
MI is computed from a 4x256
joint probability histogram.
Joint Probability Histogram
See D'Agostino, Maes, Vandermeulen &
P. Suetens. “Non-rigid Atlas-to-Image
Registration by Minimization of ClassConditional Image Entropy”. Proc.
MICCAI 2004. LNCS 3216, 2004. Pages
745-753.
Background voxels
excluded
Background Voxels are
Excluded
An intensity threshold is found by
fitting image intensities to a
mixture of two Gaussians. This
threshold is used to exclude most
of the voxels containing only air.
Spatially
normalised
BrainWeb
phantoms (T1,
T2 and PD)
Tissue
probability
maps of GM
and WM
Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)
Image Registration
* Figure out how to warp
one image to match
another
* Normally, all subjects’
scans are matched with a
common template
Current SPM approach
* Only about 1000 parameters.
* Unable model detailed deformations
A one-to-one mapping
* Many models simply add a smooth displacement to an identity transform
* One-to-one mapping not enforced
* Inverses approximately obtained by subtracting the displacement
* Not a real inverse
Large deformation
approximation
Small deformation
approximation
DARTEL
* Parameterising the deformation
*φ(0)(x) = x
*φ(1)(x) =
*u
1
∫
u(φ(t)(x))dt
t=0
is a flow field to be estimated
Flow Field
Template
Initial
Average
Iteratively generated
from 471 subjects
Began with rigidly
aligned tissue
probability maps
Used an inverse
consistent
formulation
After a few
iterations
Final
template
Grey matter
average of 452
subjects – affine
Grey matter
average of 471
subjects
Subject 1
Subject 2
VBM results of Patients with
Alzheimer’s Disease
Both sets were smoothed with only 3mm FWHM Gaussian kernel, (FWE p>0.05)
Conventional SPM5 normalized image
DARTEL normalized image
VBM results of Patients with
Huntington’s Disease
Both sets were smoothed with only 6mm FWHM Gaussian kernel, (FWE p>0.05)
5
6
4
5
3
4
2
1
3
2
1
0
Conventional SPM5 normalized image
0
DARTEL normalized image
References
* Friston et al. Spatial registration and normalisation of images.
Human Brain Mapping 3:165-189 (1995).
* Collignon et al. Automated multi-modality image registration based on
information theory. IPMI’95 pp 263-274 (1995).
* Ashburner et al. Incorporating prior knowledge into image registration.
NeuroImage 6:344-352 (1997).
* Ashburner & Friston. Nonlinear spatial normalisation using basis functions.
Human Brain Mapping 7:254-266 (1999).
* Thévenaz et al. Interpolation revisited.
IEEE Trans. Med. Imaging 19:739-758 (2000).
* Andersson et al. Modeling geometric deformations in EPI time series.
Neuroimage 13:903-919 (2001).
* Ashburner & Friston. Unified Segmentation.
NeuroImage in press (2005).
* Ashburner. A fast diffeomorphic image reigstration algorithm. NeuroImage in
press (2007).

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