Pre-processing for “VoxelBased Morphometry”
John Ashburner
The Wellcome Trust Centre for Neuroimaging
12 Queen Square, London, UK.
Contents
• Introduction
• Segmentation
• DARTEL Registration
Voxel-based Morphometry
• Pre-process the images of lots of subjects,
to generate spatially normalised grey
matter maps of each subject.
• Smooth spatially.
• Perform voxel-wise statistics.
• Try to interpret the findings in terms of
volumetric differences.
Segment into different
tissue classes
Spatially Normalize – with scaling
by Jacobian determinant
Smooth Spatially
Mass-univariate
statistical testing
Inference via
Random Field
Theory
Smoothing
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
Possible Explanations for
Findings
Mis-classify
Mis-register
Folding
Thickening
Thinning
Mis-register
Mis-classify
Contents
• Introduction
• Segmentation
– Mixture of Gaussians
– Bias correction
– Warping to match tissue probability maps
• DARTEL Registration
Tissue Segmentation
• Circularity:
– Registration is helped by tissue classification or bias correction.
– Tissue classification helped by registration and bias correction.
– Bias correction is helped by registration and tissue classification.
• The solution is to put everything in the same generative
model.
– A MAP solution is found by repeatedly alternating among
classification, bias correction and registration steps.
• Should produce “better” results than simple serial
applications of each component.
A Generative Model
• A model of how the data may have been
generated, which comprises:
– Mixture of Gaussians (MOG)
– Bias Correction
– Non-linear Inter-subject
Registration
g
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a
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c1
y1
m
c2
y2
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c3
y3
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Mixture of Gaussians (MOG)
• Tissue classification is based on a Mixture of
Gaussians model (MOG), which represents
the intensity probability density by a number
of Gaussian distributions.
Frequency
Image Intensity
Belonging Probabilities
Belonging
probabilities are
assigned by
normalising to
one.
Non-Gaussian Intensity
Distributions
• Multiple Gaussians per tissue class allow nonGaussian intensity distributions to be modelled.
– E.g. accounting for partial volume effects
Modelling a Bias Field
Corrupted
image
Bias Field
Corrected image
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.
Deforming the Tissue Probability Maps
• Tissue
probability
images are
deformed so that
they can be
overlaid on top
of the image to
segment.
Optimisation
• The “best” parameters are those that
maximise the log-probability.
• 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
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)
Contents
• Introduction
• Segmentation
• DARTEL Registration
– Scaling and squaring
– Optimisation
– Warping GM and WM images to their
average
Parameterization
Deformations
parameterized by a
single flow field, which
is considered to be
constant in time.
Not really a proper Lie
Group.
Often referred to as a
one parameter
subgroup.
Diffeomorphic
Anatomical
Registration
Through
Exponentiated
Lie Algebra
Euler Integration
• Parameterising the deformation
• φ(0)(x) = x
1
• φ(1)(x) = ∫ u(φ(t)(x))dt
• u is a flow field to be estimated
t=0
• Scaling and squaring is used to
generate deformations.
– c.f. matrix exponentiation
Euler integration
• The differential equation is
dφ(x)/dt = u(φ(t)(x))
• By Euler integration
φ(t+h) = φ(t) + hu(φ(t))
• Equivalent to
φ(t+h) = (x + hu) o φ(t)
For (e.g) 8 time steps
Simple integration
• φ(1/8) = x + u/8
• φ(2/8) = φ(1/8) o φ(1/8)
• φ(3/8) = φ(1/8) o φ(2/8)
• φ(4/8) = φ(1/8) o φ(3/8)
• φ(5/8) = φ(1/8) o φ(4/8)
• φ(6/8) = φ(1/8) o φ(5/8)
• φ(7/8) = φ(1/8) o φ(6/8)
• φ(8/8) = φ(1/8) o φ(7/8)
7 compositions
Scaling and squaring
• φ(1/8) = x + u/8
• φ(2/8) = φ(1/8) o φ(1/8)
• φ(4/8) = φ(2/8) o φ(2/8)
• φ(8/8) = φ(4/8) o φ(4/8)
3 compositions
• Similar procedure
used for the inverse.
Starts with
φ(-1/8) = x - u/8
Scaling and squaring example
Deformations at
different times
Jacobians
• Jacobian fields can also be obtained by
scaling and squaring.
• If warps are composed by:
ϕC=ϕB○ϕA
then Jacobian matrices are obtained by:
JϕC=(JϕB○ϕA) JϕA
Jacobian determinants remain
positive (almost)
See also…
• C. Moler and C. van Loan. “Nineteen Dubious Ways to Compute the
Exponential of a Matrix, Twenty-Five Years Later”. SIAM Review
45(1):3-49 (2003).
• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A LogEuclidean Polyaffine Framework for Locally Rigid or Affine
Registration”. Proc. Of the 3rd International Workshop on
Biomedical Image Registration (WBIR'06), 2006, pp. 120-127. LNCS
vol 4057. Springer-Verlag, Utrecht, NL.
• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A LogEuclidean Framework for Statistics on Diffeomorphisms”. Proc. of
the 9th International Conference on Medical Image Computing and
Computer Assisted Intervention (MICCAI'06), 2006, pp. 924-931.
LNCS 4190. Springer-Verlag, Berlin, Germany.
• M. Hernandez, M. N. Bossa, and S. Olmos. “Registration of
anatomical images using geodesic paths of diffeomorphisms
parameterized with stationary vector fields”. IEEE workshop on
Math. Meth. in Biom. Image Anal. (MMBIA’07), 2007.
Contents
• Introduction
• Segmentation
• DARTEL Registration
– Scaling and squaring
– Optimisation
– Warping GM and WM images to their
average
Multinomial Likelihood Term
• Model is multinomial for matching tissue
class images.
-log p(t|μ,ϕ) = -ΣjΣk tjk log(μk(ϕj))
t
– individual GM, WM and background
μ
– template GM, WM and background
ϕ
– deformation
• A general purpose template should not
have regions where log(μ) is –Inf.
Prior Term
• ½uTHu
• DARTEL has three different models for H
– Membrane energy
– Linear elasticity
– Bending energy
• H is very sparse
An example H for 2D
registration of 6x6
images (linear
elasticity)
Regularization models
“Membrane energy”
Images registered using a small deformation approximation
“Bending energy”
Optimization
• Uses Gauss-Newton
– Requires a matrix solution to a very large set
of equations at each iteration
u(k+1) = u(k) - (H+A)-1 b
– b are the first derivatives of objective function
– A is a sparse matrix of second derivatives
– Computed efficiently, making use of scaling
and squaring
Relaxation
• To solve Mx = c
Split M into E and F, where
• E is easy to invert
• F is more difficult
• If M is diagonally dominant (membrane energy):
x(k+1) = E-1(c – F x(k))
• Otherwise regularize (bending or linear elastic
energy):
x(k+1) = x(k) + (E+sI)-1(c – M x(k))
– Diagonal dominance is when |mii| > Σi≠j |mij|
2nd derivs of
prior term
2nd derivs of
likelihood
term
M = H+A = E+F
Easy to invert
Difficult to
invert
Highest resolution
Full Multi-Grid
Lowest
resolution
A
•Prolongation of low resolution solution to current resolution.
•Add this to existing solution.
•Perform a few iterations of relaxation.
•Restrict residuals down to lower resolution.
B
•Prolongation of low resolution solution to current resolution.
•Add this to existing solution at current resolution.
•Perform a few iterations of relaxation.
•Prolongation of solution to higher resolution.
C
•Restrict high resolution residuals to current resolution.
•Perform a few iterations of relaxation.
•Restrict residuals down to lower resolution.
E
•Restrict higher resolution residuals to current resolution.
•Obtain exact solution by matrix inversion.
•Prolongation of solution to higher resolution.
See also…
• W. H. Press, S. A. Teukolsky, W. T.
Vetterling and B. P. Flannery. Numerical
Recipes in C (Second Edition). Cambridge
University Press, Cambridge, UK. 1992.
– Chapter 15, Section 5 explains GaussNewton optimization (Levenberg-Marquardt
without the regularisation).
– Chapter 19, Section 6 explains the basics of
multi-grid methods.
Contents
• Introduction
• Segmentation
• DARTEL Registration
– Scaling and squaring
– Optimisation
– Warping GM and WM images to their
average
Template
Generation
Initial
Average
Iteratively
generated from
471 subjects.
After a few
Began with rigidly iterations
aligned tissue
probability maps.
Regularization
lighter for later
iterations.
Final
template
Generative Model
• p(ϕ1,t1, ϕ2,t2, ϕ3,t3,… μ)
= p(t1,ϕ1|μ) p(t2,ϕ2|μ) p(t3,ϕ3|μ) … p(μ)
• = p(t1|ϕ1,μ) p(ϕ1) p(t2|ϕ2,μ) p(ϕ2)… p(μ)
ϕ1
t1
• MAP solution obtained
for template.
t
• Requires p(μ)
ϕ2
μ
t2
5
ϕ5
t4
ϕ4
t3
ϕ3
Laplacian Smoothness Priors on
template
2D
3D
Template modelled as softmax of a Gaussian process
μk(x) = exp(ak(x))/(Σj exp(aj(x)))
MAP solution determined for a, by Gauss-Newton optimisation,
using multi-grid.
ML and MAP templates from 6
subjects
Nonlinearly aligned
Rigidly aligned
ML
MAP
log
471 Subject Average
471 Subject Average
471 Subject Average
Subject 1
471 Subject Average
Subject 2
471 Subject Average
Subject 3
471 Subject Average
Preprocessing with
DARTEL
u
Hu
Variable velocity framework (as in LDDMM)
“Initial
momentum”
Variable velocity framework (as in LDDMM)
“Initial
momentum”
Determining amount of
regularisation
• Matrices too big for
Bayesian variance
component
estimation.
• Used crossvalidation.
• Smooth an image
by different
amounts, see how
well it predicts other
images:
Nonlinear registered
log p(t|μ) = ΣjΣk tjk log(μjk)
Rigidly aligned