DARTEL John Ashburner 2008
Mar 28, 2015
DARTEL
John Ashburner
2008
Overview
• Motivation– Dimensionality– Inverse-consistency
• Principles
• Geeky stuff
• Example
• Validation
• Future directions
Motivation
• More precise inter-subject alignment– Improved fMRI data analysis
• Better group analysis• More accurate localization
– Improve computational anatomy• More easily interpreted VBM• Better parameterization of brain shapes
– Other applications• Tissue segmentation• Structure labeling
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 Small deformation
approximation
Overview
• Motivation
• Principles
• Optimisation
• Group-wise Registration
• Validation
• Future directions
Principles
DiffeomorphicAnatomicalRegistrationThroughExponentiatedLie Algebra
Deformations parameterized by a single flow field, which is considered to be constant in time.
DARTEL
• Parameterising the deformation
• φ(0)(x) = x
• φ(1)(x) = ∫ u(φ(t)(x))dt• u is a flow field to be estimated
t=0
1
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)
Flow Field
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 withφ(-1/8) = x - u/8
Scaling and squaring example
DARTEL
Jacobian determinants remain positive
Overview
• Motivation
• Principles
• Optimisation– Multi-grid
• Group-wise Registration
• Validation
• Future directions
Registration objective function
• Simultaneously minimize the sum of – Likelihood component
• From the sum of squares difference
• ½∑i(g(xi) – f(φ(1)(xi)))2
• φ(1) parameterized by u
– Prior component• A measure of deformation roughness
• ½uTHu
Regularization model
• 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
Optimisation
• Uses Levenberg-Marquardt– 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 = cSplit M into E and F, where
• E is easy to invert• F is more difficult
• Sometimes: x(k+1) = E-1(c – F x(k))• Otherwise: x(k+1) = x(k) + (E+sI)-1(c – M x(k))
• Gauss-Siedel when done in place.• Jacobi’s method if not
• Fits high frequencies quickly, but low frequencies slowly
H+A = E+F
Highest resolution
Lowest resolution
Full Multi-Grid
Overview
• Motivation
• Principles
• Optimisation
• Group-wise Registration– Simultaneous registration of GM & WM– Tissue probability map creation
• Validation
• Future directions
Generative Models for Images
• Treat the template as a deformable probability density.– Consider the intensity distribution at each
voxel of lots of aligned images.• Each point in the template represents a probability
distribution of intensities.
– Spatially deform this intensity distribution to the individual brain images.
• Likelihood of the deformations given by the template (assuming spatial independence of voxels).
Generative models of anatomy
• Work with tissue class images.
• Brains of differing shapes and sizes.• Need strategies to encode such variability.
Automaticallysegmentedgrey matter
images.
Simultaneous registration of GM to GM and WM to WM
Grey matter
White matter
Grey matter
White matter
Grey matter
White matter
Grey matter
White matter
Grey matter
White matterTemplate
Subject 1
Subject 2
Subject 3
Subject 4
Template Creation
• Template is an average shaped brain.– Less bias in subsequent analysis.
• Iteratively created mean using DARTEL algorithm.– Generative model of data.– Multinomial noise model. Grey matter
average of 471 subjects
White matter average of 471 subjects
μ
t1
ϕ1
t2
ϕ2
t3
ϕ3
t4 ϕ4
t5
ϕ5
Average Shaped Template
• For CA, work in the tangent space of the manifold, using linear approximations.– Average-shaped templates give less bias, as the
tangent-space at this point is a closer approximation.• For spatial normalisation of fMRI, warping to a
more average shaped template is less likely to cause signal to disappear.– If a structure is very small in the template, then it will
be very small in the spatially normalised individuals.• Smaller deformations are needed to match with
an average-shaped template.– Smaller errors.
Average shaped templates
Linear Average
Average on Riemannian manifold
(Not on Riemannian manifold)
TemplateInitial
Average
After a few iterations
Final template
Iteratively generated from 471 subjects
Began with rigidly aligned tissue probability maps
Used an inverse consistent formulation
Grey matter average of 452 subjects – affine
Grey matter average of 471 subjects
Multinomial Model
• Current DARTEL 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.
Laplacian Smoothness Priors on template
2DNicely scale invariant
3DNot quite scale invariant – but probably close enough
Smoothing by solving matrix equations using multi-grid
Template modelled as softmax of a Gaussian process
μk(x) = exp(ak(x))/(Σj exp(aj(x)))
Rather than compute mean images and convolve with a Gaussian, the smoothing is done by maximising a log-likelihood for a MAP solution.
Note that Jacobian transformations are required (cf “modulated VBM”) to properly account for expansion/contraction during warping.
Determining amount of regularisation
• Matrices too big for REML estimates.
• Used cross-validation.
• Smooth an image by different amounts, see how well it predicts other images:
Rigidly aligned
Nonlinear registered
log p(t|μ) = ΣjΣk tjk log(μjk)
ML and MAP templates from 6 subjects
Nonlinear Registered Rigid registered
log
MAP
ML
Overview
• Motivation
• Principles
• Optimisation
• Group-wise Registration
• Validation– Sex classification– Age regression
• Future directions
Validation
• There is no “ground truth”• Looked at predictive accuracy
– Can information encoded by the method make predictions?
• Registration method blind to the predicted information• Could have used an overlap of fMRI results
– Chose to see whether ages and sexes of subjects could be predicted from the deformations
• Comparison with small deformation model
Training and Classifying
ControlTraining Data
PatientTraining Data
?
?
??
Classifying
Controls
Patients
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??
y=f(aTx+b)
Support Vector Classifier
Support Vector Classifier (SVC)
SupportVector
SupportVector
Support
Vector
a is a weighted linear combination of the support vectors
Nonlinear SVC
Support-vector classification
• Guess sexes of 471 subjects from brain shapes – 207 Females / 264 Males
• Use a random sample of 400 for training.
• Test on the remaining 71.
• Repeat 50 times.
Sex classification results
• Small Deformation– Linear classifier
• 87.0% correct• Kappa = 0.736
– RBF classifier• 87.1% correct• Kappa = 0.737
• DARTEL– Linear classifier
• 87.7% correct• Kappa = 0.749
– RBF classifier• 87.6% correct• Kappa = 0.748
An unconvincing improvement
Regression
23
26
30
29
18
32
40
Relevance-vector regression
• A Bayesian method, related to SVMs– Developed by Mike Tipping
• Guess ages of 471 subjects from brain shapes.
• Use a random sample of 400 for training.
• Test on the remaining 71.
• Repeat 50 times.
Age regression results
• Small deformation– Linear regression
• RMS error = 7.55• Correlation = 0.836
– RBF regression• RMS error = 6.68• Correlation = 0.856
• DARTEL– Linear regression
• RMS error = 7.90• Correlation = 0.813
– RBF regression• RMS error = 6.50• Correlation = 0.867
An unconvincing improvement(slightly worse for linear regression)
Overview
• Motivation
• Principles
• Optimisation
• Group-wise Registration
• Validation
• Future directions
Future directions
• Compare with variable velocity methods– Beg’s LDDMM algorithm.
• Classification/regression from “initial momentum”.
• Combine with tissue classification model.
• Develop a proper EM framework for generating tissue probability maps.
u
Hu
“Initial momentum”
Variable velocity framework (as in LDDMM)
“Initial momentum”
Variable velocity framework (as in LDDMM)
Thank you