1. DARTEL
John Ashburner
2008

2. Overview Motivation Principles Geeky stuff Example Validation
Dimensionality
Inverse-consistency
Principles
Geeky stuff
Example
Validation
Future directions

3. 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

4. Image Registration Figure out how to warp one image to match another
Normally, all subjects’ scans are matched with a common template

5. Current SPM approach Only about 1000 parameters.
Unable model detailed deformations

6. Small deformation approximation
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

7. Overview Motivation Principles Optimisation Group-wise Registration
Validation
Future directions

8. Principles Diffeomorphic Anatomical Registration Through Exponentiated
Lie Algebra
Deformations parameterized by a single flow field, which is considered to be constant in time.

9. φ(1)(x) = ∫ u(φ(t)(x))dt u is a flow field to be estimated
DARTEL
Parameterising the deformation
φ(0)(x) = x
φ(1)(x) = ∫ u(φ(t)(x))dt
u is a flow field to be estimated 1
t=0

10. Euler integration dφ(x)/dt = u(φ(t)(x)) φ(t+h) = φ(t) + hu(φ(t))
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)

11. Flow Field

12. For (e.g) 8 time steps Simple integration φ(1/8) = x + u/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

13. Scaling and squaring example

14. DARTEL

15. Jacobian determinants remain positive

16. Overview Motivation Principles Optimisation Group-wise Registration
Multi-grid
Group-wise Registration
Validation
Future directions

17. 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

18. 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)

19. Regularization models

20. 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

21. Relaxation To solve Mx = c Sometimes: x(k+1) = E-1(c – F x(k))
Split 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

22. H+A = E+F

23. Highest resolution
Full Multi-Grid
Lowest resolution

24. Overview Motivation Principles Optimisation Group-wise Registration
Simultaneous registration of GM & WM
Tissue probability map creation
Validation
Future directions

25. 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).

26. Generative models of anatomy
Work with tissue class images.
Brains of differing shapes and sizes.
Need strategies to encode such variability.
Automatically
segmented
grey matter
images.

27. Simultaneous registration of GM to GM and WM to WM
Grey matter
White matter
Subject 1
Grey matter
White matter
Subject 3
Grey matter
White matter
Grey matter
White matter
Template
Grey matter
White matter
Subject 2
Subject 4

28. 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 μ t1 ϕ1 t2 ϕ2 t3 ϕ3 t4 ϕ4 t5 ϕ5
White matter average of 471 subjects

29. 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.

30. Average shaped templates
Average on Riemannian manifold
Linear Average
(Not on Riemannian manifold)

31. 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

32. Grey matter average of 452 subjects – affine

33. Grey matter average of 471 subjects

34. Multinomial Model log p(t|μ,ϕ) = ΣjΣk tjk log(μk(ϕj))
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.

35. Laplacian Smoothness Priors on template2D
Nicely scale invariant 3D
Not quite scale invariant – but probably close enough

36. μk(x) = exp(ak(x))/(Σj exp(aj(x)))
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.
Smoothing by solving matrix equations using multi-grid

37. 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:
Nonlinear registered
Rigidly aligned
log p(t|μ) = ΣjΣk tjk log(μjk)

38. ML and MAP templates from 6 subjects
Nonlinear Registered
Rigid registered ML
MAP
log

49. Overview Motivation Principles Optimisation Group-wise Registration
Validation
Sex classification
Age regression
Future directions

50. 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

51. Training and Classifying?
Control
Training Data ? ? ?
Patient
Training Data

52. Classifying ?
Controls ? ? ?
Patients
y=f(aTx+b)

53. Support Vector Classifier

54. Support Vector Classifier (SVC)
a is a weighted linear combination of the support vectors
Support
Vector
Support
Vector

55. Nonlinear SVC

56. 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.

57. 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

58. Regression 40 30 23 29 26 18 32

59. 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.

60. 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)

62. Overview Motivation Principles Optimisation Group-wise Registration
Validation
Future directions

63. 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.

65. u Hu

66. Variable velocity framework (as in LDDMM)
“Initial momentum”

67. Variable velocity framework (as in LDDMM)
“Initial momentum”

68. Thank you