DARTEL

1. DARTEL John Ashburner 2008

2. Overview • Motivation • 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. 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. 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 • 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) oφ(t)

11. Flow Field

12. Simple integration φ(1/8) = x + u/8 φ(2/8) = φ(1/8)oφ(1/8) φ(3/8) = φ(1/8)oφ(2/8) φ(4/8) = φ(1/8)oφ(3/8) φ(5/8) = φ(1/8)oφ(4/8) φ(6/8) = φ(1/8)oφ(5/8) φ(7/8) = φ(1/8)oφ(6/8) φ(8/8) = φ(1/8)oφ(7/8) 7 compositions Scaling and squaring φ(1/8) = x + u/8 φ(2/8) = φ(1/8)oφ(1/8) φ(4/8) = φ(2/8)oφ(2/8) φ(8/8) = φ(4/8)oφ(4/8) 3 compositions Similar procedure used for the inverse. Starts with φ(-1/8) = x - u/8 For (e.g) 8 time steps

13. Scaling and squaring example

14. DARTEL

15. Jacobian determinants remain positive

16. Overview • Motivation • Principles • Optimisation • 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)-1b • 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 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. Grey matter Grey matter Grey matter Grey matter White matter White matter White matter White matter Simultaneous registration of GM to GM and WM to WM Subject 1 Subject 3 Grey matter White matter Template Subject 2 Subject 4

28. ϕ1 t1 ϕ2 μ t2 t5 ϕ3 t3 ϕ5 t4 ϕ4 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

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 • 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 template 2D Nicely scale invariant 3D Not quite scale invariant – but probably close enough

36. 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