DARTEL John Ashburner 2008

Overview Motivation Principles Geeky stuff Example Validation 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

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

Overview Motivation Principles Optimisation Group-wise Registration Validation Future directions

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

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

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)

Flow Field

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

Scaling and squaring example

DARTEL

Jacobian determinants remain positive

Overview Motivation Principles Optimisation Group-wise Registration 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 = 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

H+A = E+F

Highest resolution Full Multi-Grid Lowest resolution

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. Automatically segmented grey matter images.

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

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

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 Average on Riemannian manifold Linear Average (Not on Riemannian manifold)

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

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.

Laplacian Smoothness Priors on template 2D Nicely scale invariant 3D Not quite scale invariant – but probably close enough

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

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)

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

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 ? Control Training Data ? ? ? Patient Training Data

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

Support Vector Classifier

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

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 40 30 23 29 26 18 32

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

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

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

Thank you