3 Motivation More precise inter-subject alignment Improved fMRI data analysisBetter group analysisMore accurate localizationImprove computational anatomyMore easily interpreted VBMBetter parameterization of brain shapesOther applicationsTissue segmentationStructure 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 mappingMany models simply add a smooth displacement to an identity transformOne-to-one mapping not enforcedInverses approximately obtained by subtracting the displacementNot a real inverseSmall deformation approximation
12 For (e.g) 8 time steps Simple integration φ(1/8) = x + u/8 7 compositionsScaling and squaringφ(1/8) = x + u/8φ(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 compositionsSimilar procedure used for the inverse.Starts withφ(-1/8) = x - u/8
17 Registration objective function Simultaneously minimize the sum ofLikelihood componentFrom the sum of squares difference½∑i(g(xi) – f(φ(1)(xi)))2φ(1) parameterized by uPrior componentA measure of deformation roughness½uTHu
18 Regularization model DARTEL has three different models for H Membrane energyLinear elasticityBending energyH is very sparseAn example H for 2D registration of 6x6 images (linear elasticity)
20 Optimisation Uses Levenberg-Marquardt Requires a matrix solution to a very large set of equations at each iterationu(k+1) = u(k) - (H+A)-1 bb are the first derivatives of objective functionA is a sparse matrix of second derivativesComputed 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, whereE is easy to invertF is more difficultSometimes: 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 notFits high frequencies quickly, but low frequencies slowly
24 Overview Motivation Principles Optimisation Group-wise Registration Simultaneous registration of GM & WMTissue probability map creationValidationFuture 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.Automaticallysegmentedgrey matterimages.
27 Simultaneous registration of GM to GM and WM to WM Grey matterWhite matterSubject 1Grey matterWhite matterSubject 3Grey matterWhite matterGrey matterWhite matterTemplateGrey matterWhite matterSubject 2Subject 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 subjectsμt1ϕ1t2ϕ2t3ϕ3t4ϕ4t5ϕ5White 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 manifoldLinear Average(Not on Riemannian manifold)
31 Template Initial Average Iteratively generated from 471 subjects Began with rigidly aligned tissue probability mapsUsed an inverse consistent formulationAfter a few iterationsFinal template
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 backgroundμ – template GM, WM and backgroundϕ – deformationA general purpose template should not have regions where log(μ) is –Inf.
35 Laplacian Smoothness Priors on template 2DNicely scale invariant3DNot quite scale invariant – but probably close enough
36 μk(x) = exp(ak(x))/(Σj exp(aj(x))) Template modelled as softmax of a Gaussian processμk(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 registeredRigidly alignedlog p(t|μ) = ΣjΣk tjk log(μjk)
38 ML and MAP templates from 6 subjects Nonlinear RegisteredRigid registeredMLMAPlog
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 informationCould have used an overlap of fMRI resultsChose to see whether ages and sexes of subjects could be predicted from the deformationsComparison with small deformation model
51 Training and Classifying ?ControlTraining Data???PatientTraining Data
59 Relevance-vector regression A Bayesian method, related to SVMsDeveloped by Mike TippingGuess 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 deformationLinear regressionRMS error = 7.55Correlation = 0.836RBF regressionRMS error = 6.68Correlation = 0.856DARTELLinear regressionRMS error = 7.90Correlation = 0.813RBF regressionRMS error = 6.50Correlation = 0.867An unconvincing improvement (slightly worse for linear regression)
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.
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