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Experiments on a New Inter- Subject Registration Method John Ashburner 2007

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Abstract The objective of this work was to devise a more precise method of inter- subject brain image registration than those currently available in the SPM software. This involved a model with many more degrees of freedom, but which still enforces a one-to-one mapping. Speed considerations were also important. The result is an approach that models each warp by single velocity field. These are converted to deformations by a scaling and squaring procedure, and the inverses can be generated in a similar way. Registration is via a Levenberg-Marquardt optimization strategy, which uses a full multi-grid algorithm to rapidly solve the necessary equations. The method has been used for warping images of 471 subjects. This involved simultaneously matching grey matter with a grey matter template, and white matter with a white matter template. After every few iterations, the templates were re-generated from the means of the warped individual images. Evaluations involved applying pattern recognition procedures to the resulting deformations, in order to assess how well information such as the ages and sexes of the subjects could be predicted from the encoded deformations. A slight improvement in prediction accuracy was obtained when compared to a similar procedure using a small deformation model.

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

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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

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

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

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A simple 2D example Individual brain Warped Individual Reference

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Residual Differences Individual brain Warped Individual

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Expansion and contraction Relative volumes encoded by Jacobian determinants of deformation

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Tissue volume comparisons Warped grey matter Jacobian determinants Absolute grey matter volumes

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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

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Overview Motivation Principles Geeky stuff Example Validation Future directions

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

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DARTEL Parameterizing 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

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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)

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Flow Field

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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 with φ (-1/8) = x - u/8

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Scaling and squaring example

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DARTEL

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Jacobian determinants remain positive

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Overview Motivation Principles Geeky stuff –Feel free to sleep Example Validation Future directions

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Registration objective function Simultaneously minimize the sum of –Likelihood component From the sum of squares difference ½ i ( g(x i ) – f(φ (1) (x i )) ) 2 φ (1) parameterized by u –Prior component A measure of deformation roughness ½u T Hu

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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)

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Regularization models

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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

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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. Jacobis method if not Fits high frequencies quickly, but low frequencies slowly

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H+A = E+F

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Highest resolution Lowest resolution Full Multi-Grid

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Overview Motivation Principles Geeky stuff Example –Simultaneous registration of GM & WM –Tissue probability map creation Validation Future directions

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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 matter Template Subject 1 Subject 2 Subject 3 Subject 4

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Template Initial Average After a few iterations Final template Iteratively generated from 471 subjects Began with rigidly aligned tissue probability maps Used an inverse consistent formulation

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Grey matter average of 452 subjects – affine

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Grey matter average of 471 subjects

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White matter average of 471 subjects

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Initial GM images

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Warped GM images

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Overview Motivation Principles Geeky stuff Example Validation –Sex classification –Age regression Future directions

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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

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Training and Classifying Control Training Data Patient Training Data ? ? ? ?

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Classifying Controls Patients ? ? ? ? y=f(a T x+b)

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Support Vector Classifier

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

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Some Equations Linear classification is by y = f(a T x + b) –where a is a weighting vector, x is the test data, b is an offset, and f(.) is a thresholding operation a is a linear combination of SVs a = i w i x i So y = f( i w i x i T x + b)

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Going Nonlinear Nonlinear classification is by y = f( i w i (x i,x)) –where (x i,x) is some function of x i and x. e.g. RBF classification (x i,x) = exp(-||x i -x|| 2 /(2 2 )) Requires a matrix of distance measures (metrics) between each pair of images.

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Nonlinear SVC

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Cross-validation Methods must be able to generalise to new data Various control parameters –More complexity -> better separation of training data –Less complexity -> better generalisation Optimal control parameters determined by cross-validation –Test with data not used for training –Use control parameters that work best for these data

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Two-fold Cross-validation Use half the data for training. and the other half for testing.

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Two-fold Cross-validation Then swap around the training and test data.

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Leave One Out Cross-validation Use all data except one point for training. The one that was left out is used for testing.

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Leave One Out Cross-validation Then leave another point out. And so on...

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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.

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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

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Regression 23 26 30 29 18 32 40

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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.

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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)

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Overview Motivation Principles Geeky stuff Example Validation Future directions

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Compare with variable velocity methods –Begs LDDMM algorithm Classification/regression from initial momentum Combine with SPM5 segmentation model –Similar to Emiliano DAgostinos method Develop a proper EM framework for generating tissue probability maps

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u Hu

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Initial momentum Variable velocity framework

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Initial momentum Variable velocity framework

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