Experiments on a New Inter-Subject Registration Method

John Ashburner

2007

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.

Overview

• Motivation– 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

A simple 2D example

Individual brain

Warped Individual

Reference

Residual Differences

Individual brain

Warped Individual

Expansion and

contraction

• Relative volumes encoded by Jacobian determinants of deformation

Tissue volume comparisons

Warped grey matter

Jacobian determinants

Absolute grey matter

volumes

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

• Geeky stuff

• Example

• Validation

• Future directions

Principles

DiffeomorphicAnatomicalRegistrationThroughExponentiatedLie Algebra

Deformations parameterized by a single flow field, which is considered to be constant in time.

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

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)

Flow Field

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 withφ(-1/8) = x - u/8

Scaling and squaring example

DARTEL

Jacobian determinants remain positive

Overview

• Motivation

• Principles

• Geeky stuff– Feel free to sleep

• Example

• 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 = cSplit 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

Lowest resolution

Full Multi-Grid

Overview

• Motivation

• Principles

• Geeky stuff

• Example– Simultaneous registration of GM & WM– Tissue probability map creation

• Validation

• Future directions

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 matterTemplate

Subject 1

Subject 2

Subject 3

Subject 4

TemplateInitial

Average

After a few iterations

Final template

Iteratively generated from 471 subjects

Began with rigidly aligned tissue probability maps

Used an inverse consistent formulation

Grey matter average of 452 subjects – affine

Grey matter average of 471 subjects

White matter average of 471 subjects

Initial GM images

Warped GM images

Overview

• Motivation

• Principles

• Geeky stuff

• Example

• 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

ControlTraining Data

PatientTraining Data

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??

Classifying

Controls

Patients

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??

y=f(aTx+b)

Support Vector Classifier

Support Vector Classifier (SVC)

SupportVector

SupportVector

Support

Vector

a is a weighted linear combination of the support vectors

Some Equations

• Linear classification is by y = f(aTx + 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 wi xi

• So y = f(i wi xiTx + b)

Going Nonlinear

• Nonlinear classification is by

y = f(i wi (xi,x))– where (xi,x) is some function of xi and x.

• e.g. RBF classification (xi,x) = exp(-||xi-x||2/(22))

• Requires a matrix of distance measures (metrics) between each pair of images.

Nonlinear SVC

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

Two-fold Cross-validation

Use half the data for training.

and the other half for testing.

Two-fold Cross-validation

Then swap around the training and test data.

Leave One Out Cross-validation

Use all data except one point for training.

The one that was left out is used for testing.

Leave One Out Cross-validation

Then leave another point out.

And so on...

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

23

26

30

29

18

32

40

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

• Geeky stuff

• Example

• Validation

• Future directions

Future directions

• Compare with variable velocity methods– Beg’s LDDMM algorithm

• Classification/regression from “initial momentum”

• Combine with SPM5 segmentation model– Similar to Emiliano D’Agostino’s method

• Develop a proper EM framework for generating tissue probability maps

u

Hu

“Initial momentum”

Variable velocity framework

“Initial momentum”

Variable velocity framework

Thank you