Pre-processing for “Voxel-Based Morphometry”

John AshburnerThe Wellcome Trust Centre for Neuroimaging12 Queen Square, London, UK.

Contents

• Introduction

• Segmentation

• DARTEL Registration

Voxel-based Morphometry

• Pre-process the images of lots of subjects, to generate spatially normalised grey matter maps of each subject.

• Smooth spatially.

• Perform voxel-wise statistics.

• Try to interpret the findings in terms of volumetric differences.

Segment into different tissue classes

Spatially Normalize – with scaling by Jacobian determinant

Smooth Spatially

Mass-univariate statistical testing

Inference via Random Field

Theory

Smoothing

Before convolution Convolved with a circleConvolved with a Gaussian

Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI).

Possible Explanations for Findings

ThickeningThinning

Folding

Mis-classify

Mis-classify

Mis-register

Mis-register

Contents

• Introduction

• Segmentation– Mixture of Gaussians– Bias correction– Warping to match tissue probability maps

• DARTEL Registration

Tissue Segmentation

• Circularity:– Registration is helped by tissue classification or bias correction.– Tissue classification helped by registration and bias correction.– Bias correction is helped by registration and tissue

classification.

• The solution is to put everything in the same generative model.– A MAP solution is found by repeatedly alternating among

classification, bias correction and registration steps.

• Should produce “better” results than simple serial applications of each component.

A Generative Model

• A model of how the data may have been generated, which comprises:– Mixture of Gaussians (MOG)– Bias Correction– Non-linear Inter-subject

Registrationy1c1

y2

y3

c2

c3

C

CyIcI

Mixture of Gaussians (MOG)• Tissue classification is based on a Mixture of

Gaussians model (MOG), which represents the intensity probability density by a number of Gaussian distributions.

Image Intensity

Frequency

Belonging Probabilities

Belonging probabilities are assigned by normalising to one.

Non-Gaussian Intensity Distributions

• Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled.– E.g. accounting for partial volume effects

Modelling a Bias Field

Corrupted image

Corrected imageBias Field

Tissue Probability Maps

• Tissue probability maps (TPMs) are used instead of the proportion of voxels in each Gaussian as the prior.

ICBM Tissue Probabilistic Atlases. These tissue probability maps are kindly provided by the International Consortium for Brain Mapping, John C. Mazziotta and Arthur W. Toga.

Deforming the Tissue Probability Maps

• Tissue probability images are deformed so that they can be overlaid on top of the image to segment.

Optimisation

• The “best” parameters are those that maximise the log-probability.

• Optimisation involves finding them.

• Begin with starting estimates, and repeatedly change them so that the objective function decreases each time.

Steepest Descent

Start

Optimum

Alternate between optimising different groups

of parameters

Tissue probability

maps of GM and WM

Spatially normalised BrainWeb

phantoms (T1, T2 and PD)

Cocosco, Kollokian, Kwan & Evans. “BrainWeb: Online Interface to a 3D MRI Simulated Brain Database”. NeuroImage 5(4):S425 (1997)

Contents

• Introduction

• Segmentation

• DARTEL Registration– Scaling and squaring– Optimisation– Warping GM and WM images to their

average

Parameterization

DiffeomorphicAnatomicalRegistrationThroughExponentiatedLie Algebra

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

Not really a proper Lie Group.Often referred to as a one parameter subgroup.

Euler Integration• Parameterising the deformation

• φ(0)(x) = x• φ(1)(x) = ∫ u(φ(t)(x))dt• u is a flow field to be estimated

• Scaling and squaring is used to generate deformations.– c.f. matrix exponentiation

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)

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

Deformations at different times

Jacobians

• Jacobian fields can also be obtained by scaling and squaring.

• If warps are composed by:ϕC=ϕB○ϕA

then Jacobian matrices are obtained by:JϕC=(JϕB○ϕA) JϕA

Jacobian determinants remain positive (almost)

See also…• C. Moler and C. van Loan. “Nineteen Dubious Ways to Compute the

Exponential of a Matrix, Twenty-Five Years Later”. SIAM Review 45(1):3-49 (2003).

• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A Log-Euclidean Polyaffine Framework for Locally Rigid or Affine Registration”. Proc. Of the 3rd International Workshop on Biomedical Image Registration (WBIR'06), 2006, pp. 120-127. LNCS vol 4057. Springer-Verlag, Utrecht, NL.

• V. Arsigny, O. Commowick, X. Pennec and N. Ayache. “A Log-Euclidean Framework for Statistics on Diffeomorphisms”. Proc. of the 9th International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI'06), 2006, pp. 924-931. LNCS 4190. Springer-Verlag, Berlin, Germany.

• M. Hernandez, M. N. Bossa, and S. Olmos. “Registration of anatomical images using geodesic paths of diffeomorphisms parameterized with stationary vector fields”. IEEE workshop on Math. Meth. in Biom. Image Anal. (MMBIA’07), 2007.

Contents

• Introduction

• Segmentation

• DARTEL Registration– Scaling and squaring– Optimisation– Warping GM and WM images to their

average

Multinomial Likelihood Term

• 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.

Prior Term

• ½uTHu• 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“Membrane energy”

“Bending energy”Images registered using a small deformation approximation

Optimization

• Uses Gauss-Newton– 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

• If M is diagonally dominant (membrane energy):

x(k+1) = E-1(c – F x(k))• Otherwise regularize (bending or linear elastic

energy):

x(k+1) = x(k) + (E+sI)-1(c – M x(k))– Diagonal dominance is when |mii| > Σi≠j |mij|

M = H+A = E+F

2nd derivs of prior term

2nd derivs of likelihood term

Easy to invert

Difficult to invert

Highest resolution

Lowest resolution

Full Multi-Grid

A

•Prolongation of low resolution solution to current resolution.•Add this to existing solution.•Perform a few iterations of relaxation.•Restrict residuals down to lower resolution.

B

•Prolongation of low resolution solution to current resolution.•Add this to existing solution at current resolution.•Perform a few iterations of relaxation.•Prolongation of solution to higher resolution.

C

•Restrict high resolution residuals to current resolution.•Perform a few iterations of relaxation.•Restrict residuals down to lower resolution.

E

•Restrict higher resolution residuals to current resolution.•Obtain exact solution by matrix inversion.•Prolongation of solution to higher resolution.

See also…

• W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery. Numerical Recipes in C (Second Edition). Cambridge University Press, Cambridge, UK. 1992.– Chapter 15, Section 5 explains Gauss-Newton

optimization (Levenberg-Marquardt without the regularisation).

– Chapter 19, Section 6 explains the basics of multi-grid methods.

Contents

• Introduction

• Segmentation

• DARTEL Registration– Scaling and squaring– Optimisation– Warping GM and WM images to their

average

Template Generation Initial

Average

After a few iterations

Final template

Iteratively generated from 471 subjects.

Began with rigidly aligned tissue probability maps.

Regularization lighter for later iterations.

Generative Model

• p(ϕ1,t1, ϕ2,t2, ϕ3,t3,… μ)= p(t1,ϕ1|μ) p(t2,ϕ2|μ) p(t3,ϕ3|μ) … p(μ)

• = p(t1|ϕ1,μ) p(ϕ1) p(t2|ϕ2,μ) p(ϕ2)… p(μ)

• MAP solution obtainedfor template.

• Requires p(μ)

μ

t1

ϕ1

t2

ϕ2

t3

ϕ3

t4 ϕ4

t5

ϕ5

Laplacian Smoothness Priors on template

2D

3D

Template modelled as softmax of a Gaussian process

μk(x) = exp(ak(x))/(Σj exp(aj(x)))

MAP solution determined for a, by Gauss-Newton optimisation, using multi-grid.

ML and MAP templates from 6 subjects

Nonlinearly aligned Rigidly aligned

log

MAP

ML

471 Subject Average

471 Subject Average

471 Subject Average

Subject 1

471 Subject Average

Subject 2

471 Subject Average

Subject 3

471 Subject Average

Preprocessing with DARTEL

u

Hu

“Initial momentum”

Variable velocity framework (as in LDDMM)

“Initial momentum”

Variable velocity framework (as in LDDMM)

Determining amount of regularisation

• Matrices too big for Bayesian variance component estimation.

• Used cross-validation.

• Smooth an image by different amounts, see how well it predicts other images:

Rigidly aligned

Nonlinear registered

log p(t|μ) = ΣjΣk tjk log(μjk)