Computational Anatomy:
VBM and Alternatives
Motivation for Computational Anatomy
* See Wednesday’s symposium 13:30-15:00
* Cortical Fingerprinting: What Anatomy Can Tell Us
About Functional Architecture
* There are many ways of examining brain
structure. Depends on:
* The question you want to ask
* The data you have
* The available software
Overview
* Volumetric differences
* Serial Scans
* Jacobian Determinants
*
*
*
*
Voxel-based Morphometry
Multivariate Approaches
Difference Measures
Another approach
Deformation Field
Original
Warped
Deformation field
Template
Jacobians
Jacobian Matrix (or just “Jacobian”)
Jacobian Determinant (or just “Jacobian”) - relative volumes
Serial Scans
Early
Late
Difference
Data from the
Dementia Research
Group, Queen Square.
Regions of expansion and contraction
* Relative
volumes
encoded in
Jacobian
determinants.
Rigid Registration Software Packages
* AIR: Automated Image Registration
http://bishopw.loni.ucla.edu/AIR5/
* FLIRT: FMRIB’s Linear Image Registration Tool
http://www.fmrib.ox.ac.uk/fsl/flirt/
* MNI_AutoReg
http://www.bic.mni.mcgill.ca/users/louis/MNI_AUTOREG_home/readme/
* SPM
http://www.fil.ion.ucl.ac.uk/spm
* VTK CISG Registration Toolkit
http://www.image-registration.com/
...and many others...
Nonlinear Registration Software
Only listing public software that can (probably) estimate
detailed warps suitable for longitudinal analysis.
* HAMMER
http://oasis.rad.upenn.edu/sbia/
* MNI_ANIMAL Software Package
http://www.bic.mni.mcgill.ca/users/louis/MNI_ANIMAL_home/readme/
* SPM2
http://www.fil.ion.ucl.ac.uk/spm
* VTK CISG Registration Toolkit
http://www.image-registration.com/
…there is much more software that is less readily available...
Late
Warped early
Early
Difference
Late CSF
Relative volumes
Early CSF
CSF “modulated” by
relative volumes
Late CSF - modulated CSF
Late CSF - Early CSF
Smoothed
Smoothing
Smoothing is done by convolution.
Each voxel after smoothing effectively
becomes the result of applying a weighted
region of interest (ROI).
Before convolution
Convolved with a circle
Convolved with a Gaussian
Overview
* Volumetric differences
* Voxel-based Morphometry
* Method
* Interpretation Issues
* Multivariate Approaches
* Difference Measures
* Another approach
Voxel-Based Morphometry
* I. C. Wright et al. A Voxel-Based Method for the
Statistical Analysis of Gray and White Matter Density
Applied to Schizophrenia. NeuroImage 2:244-252 (1995).
* I. C. Wright et al. Mapping of Grey Matter Changes in
Schizophrenia. Schizophrenia Research 35:1-14 (1999).
* J. Ashburner & K. J. Friston. Voxel-Based Morphometry The Methods. NeuroImage 11:805-821 (2000).
* J. Ashburner & K. J. Friston. Why Voxel-Based
Morphometry Should Be Used. NeuroImage 14:1238-1243
(2001).
* C. D. Good et al. Automatic Differentiation of Anatomical
Patterns in the Human Brain: Validation with Studies of
Degenerative Dementias. NeuroImage 17:29-46 (2002).
Voxel-Based Morphometry
* Produce a map of statistically significant differences
among populations of subjects.
* e.g. compare a patient group with a control group.
* or identify correlations with age, test-score etc.
* The data are pre-processed to sensitise the tests to
regional tissue volumes.
* Usually grey or white matter.
* Can be done with SPM package, or e.g.
* HAMMER and FSL
http://oasis.rad.upenn.edu/sbia/
http://www.fmrib.ox.ac.uk/fsl/
Pre-processing for Voxel-Based
Morphometry (VBM)
VBM Preprocessing in SPM5b
* Segmentation in SPM5b also estimates a spatial
transformation that can be used for spatially
normalising images.
* It uses a generative model, which involves:
* Mixture of Gaussians (MOG)
* Bias Correction Component
* Warping (Non-linear Registration) Component
Mixture of Gaussians
g
a0
a
Ca
c1
y1
m
c2
y2
s2
c3
y3
b
cI
yI
Cb
b0
Bias Field
g
a0
a
Ca
y
r(b)
c1
y1
m
c2
y2
s2
c3
y3
b
cI
yI
Cb
y r(b)
b0
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.
“Mixing Proportions”
g
a0
a
Ca
c1
y1
m
c2
y2
s2
c3
y3
b
cI
yI
Cb
b0
Deforming the Tissue Probability Maps
* Tissue probability
maps are deformed
according to
parameters a.
g
a0
a
Ca
c1
y1
m
c2
y2
s2
c3
y3
b
cI
yI
Cb
b0
SPM5b Pre-processed data for four subjects
Warped, Modulated Grey Matter
12mm FWHM Smoothed Version
Statistical Parametric Mapping…
–
group 1
voxel by voxel
modelling

parameter estimate
standard error
=
statistic image
or
SPM
group 2
Validity of the statistical tests in SPM
* Residuals are not normally distributed.
* Little impact on uncorrected statistics for
experiments comparing groups.
* Invalidates experiments that compare one subject
with a group.
* Corrections for multiple comparisons.
* Mostly valid for corrections based on peak heights.
* Not valid for corrections based on cluster extents.
* SPM makes the inappropriate assumption that the
smoothness of the residuals is stationary.
* Bigger blobs expected in smoother regions.
Interpretation Problem
* What do the blobs really mean?
* Unfortunate interaction between the algorithm's spatial
normalization and voxelwise comparison steps.
* Bookstein FL. "Voxel-Based Morphometry" Should Not Be
Used with Imperfectly Registered Images. NeuroImage
14:1454-1462 (2001).
* W.R. Crum, L.D. Griffin, D.L.G. Hill & D.J. Hawkes. Zen
and the art of medical image registration: correspondence,
homology, and quality. NeuroImage 20:1425-1437 (2003).
* N.A. Thacker. Tutorial: A Critical Analysis of Voxel-Based
Morphometry. http://www.tina-vision.net/docs/memos/2003011.pdf
Some Explanations of the Differences
Mis-classify
Mis-register
Folding
Thickening
Thinning
Mis-register
Mis-classify
Cortical Thickness Mapping
* Direct measurement of cortical thickness may be
better for studying neuro-degenerative diseases
* http://surfer.nmr.mgh.harvard.edu/
* http://brainvoyager.com/
* Some example references
* B. Fischl & A.M. Dale. Measuring Thickness of the Human Cerebral Cortex from Magnetic
Resonance Images. PNAS 97(20):11050-11055 (2000).
* S.E. Jones, B.R. Buchbinder & I. Aharon. Three-dimensional mapping of cortical thickness
using Laplace's equation. Human Brain Mapping 11 (1): 12-32 (2000).
* J.P. Lerch et al. Focal Decline of Cortical Thickness in Alzheimer’s Disease Identified by
Computational Neuroanatomy. Cereb Cortex (2004).
* Narr et al. Mapping Cortical Thickness and Gray Matter Concentration in First Episode
Schizophrenia. Cerebral Cortex (2005).
* Thompson et al. Abnormal Cortical Complexity and Thickness Profiles Mapped in Williams
Syndrome. Journal of Neuroscience 25(16):4146-4158 (2005).
Overview
* Volumetric differences
* Voxel-based Morphometry
* Multivariate Approaches
* Scan Classification
* Cross-Validation
* Difference Measures
* Another approach
Multivariate Approaches
* Z. Lao, D. Shen, Z. Xue, B. Karacali, S. M. Resnick and C.
Davatzikos. Morphological classification of brains via
high-dimensional shape transformations and machine
learning methods. NeuroImage 21(1):46-57, 2004.
* C. Davatzikos. Why voxel-based morphometric analysis
should be used with great caution when characterizing
group differences. NeuroImage 23(1):17-20, 2004.
* K. J. Friston and J. Ashburner. Generative and
recognition models for neuroanatomy. NeuroImage
23(1):21-24, 2004.
“Globals” for VBM
* Shape is multivariate
* Dependencies among
volumes in different
regions
* SPM is mass univariate
* “globals” used as a
compromise
* Can be either ANCOVA or
proportional scaling
Where should any
difference between the two
“brains” on the left and that
on the right appear?
Multivariate Approaches
* An alternative to mass-univariate testing (SPMs)
* Generate a description of how to separate groups of
subjects
* Use training data to develop a classifier
* Use the classifier to diagnose test data
* Data should be pre-processed so that clinically
relevant features are emphasised
* use existing knowledge
Training and Classifying
?
?
Patient
Training Data
Control
Training Data
?
?
Classifying
?
?
Patients
Controls
?
?
y=f(wTx+w0)
Difference between means
m2
m1
w  m2-m1
Does not take account of
variances and covariances
Fisher’s Linear Discriminant
w  S-1(m2-m1 )
Curse of dimensionality !
Support Vector Classifier (SVC)
Support Vector Classifier (SVC)
Support
Vector
Support
Vector
Support
Vector
w is a weighted linear
combination of the
support vectors
Going Nonlinear
* Linear classification is by y = f(wTx + w0)
* where w is a weighting vector, x is the test data, w0 is an offset,
and f(.) is a thresholding operation
* w is a linear combination of SVs w = Si ai xi
* So y = f(Si ai xiTx + w0)
* Nonlinear classification is by
y = f(Si ai (xi,x) + w0)
* where (xi,x) is some function of xi and x.
* e.g. RBF classification (xi,x) = exp(-||xi-x||2/(2s2))
Nonlinear SVC
Over-fitting
Test data
A simpler model can often do better...
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 crossvalidation
* 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...
Regression (e.g. against age)
Other Considerations
* Should really take account of Bayes Rule:
P(sick | data)
=
P(data | sick) x P(sick)
P(data | sick) x P(sick) + P(data | healthy) x P(healthy)
Requires prior probabilities
* Sometimes decisions should be weighted using
Decision Theory
* Utility Functions/Risk
* e.g. a false negative may be more serious than a false positive
Overview
*
*
*
*
Volumetric differences
Voxel-based Morphometry
Multivariate Approaches
Difference Measures
* Derived from Deformations
* Derived from Deformations + Residuals
* Another approach
Distance Measures
* Kernel-based classifiers (such as SVC) use measures
of distance between data points (scans).
* I.e. measure of how different each scan is from each other
scan.
* The measure is likely to depend on the application.
Deformation Distance Summary
•Deformations can be considered within a small or
large deformation setting.
•Small deformation setting is a linear approximation.
•Large deformation setting accounts for the nonlinear nature of
deformations.
•Uses Lie Group Theory.
• Miller, Trouvé, Younes “On the Metrics and Euler-Lagrange
Equations of Computational Anatomy”. Annual Review of
Biomedical Engineering, 4:375-405 (2003) plus supplement
• Beg, Miller, Trouvé, L. Younes. “Computing Large Deformation
Metric Mappings via Geodesic Flows of Diffeomorphisms”. Int.
J. Comp. Vision, 61:1573-1405 (2005)
Tilak Ratnanather gave me the following two slides…
Computing the geodesic: problem statement
I0: Template I1:Target

Problem Statement: Given I0 and I1 , compute v such that
1


arg min   Lv ( y , t ), Lv( y , t ) dydt   I0 ( ( y,1))  I1 dy
v
0

 1
t 1
where
( y , t )   y ( y, t )v( y, t )
t
1
2
Metrics on 3D Hippocampus in
Neuro-psychiatric Disorders.
Data from the lab. of Dr. Csernansky,
Washington University, St Louis.
3D Hippocampus: Young to Schizophrenia
Young
1.386
2.541
3.696
4.620
Schizophrenia
3D Hippocampus: Young to Alzheimer’s
Young
1.430
2.621
3.813
4.766
Alzheimer’s
Accuracy of Automated Volumetric
Inter-subject Registration
Sulcal misregistration
12
Distance (mm)
10
8
6
4
2
0
A
D
M
Method
P
R
SPM2
Hellier et al. Inter subject registration of functional and anatomical data using SPM.
MICCAI'02 LNCS 2489 (2002)
Hellier et al. Retrospective evaluation of inter-subject brain registration. MIUA (2001)
One-to-One Mappings
* One-to-one mappings
break down beyond a
certain scale
* The concept of a
single “best” mapping
may become
meaningless at higher
resolution
Pictures taken from
http://www.messybeast.com/freak-face.htm
A Combined Distance Measure
* Exact registration may not be possible.
* Base distance measures on deformations plus
residuals after registration.
* Could use a related framework to that used for
registering/segmenting.
* Distance measures should be adjusted based on user
expertise.
* E.g. Some brain regions may be more informative than
others, so give them more weighting.
* Differences may be focal or more global
* Could use some sort of high- or low-pass filtering.
Overview
*
*
*
*
*
Volumetric differences
Voxel-based Morphometry
Multivariate Approaches
Difference Measures
Another approach
Anatomist/BrainVISA Framework
* Free software available from:
http://brainvisa.info/
* Automated identification and labelling of sulci etc.
* These could be used to help spatial normalisation etc.
* Can do morphometry on sulcal areas, etc
* J.-F. Mangin, D. Rivière, A. Cachia, E. Duchesnay, Y. Cointepas, D.
Papadopoulos-Orfanos, D. L. Collins, A. C. Evans, and J. Régis. ObjectBased Morphometry of the Cerebral Cortex. IEEE Trans. Medical Imaging
23(8):968-982 (2004)
Design of an artificial neuroanatomist
Elementary
folds
3D
retina
Fields of
view of
neural nets
Bottom-up
flow
Sulci
Correlates of handedness
14 subjects
128 subjects
Central sulcus
surface is larger
in dominant hemisphere
Handedness correlates : localization after affine
normalization
Some of the potentially interesting
posters
* (#728 T-PM ) A Matlab-based toolbox to facilitate multi-voxel pattern
classification of fMRI data.
* (#699 T-AM ) Pattern classification of hippocampal shape analysis in a
study of Alzheimer's Disease
* (#697 M-AM ) Metric distances between hippocampal shapes predict
different rates of shape changes in dementia of Alzheimer type and
nondemented subjects: a validation study
* (#721 M-PM ) Unbiased Diffeomorphic Shape and Intensity Template
Creation: Application to Canine Brain
* (#171 T-AM ) A Population-Average, Landmark- and Surface-based
(PALS) Atlas of Human Cerebral Cortex
* (#70 M-PM ) Cortical Folding Hypotheses: What can be inferred from
shape?
* (#714 T-AM ) Shape Analysis of Neuroanatomical Structures Based on
Spherical Wavelets