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Computational Anatomy & Statistical Shape Models John Ashburner john@fil.ion.ucl.ac.uk Functional Imaging Lab, 12 Queen Square, London, UK. Why?. The Wellcome Trust is keen that there is a translational component to the work in the FIL. E.g. develop some potentially useful diagnostic stuff.

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slide1

Computational Anatomy&Statistical Shape ModelsJohn Ashburnerjohn@fil.ion.ucl.ac.ukFunctional Imaging Lab, 12 Queen Square, London, UK.

slide2
Why?
  • The Wellcome Trust is keen that there is a translational component to the work in the FIL.
    • E.g. develop some potentially useful diagnostic stuff.
  • For proper generative models of brain shape differences.
    • More accurate spatial normalisation.
    • More accurate shape characterisations.
  • To use these models for proper characterisation of population differences.
    • These may be multivariate.
  • Join the mainstream.
    • How do more established fields of biology compare shapes?
slide7
NeuroImage Volume 23, Supplement 1, Pages CO2-S299 (2004)

Mathematics in Brain Imaging

Edited by P.M. Thompson, M.I. Miller, T. Ratnanather, R.A. Poldrack and T.E. Nichols

  • Mapping cortical change in Alzheimer's disease, brain development, and schizophrenia Paul M. Thompson, Kiralee M. Hayashi, Elizabeth R. Sowell, Nitin Gogtay, Jay N. Giedd, Judith L. Rapoport, Greig I. de Zubicaray, Andrew L. Janke, Stephen E. Rose, James Semple et al.
  • Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms Michael I. Miller
  • Geometric strategies for neuroanatomic analysis from MRI James S. Duncan, Xenophon Papademetris, Jing Yang, Marcel Jackowski, Xiaolan Zeng and Lawrence H. Staib
  • Variational, geometric, and statistical methods for modeling brain anatomy and function Olivier Faugeras, Geoffray Adde, Guillaume Charpiat, Christophe Chefd'Hotel, Maureen Clerc, Thomas Deneux, Rachid Deriche, Gerardo Hermosillo, Renaud Keriven, Pierre Kornprobst et al.
  • Computational anatomy and neuropsychiatric disease: probabilistic assessment of variation and statistical inference of group difference, hemispheric asymmetry, and time-dependent change John G. Csernansky, Lei Wang, Sarang C. Joshi, J. Tilak Ratnanather and Michael I. Miller
  • Sequence-independent segmentation of magnetic resonance images Bruce Fischl, David H. Salat, André J.W. van der Kouwe, Nikos Makris, Florent Ségonne, Brian T. Quinn and Anders M. Dale
  • Expert knowledge-guided segmentation system for brain MRI Alain Pitiot, Hervé Delingette, Paul M. Thompson and Nicholas Ayache
  • Surface-based approaches to spatial localization and registration in primate cerebral cortex David C. Van Essen
  • Cortical surface segmentation and mapping Duygu Tosun, Maryam E. Rettmann, Xiao Han, Xiaodong Tao, Chenyang Xu, Susan M. Resnick, Dzung L. Pham and Jerry L. Prince
  • Cortical cartography using the discrete conformal approach of circle packings Monica K. Hurdal and Ken Stephenson
  • A framework to study the cortical folding patterns J.-F. Mangin, D. Rivière, A. Cachia, E. Duchesnay, Y. Cointepas, D. Papadopoulos-Orfanos, P. Scifo, T. Ochiai, F. Brunelle and J. Régis
  • Geodesic estimation for large deformation anatomical shape averaging and interpolation Brian Avants and James C. Gee
  • Unbiased diffeomorphic atlas construction for computational anatomy S. Joshi, Brad Davis, Matthieu Jomier and Guido Gerig
  • Statistics on diffeomorphisms via tangent space representations M. Vaillant, M.I. Miller, L. Younes and A. Trouvé
  • Soliton dynamics in computational anatomy Darryl D. Holm, J. Tilak Ratnanather, Alain Trouvé and Laurent Younes
  • Implicit brain imaging Facundo Mémoli, Guillermo Sapiro and Paul Thompson
training and classifying

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Training and Classifying

Control

Training Data

Patient

Training Data

classifying

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Classifying

Controls

Patients

y=f(aTx+b)

support vector classifier svc
Support Vector Classifier (SVC)

a is a weighted linear combination of the support vectors

Support

Vector

Support

Vector

Support

Vector

some equations
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 = Si wixi
  • So y = f(Si wixiTx + b)
going nonlinear
Going Nonlinear
  • Nonlinear classification is by

y = f(Si wi(xi,x))

    • where (xi,x) is some function of xiand x.
      • e.g. RBF classification (xi,x) = exp(-||xi-x||2/(2s2))
  • Requires a matrix of distance measures (metrics) between each pair of images.
what is a metric
What is a Metric?

A

B

  • Positive
    • Dist(A,B) ≥ 0
    • Dist(A,A) = 0
  • Symmetric
    • Dist(A,B) = Dist(B,A)
  • Satisfy triangle inequality
    • Dist(A,B)+Dist(B,C) ≥ Dist(A,C)

C

concise representations
Concise representations
  • Information reduction/compression
  • Most parsimonious representation - best generalisation
    • Occam’s Razor
  • Registration compresses data
    • signal is partitioned into
      • deformations
      • residuals
the small deformation setting
The Small deformation setting
  • Most “nonlinear” registration is done in the small-deformation setting.
  • Involves adding a smooth displacement field to an identity transform: y = x + u
    • No one-to-one constraint
  • Inverse from: x = y - u
    • Can be a poor approximation to the real inverse
  • Adding and subtracting displacements doesn’t work properly
  • Smoothing and averaging displacements doesn’t work properly
  • Not the most parsimonious model
  • Unrealistic generative model
slide21

Forward transform

Small def. approx. to backward transform

Backward transform

Small def. approx. to forward transform

illustrating some concepts with rotations
Illustrating some concepts with rotations
  • Consider a 2D rotation y=Rx, where
  • This can be formulated as the solution of a differential equation at time t=1
    • x1(t) = x2(t)
    • x2(t) = -x1(t)
  • or
    • x(t) = Ax(t), where
exponentials
Exponentials
  • The solution can be obtained by
  • The exponential is defined as:
  • There are many ways of computing R from A, but one of the easiest is by scaling and squaring
averaging rotations
Averaging rotations
  • It makes no sense to average the rotation matrices themselves.
    • The result may not be a rotation
  • The elements of a rotationmatrix lie on a manifold.
  • Average by minimising thesum of squared distancestangential to the manifold
  • Distance derived fromvelocity (distance travelled inunit time)
  • Shortest distances are geodesics,which require constant velocity

r12

r11

groups
Groups
  • 2D rotation matrices form a Liegroup under multiplication (SO2).
  • Group Requirements
    • Composition of group members is another group member
    • The members have inverses
    • There is an identity member
    • The composition operations are associative
  • Lie Group requirements
    • Continuous and differentiable manifold
3d rotations
3D Rotations
  • 3D rotations defined by
  • These do not commute (R1R2≠R2R1)
  • Similarly exp(A1) exp(A2) ≠exp(A2) exp(A1)
  • Both differ from exp(A1+A2)
    • Makes life more difficult
    • Iterative schemes needed for averaging etc.
lie algebra
Lie Algebra
  • A would be known as the Lie algebra of the rotation matrix.
  • The amount of non-commutativity is measured by the Lie bracket
    • Results fromcurvature of themanifold
    • [A,B] = AB-BA

eB

e2(AB-BA)

eA

eA

eB

how much rotation is in a rotation matrix
How much rotation is in a rotation matrix?
  • Given two rotation matrices, R and S. The relative difference between the rotations can be found by computing

C = log (R-1S)

and then computing the RMS of C.

(12+ 22 + 32)1/2

cartan decomposition
Cartan decomposition
  • A matrix can be decomposed into

A = (A+AT)/2 + (A-AT)/2

    • (A+AT)/2 is symmetric
      • Encodes zooms and shears
    • (A-AT)/2 is skew symmetric
      • Encodes rigid rotations
  • A can be converted into a column vector (a)
  • There is a matrix L, such that La gives the elements of (A-AT)/2.
  • The square of the rotation angle is then given by (La)T(La) = aT(LTL)a
    • This excludes any zooming and shearing from the measure
  • Similarly – and more usefully - the amount of zooming and shearing can be computed in a way that is independent of the rotations.
slide36

Diffeomorphisms have curved trajectories (variable velocity) if followed in the Eulerian reference frame (fixed).

If followed within the Lagrangian frame (moves over time), they appear to have constant velocity.

partial differential equations
Partial Differential Equations

Model one image as it deforms to match another.

x(t) = u(x(t))

x(1) = eu (x(0))

matrix representations of diffeomorphisms
Matrix representations of diffeomorphisms

x(1) = eUx(0)

x(0) = e-Ux(1)

For large k

eU ≈ (I+U/k)k

compositions
Compositions

Large deformations generated from compositions of small deformations

S1 = S1/8oS1/8oS1/8oS1/8oS1/8oS1/8oS1/8oS1/8

Recursive formulation

S1 = S1/2oS1/2,S1/2 = S1/4oS1/4,S1/4 = S1/8oS1/8

Small deformation

approximation

S1/8 ≈ I + U/8

the shape metric
The shape metric
  • Don’t use the straight distance (i.e. √uTu)
  • Distance = √uTLTLu
  • What’s the best form of L?
    • Membrane Energy
    • Bending Energy
    • Linear Elastic Energy
consistent registration
Consistent registration

Register to a mean shaped image

A

B

A

B

µ

C

C

Totally impractical for lots of scans

Problem: How can the distance between e.g. A and B be computed? Inverse exponentiating is iterative and slow.

baker campbell hausdorff series
Baker-Campbell-Hausdorff series
  • Exp-1(Exp(A)Exp(B))= A+B +[A,B]/2 +[A,[A,B]]/12-[B,[A,B]]/12 -[B,[A,[A,B]]]/48-[A,[B,[A,B]]]/48

+ …

    • Where [A,B] is the Lie bracket applied to flow fields

Sometimes unstable. Looks like proper nonlinear methods would be impractical.

eB

e2(AB-BA)

eA

eA

eB

alternative strategy
Alternative strategy
  • Assume the manifold is locally flat around some point (the template image.
    • The results depend on the point on the manifold that is chosen.
    • Ideally use a mean shape as the template.
    • Best approximation.
visualisation

Controls

Patients

y=f(aTx+b)

Visualisation
  • The results of multivariate analyses are difficult to visualise
  • For linear classifiers, it can be done by “caricaturing” the difference
    • E.g. for the separation of two groups, it would be possible to show two exaggerated versions of the mean image.
slide52

t=-1.0

t=-0.75

t=-0.5

t=0.25

t=0.0

t=-0.25

t=1.0

t=0.5

t=0.75

slide56

But let’s just say that it shows potential…

Average of 452 images

2D average of 471 images

Registration of each 2D image takes about 3 seconds per iteration, and about 16 iterations. I see no problems scaling it to 3D.

Only affine registered

over fitting
Over-fitting

Test data

A simpler model can often do better...

cross validation
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
Two-fold Cross-validation

Use half the data for training.

and the other half for testing.

two fold cross validation1
Two-fold Cross-validation

Then swap around the training and test data.

leave one out cross validation
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 validation1
Leave One Out Cross-validation

Then leave another point out.

And so on...

interpretation
Interpretation??
  • Significance assessed from accuracy based on cross-validation.
  • Main problems:
    • No simple interpretation.
    • Mechanism of classification is difficult to visualise
      • especially for nonlinear classifiers
    • Difficult to understand (not like blobs)
  • May be able to use the separation to derive simple (and more publishable hypotheses).
group theory
Diffeomorphisms (smooth continuous one-to-one mappings) form a Group.

Closure

AoB remains in the same group.

Associativity

(AoB)oC = Ao(BoC)

Identity

Identity transform I exists.

Inverse

A-1 exists, and A-1oA=AoA-1 = I

It is a Lie Group.

The group of diffeomorphisms constitute a smooth manifold.

The operations are differentiable.

Group Theory
lie groups
Simple Lie Groups include various classes of affine transform matrices.

E.g. SO(2) : Special Orthogonal 2D (rigid-body rotation in 2D).

Manifold is a circle

Lie Algebra is exponentiated to give Lie group. For square matrices, this involves a matrix exponential.

Lie Groups
relevance to diffeomorphisms
Parameterise with velocities, rather than displacements.

Velocities are the Lie Algebra. These are exponentiated to a deformation by recursive application of tiny displacements, over a period of time=0..1.

A(1) = A(1/2)oA(1/2)

A(1/2) = A(1/4) oA(1/4)

Don’t actually use matrices.

For tiny deformations, things are almost linear.

x(1/1024)x(0) + vx/1024

y(1/1024)y(0) + vy/1024

z(1/1024)z(0) + vz/1024

Recursive application by

x(1/2) = x(1/4) (x(1/4), y(1/4),z(1/4))

y(1/2) = y(1/4) (x(1/4), y(1/4),z(1/4))

z(1/2) = z(1/4) (x(1/4), y(1/4),z(1/4))

Relevance to Diffeomorphisms
working with diffeomorphisms
Averaging Warps.

Distances on the manifold are given by geodesics.

Average of a number of deformations is a point on the manifold with the shortest sum of squared geodesic distances.

E.g. average position of London, Sydney and Honolulu.

Inversion.

Negate the velocities, and exponentiate.

x(1/1024)x(0) - vx/1024

y(1/1024)y(0) - vy/1024

z(1/1024)z(0) - vz/1024

Priors for registration

Based on smoothness of the velocities.

Velocities relate to distances from origin.

Working with Diffeomorphisms