ENS Workshop John Ashburner [email protected] Functional Imaging Lab, 12 Queen Square, London, UK.

ENS WorkshopJohn Ashburner

[email protected] Imaging Lab, 12 Queen

Square, London, UK.

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

o Linear classification is by y = f(aTx + b)o where a is a weighting vector, x is the test data, b is an

offset, and f(.) is a thresholding operation

o a is a linear combination of SVs a = i wi xi

o So y = f(i wi xiTx + b)

Going Nonlinear

o Nonlinear classification is by

y = f(i wi (xi,x))

o where (xi,x) is some function of xi and x.

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

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

Nonlinear SVC

What is a Metric?

o Positiveo Dist(A,B) ≥ 0o Dist(A,A) = 0

o Symmetrico Dist(A,B) = Dist(B,A)

o Satisfy triangle inequalityo Dist(A,B)+Dist(B,C) ≥ Dist(A,C)

AB

C

Concise representations

o Information reduction/compressiono Most parsimonious representation - best

generalisationo Occam’s Razor

o Registration compresses datao signal is partitioned into

odeformationsoresiduals

©Friston

Nonlinear Registration

Mapping

How could DTI help?

Small Deformations

Diffeomorphisms

Partial Differential Equations

Model one image as it deforms to match another.

x(t) = V x(t)It’s a bit like DCM but with much bigger V matrices(about 10,000,000 x 10,000,000 – instead of about 4x4).

x(t+1) = eV x(t)

Matrix representations of diffeomorphisms

x(1) = eV x(0)x(0) = e-V x(1)

For large k

eV ≈ (I+V/k)k

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 deformationapproximation

S1/8 ≈ I + V/8

The shape metric

o Don’t use the straight distance (i.e. √vTv)

oDistance = √vTLTLv

o What’s the best form of L?o Membrane Energyo Bending Energyo Linear Elastic Energy

Consistent registration

A B

C

A B

C

µ

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.

Register to a mean shaped image

Metrics from residuals

o Measures of difference between tensors.

o Relates to objective functions used for image registration.

o Can the same principles be used?

Over-fitting

Test data

A simpler model can often do better...

Cross-validation

o Methods must be able to generalise to new datao Various control parameters

o More complexity -> better separation of training datao Less complexity -> better generalisation

o Optimal control parameters determined by cross-validationo Test with data not used for trainingo 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...

Interpretation??

o Significance assessed from accuracy based on cross-validation.

o Main problems:o No simple interpretation.o Mechanism of classification is difficult to visualise

o especially for nonlinear classifiers

o Difficult to understand (not like blobs)

o May be able to use the separation to derive simple (and more publishable hypotheses).

Group Theory

o Diffeomorphisms (smooth continuous one-to-one mappings) form a Group.o Closure

o AoB remains in the same group.

o Associativityo (AoB)oC = Ao(BoC)

o Identityo Identity transform I exists.

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

o It is a Lie Group.o The group of

diffeomorphisms constitute a smooth manifold.

o The operations are differentiable.

Lie Groups

o Simple Lie Groups include various classes of affine transform matrices.o E.g. SO(2) : Special

Orthogonal 2D (rigid-body rotation in 2D).

o Manifold is a circle

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

Relevance to Diffeomorphisms

o Parameterise with velocities, rather than displacements.

o Velocities are the Lie Algebra. These are exponentiated to a deformation by recursive application of tiny displacements, over a period of time=0..1.o A(1) = A(1/2) oA(1/2)

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

o Don’t actually use matrices.

o For tiny deformations, things are almost linear.o x(1/1024) x(0) + vx/1024

o y(1/1024) y(0) + vy/1024

o z(1/1024) z(0) + vz/1024

o Recursive application byo x(1/2) = x(1/4) (x(1/4), y(1/4),z(1/4))o y(1/2) = y(1/4) (x(1/4), y(1/4),z(1/4))o z(1/2) = z(1/4) (x(1/4), y(1/4),z(1/4))

Working with Diffeomorphisms

o Averaging Warps.o Distances on the manifold

are given by geodesics.o Average of a number of

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

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

o Inversion.o Negate the velocities, and

exponentiate.o x(1/1024) x(0) - vx/1024

o y(1/1024) y(0) - vy/1024

o z(1/1024) z(0) - vz/1024

o Priors for registrationo Based on smoothness of the

velocities.o Velocities relate to distances

from origin.