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ENS WorkshopJohn Ashburnerjohn fil ion ucl ac ukFunctional Imaging Lab 12 QueenSquare London UK .
Training and ClassifyingTraining DataTraining Data ClassifyingPatients .
y f aTx b Support Vector Classifier Support Vector Classifier SVC Suppor a is a weighted linearcombination of the.
support vectors Some Equationso Linear classification is by y f aTx b o where a is a weighting vector x is the test data b is anoffset and f is a thresholding operation.
o a is a linear combination of SVs a i wi xio So y f i wi xiTx b Going Nonlinearo Nonlinear classification is byy f i wi xi x .
o where xi x is some function of xi and x o e g RBF classification xi x exp xi x 2 2 2 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 Concise representations.
o Information reduction compressiono Most parsimonious representation bestgeneralisationo Occam s Razoro Registration compresses data.
o signal is partitioned intoo deformationso residuals Nonlinear Registration How could DTI help .
Small Deformations Diffeomorphisms Partial Differential EquationsModel 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 e x t V Matrix representationsof diffeomorphisms.
x 1 eV x 0 x 0 e V x 1 For large keV I V k k Compositions.
Large deformations generated from compositions of smalldeformationsS1 S1 8oS1 8oS1 8oS1 8oS1 8oS1 8o... Recursive formulationS1 S1 2oS1 2 S1 2 S1 4oS1 4 S1 4 S1 8oS1 8.
Small deformationapproximationS1 8 I V 8 The shape metrico Don t use the straight distance i e vTv .
Distance vTLTLvo What s the best form of L o Membrane Energyo Bending Energyo Linear Elastic Energy.
Consistent registrationRegister to a meanshaped imageimpractical Problem How can the distancefor lots of between e g A and B be computed .
scans Inverse exponentiating is iterative and Metrics from residualso Measures of differencebetween tensors o Relates to objective.
functions used for imageregistration o Can the same principles Over fittingA simpler model can often do better .
Cross validationo Methods must be able to generalise to new datao Various control parameterso 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 forand the other half for Two fold Cross validationThen swap around thetraining and test data .
Leave One Out Cross validationUse all data except onepoint for training The one that was leftout is used for.
Leave One Out Cross validationThen leave anotherpoint out And so on Interpretation .
o Significance assessed from accuracy based oncross validation o Main problems o No simple interpretation o Mechanism of classification is difficult to visualise.
o especially for nonlinear classifierso Difficult to understand not like blobs o May be able to use the separation to derivesimple and more publishable hypotheses Group Theory.
o Diffeomorphisms smooth o It is a Lie Group continuous one to one o The group ofmappings form a Group diffeomorphismso Closure constitute a smootho AoB remains in the same manifold .
group o The operations areo Associativity differentiable o AoB oC Ao BoC o Identityo Identity transform I exists .
o A 1 exists and A 1oA AoA 1 I Lie Groupso Simple Lie Groups o Lie Algebra isinclude various classes exponentiated to giveof affine transform Lie group For square.
matrices matrices this involves ao E g SO 2 Special matrix exponential Orthogonal 2D rigid bodyrotation in 2D o Manifold is a circle.
Relevance to Diffeomorphismso Parameterise with o Don t actually usevelocities rather than matrices displacements o For tiny deformations o Velocities are the Lie things are almost linear .
Algebra These are o x 1 1024 x 0 vx 1024exponentiated to a o y 1 1024 y 0 vy 1024deformation by recursiveo z 1 1024 z 0 vz 1024application of tiny.
displacements over a o Recursive application byperiod of time 0 1 o x 1 2 x 1 4 x 1 4 y 1 4 z 1 4 o A 1 A 1 2 oA 1 2 o y 1 2 y 1 4 x 1 4 y 1 4 z 1 4 o A 1 2 A 1 4 oA 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 Inversion o Distances on the manifold o Negate the velocities andare given by geodesics exponentiate o x 1 1024 x 0 vx 1024o Average of a number of.
o y 1 1024 y 0 vy 1024deformations is a point ono z 1 1024 z 0 vz 1024the manifold with theshortest sum of squared.
geodesic distances o Priors for registrationo E g average position of o Based on smoothness of thevelocities London Sydney and o Velocities relate to distancesHonolulu from origin .
John Ashburner [email protected] Functional Imaging Lab, 12 Queen Square, London, UK. Training and Classifying Classifying Support Vector Classifier Support Vector Classifier (SVC) 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 wi xi So ...