IBIM Update

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IBIM Update

• Anil Rao
• Imperial College London

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New Registrations

• B-Spline FFD Algorithm used to register labelled
  images
• Subject 01014 (1650) reference subject
• Transformation consists of affine and local
  transformations
• 93 subjects, 01001-01037,02001-02038,03001-03018
• Canonical correlation analysis and weighted
  partial least squares predictions repeated
• More subjects and different registrations (using
  labelled images rather than T1)

12 of 37
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New Canonical Correlations Analysis

• Pairwise CCA analysis performed of 17 brain
  structures over 93 data sets
• Separate PCA of each structure used to reduce
  dimensionality first
• First 13 modes (gt95) of each structure retained
• CCA then performed on each pair of structures
• Average correlations (0-1) between structures
  calculated

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Results- Correlation image
Structure j
Structure i
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Results- Correlation image (Old)
Structure j
Structure i
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Results- Most Correlated Structures
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Results- Most Correlated Structures
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Results- Most Correlated Structures
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Left Thalamus, Right Thalamus
Mode1Front
Mode1 Top
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Left Thalamus, Right Thalamus
Mode2Front
Mode2 Top
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Right Pallidum, Right Putamen
Mode1
Mode2
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Using Weighted Partial Least Squares to predict
Brain structures

• Anil Rao
• Department of Computing
• Imperial College London

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Evaluation of CCA

• Canonical Correlation Analysis
• Identifies most correlated modes of X and Y and
  gives corresponding correlation coefficient
• Coordinates of highly correlated modes can then
  be predicted for Y, given corresponding
  coordinates for X
• Poorly correlated modes cannot be used for
  prediction, so whole of Y cannot be predicted
  from X
• Partial Least Squares Regression (PLSR)
• Predicts whole of Y in one step

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Partial Least Squares Regression (1)

• Idea
• X (structure 1) is set of predictor signals x, Y
  (structure 2) is set of response signals y
• Produce a linear model
• are mean-centred, normalized by
  standard deviation
• Directions in X sought that describe most
  variation in Y
• C then used to predict instance of Y from that of
  X
• Implementation
• NIPALS algorithm
• Iterative technique for calculating C
• Inputs N datasets of X (dimension p),Y
  (dimension q)
• Outputs C, means and standard deviations

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Partial Least Squares Regression (1)

• Idea
• X (structure 1) is set of predictor signals x, Y
  (structure 2) is set of response signals y
• Produce a linear model
• are mean-centred, normalized by standard
  deviation over X,Y
• Directions in X sought that describe most
  variation in Y
• C then used to predict instance of y from that of
  x

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Partial Least Squares Regression (2)

• Implementation
• NIPALS algorithm
• Iterative technique for calculating C, uses
  covariance matrices
• Inputs N datasets of X (dimension p),Y
  (dimension q)
• Outputs C (pxq), means and standard
  deviations of X,Y

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New Partial Least Squares Regression

• PLSR used to predict one structure using observed
  second structure over 93 data sets
• Separate PCA of each structure used to reduce
  dimensionality, retained 13 modes (95)
• PLSR performed on reduced data, predictions then
  transformed back to original basis
• Leave one out tests performed
• Errors between predicted shape and actual shape
  calculated
• Compared to distances between predicted shape and
  mean of that shape

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Correlation image (from CCA)
Structure j
Structure i
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Results left/right thalamus
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Results left/right thalamus

• Prediction of right thalamus shape better than
  mean in 32/37 cases
• Biggest improvement 7.07 to 1.81
• Worst case 1.14 to 1.72
• Cases that are worse associated with poor PCA
  fitting of unseen left thalamus
• Average prediction error smaller than mean
• Average error of mean2.60
• Average error of prediction1.23

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Results left lat ventricle /right putamen
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Results left lateral ventricle/right putamen

• Prediction of right putamen shape better than
  mean in 18/37 cases
• Biggest improvement 7.65 to 4.94
• Worst case 8.38 to 10.17
• Average prediction error similar to mean
• Average error of mean2.36
• Average error of prediction2.29
• Is poor performance related to low canonical
  correlation coefficient?
• PLSR is linear technique, low cca measures imply
  variation is not linear

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Results right pallidum /right putamen
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Results Left Putamen/Left Pallidum
Average squared distance (mm2)
Subject
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Results right pallidum/right putamen

• Prediction of right putamen shape better than
  mean in 36/37 cases
• Biggest improvement 7.65 to 0.88
• Worst case 0.96 to 1.04
• Average prediction error smaller than mean
• Average error of mean2.36
• Average error of prediction0.79
• Result better than left thal/right thal
• Average PCA fitting error much smaller for unseen
  predictor structure left thal0.75,right
  pall0.23

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Canonical correlations of brain structures

• Pairwise CCA analysis performed of 17 brain
  structures over 37 data sets
• Separate PCA of each structure used to reduce
  dimensionality first
• First 17 modes (gt95) of each structure retained
• CCA then performed on each pair of structures
• Average correlations (0-1) between structures
  calculated

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Results- Most Correlated Structures
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Results- Most Correlated Structures
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On-going work

• Running PLSR leave one tests for all pairs of
  structures
• Will compare with marginal mean as well as mean
• Publications
• Combined CCA/PLSR to MMBIA
• PLSR/Model Fitting with Julia to MICCAI

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Summary

• Performed registration evaluations using 2
  different approaches
• Fuzzy overlaps show similarities of registration
  schemes
• Bhattacharya measure show similarities of shape
  models
• Canonical correlation analysis
• Identifies correlations between structures

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Appendix
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Overlap of PDFs

• Use Bhattacharya overlap measure
• Covariance matrix estimated from model parameters

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SVD Approach

• Two models
• Create union space of vectors
• Do an SVD
• Transform model to union space
• Compare transformed models
• B(model 1, model 2) B(UTx1,UTx2) (assume both
  models have same amount of noise)

n x tu basis vectors in union space
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Noise term

• Some dimensions may have no values
• May cause problems with estimating covariance
  matrix
• Combine mean of modes not used with residual
  noise in model
• Empirically determine value of residual noise in
  model

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Hierarchical Shape Models

• PCA for a particular brain structure models
  variation of that structure across population
• We may want to model higher level relationships
  between structures across population
• We have considered 2 approaches
• Hierarchical deformation models
• Canonical correlation analysis

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Canonical Correlation Analysis

• Random vector X, split into 2 subvectors
• Covariance matrix for X written
• Consider linear combinations
• Correlation between
• Seek to find vectors such that
  is maximal subject to

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Definition

• Canonical correlations defined by a series of
  vector pairs and corresponding
  correlation
• First canonical correlate maximizes
• rth canonical correlate maximizes
  where
• rth correlate is uncorrelated with preceding
  correlates

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Computation

• Solve eigenvalue equations
• are paired normalized canonical basis
  vectors, are canonical correlations
• Convert to unit variance form

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Properties

• If
• Canonical variates invariant

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Prediction

• Correlation model can predict value
• of unobserved variable from observed
  value of
• Since