Multiscale Profile Analysis Joshua Stough MIDAG PI: Chaney, Pizer July, 2003 Joshua Stough MIDAG PI: Chaney, Pizer July, 2003

Profiles are Intensities along the Surface Normal ä Domain is radius proportional, [-.3*r,.3*r] ä centered at boundary ä Sampled at 10,11,… points ä Consider each profile as a column vector ä Domain is radius proportional, [-.3*r,.3*r] ä centered at boundary ä Sampled at 10,11,… points ä Consider each profile as a column vector

ä Goal: Define p(I|m) for segmentation using mreps. ä Two Approaches ä Correlation-Based : Substitute (normalized correlation with template) for the probability densities. ä Probabilistic : Use training cases to compute actual probability densities. ä Goal: Define p(I|m) for segmentation using mreps. ä Two Approaches ä Correlation-Based : Substitute (normalized correlation with template) for the probability densities. ä Probabilistic : Use training cases to compute actual probability densities.

Already-Implemented Correlation Methods ä Single template, applied everywhere (original Pablo) ä Characteristics -Can be efficiently implented -Prone to bad performance with outlier profiles ä Use training data to classify profiles, and use multiple templates, representatives of the classes ä Characteristics -Effectively accounts for most outliers (multi-modal, ICA potential?) -Assumes fine scale correspondence ä mixing of adjacent templates for coarser scale is non- intuitive ä Single template, applied everywhere (original Pablo) ä Characteristics -Can be efficiently implented -Prone to bad performance with outlier profiles ä Use training data to classify profiles, and use multiple templates, representatives of the classes ä Characteristics -Effectively accounts for most outliers (multi-modal, ICA potential?) -Assumes fine scale correspondence ä mixing of adjacent templates for coarser scale is non- intuitive

Multiple Templates with varying Template per boundary point (choice of template) Goals: -Determine ideal template types for a class of segmentations. -Determine at each point which template performs best. Why: -Profiles not one size fits all (space of profiles is multi- modal). Goals: -Determine ideal template types for a class of segmentations. -Determine at each point which template performs best. Why: -Profiles not one size fits all (space of profiles is multi- modal).

Algorithm for Determining Templates (m x thousands array) 1. Normalize (mean 0, rms 1) data. 2. Apply initial (analytic) templates to each column. 3. Classify according to highest response. 4. Set templates to mean of the classes of data. 5. Repeat starting at 2 until reasonable convergence. 1. Normalize (mean 0, rms 1) data. 2. Apply initial (analytic) templates to each column. 3. Classify according to highest response. 4. Set templates to mean of the classes of data. 5. Repeat starting at 2 until reasonable convergence.

Example Results

Algorithm for Determining Choice of Template per Boundary point ä At a given point on the model, consider the family of N intensity profiles. ä Apply templates to each and sum response according to template type. Result in (-N, N) ä Choose template with highest score. ä At a given point on the model, consider the family of N intensity profiles. ä Apply templates to each and sum response according to template type. Result in (-N, N) ä Choose template with highest score.

Probabilistic Approach: Multiscale Population Statistics Can Guide Segmentation Goals: -Use training cases to compute actual probability densities -Use training cases to compute actual probability densities. -Implied: effectively model the population. Why: -More correct theoretically than Correlation-Based, and hopefully better. Goals: -Use training cases to compute actual probability densities -Use training cases to compute actual probability densities. -Implied: effectively model the population. Why: -More correct theoretically than Correlation-Based, and hopefully better.

Profile Statistics Using Principal Components Analysis ä Means and principal modes of variation. ä Can be sensitive to outliers. ä Match computed according to projection onto eigen- modes, scaled by variance. ä Means and principal modes of variation. ä Can be sensitive to outliers. ä Match computed according to projection onto eigen- modes, scaled by variance.

Not Normalizing Profiles is Correlation vs. Covariance Analysis ä Profile family, with cross plots.

Cummulative Distribution on First Eigenmode ä Actual data verses gaussian data ä Conclusion: there is more structure than gaussian ä More tight than gaussian predicts ä Actual data verses gaussian data ä Conclusion: there is more structure than gaussian ä More tight than gaussian predicts

Smallest kurtosis [E(x-  ) 4 /  4 ] ä Most tightly distributed family, tending to bimodal.

Largest kurtosis ä Heaviest-tailed family, outlier prone.

Kurtosis Distribution of Profile data verses Gaussian Data ä Kurtosis, 5x3 vs. 6x  

That’s It.