2.  Movies showing Functional Data Analysis of Corneal Images
by N. Locantore, J. S. Marron, D. G. Simpson, N. Tripoli, J. T. Zhang and K. Cohen
 

Complete discussion, with full details, of this data analysis can be found in the paper "Motion Picture Analysis of Smoothing", by N. Locantore, J. S. Marron, D. G. Simpson, N. Tripoli, J. T. Zhang and K. Cohen (1999), with discussion in the journal  Test., 8, 1-65.  ( PDF version (3.42 MB)  /  ( Postscript Version (14.8 MB) ).
 

2a. Background:
These images represent radial curvature of the surface of corneas (outer surfaces of eyes).  This shape is important, since about 85% of the refraction done by the human eye happens here. A "temperature type" color scale is used, with hotter colors for more curvature, cooler colors in flatter regions.

Here are a few such images:

A cornea with fairly constant local curvature:

Strong Astigmatism (vertical ridge):

Kerataconus (an unpleasant disease, that your usual optometric corrections can't handle very well):

 
 

2b:  Population Viewpoint:
Now consider a collection of such images.  How can we understand the structure of the population.  A feeling for the difficulty of this problem comes from putting them one after the other as frames in this movie .  Do you feel the information overload?
 

2c:  Standard Principal Component Analysis:
Functional Data Analysis ideas, see Ramsay and Silverman (1997) Function Data Analysis, suggests using PCA (here applied to Zernike basis fit summarizations of the images).  An problem not addressed there is how to visualize the results (1-d methods such as overlays of projections don't work for 2-d images).  The solution is movies:
                  PC1:  Overall Curvature + Strength of Astigmatism
                  PC2:  Steeper on the top vs. the bottom    (Note the strong outlier effect!)
                  PC3:  With the Rule, vs. Against the Rule Astigmatism   (most stigmatism is vertical, but not all)
                  PC4:  ???
 

2d: Robust Principal Component Analysis:
The above movies show important features of the population, but a major worry is the influence of the outliers on the PC directions.  As you might expect from the movie from Part 2b above, deletion of outliers is ineffective, because there are too many of them.  I.e. when one gets deleted, another comes in.  I tried up to 4, then quite because that is 10% of the total data set.  This motivates a "robust resistant" approach to PCA.  The "directional search methods" available in the literature do not appear useful because of the high dimensionality, d=66.  Most robust estimates of the design matrix were not helpful, as they require 66=d<n=43.  So, we invented our own approach called "elliptical", see the paper for details.  Here are the improved versions of the above movies:
                  PC1:  Overall Curvature + Strength of Astigmatism
                  PC2:  Steeper on the top vs. the bottom    (Note the disappearance of the outlier effect!)
                  PC3:  With the Rule, vs. Against the Rule Astigmatism
                  PC4:  ???  (perhaps this is a third axis of the astigmatism?)
 
 

These movies were generated using "Cornean", a CORNEal curvature ANalysis software package written in Matlab.  For more about this, inquire by email from marron@email.unc.edu.
 
 
 

Movies Table of Contents

Marron's Home Page