Dept of Statistics and Applied Prob.

 I. Individual Corneal Images
 

A corneal image represents radial curvature of the surface of corneas (outer surface of a human eye). The radical curvature is closely related to the first and second derivatives of the corneal surface along the radius. The local curvature is usually large where the corneal image is steep,  and it is small where the corneal image is flat. The corneal 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.

Examples of some corneal images are as follows. 

a). The following  cornea is with fairly constant local curvature. Most area of the image is blue, representing small local curvatures, and only small portion of the image is with yellow color, indicating a little big larger local curvature. This corneal image should come from a person with no eye disease.
 

b). The following cornea is with strong astigmatism (vertical ridge). The local curvature is extremely large at the center, and getting smaller when moving to the edge.

c). The following corneal image comes from a person with kerataconus, an unpleasant disease. This eye disease is so serious that  your usual optometric corrections can not handle very well. This image indicates an area where the local curvature is extremely large.

 
 

II  A Population of Corneal Images
 

Sometimes, we need analyze  a collection of corneal images to understand the structure of the population.  It is usually quite difficult to get useful information if we just simply put them one after another other as frames via movie. Here is an example of such a movie. In this movie, one corneal image comes out after another. This is really a mess and it may be difficult to extract any useful information.
 

III Principal Component Analysis

To overcome the above problem,  we can use principal component analysis (PCA) of the population (Ramsay and Silverman 1997).  The PCA is designed to find out the modes of the variations of the population. The first principal component (PC1)  indicates the direction of the largest variation in the population; the PC2 the direction of the second largest variation, and so on.  To apply the PCA methodology, we first reconstruct the corneal images using the zernike basis which is  widely used in ophthalmology for image reconstructions.

A PC-based movie is constructed using  the average corneal image plus and minus some amount of a principal component, say PC1:  "average corneal image+/- k PC1". When k runs over a specified range, say, from the left end to the right end, the resulting corneal image will change from one extreme image to another, forming an animating movie.

Here is  the movie for PC1. The movie (the upper  frame) starts and returns at the average mean corneal image, and the lower frame shows how the parameter k changes over time, associated with the movie changing. It is seen that the PC1 reflects the direction of the strength of astigmatism.

The movie for PC2 is similarly constructed: "average corneal image +/- k PC2". It is seen that the PC2 reflects the contrast of the top and bottom.

The movie for PC3 is similarly constructed: "average corneal image +/- k PC3". It is seen that the PC3 reflects the contrast of "with the rule or against the rule  astigmatism. Most stigmatism is vertical but not all.

These movies may be useful for ophthalmologists to analyze  the variability of a group of corneal images. 
 

IV More Details and Acknowledgements