One-Dimensional Cornea

We presume that the cornea is a convex surface, as shown in Figure 2. The cornea may be described by some unknown function . Using different notation, each point on the corneal surface may be described by the vector. Our known grid pattern is the single curve . We will describe each point on the grid pattern by the vector .

Rays of light emanate from each point, , on the grid pattern (see Figure 2). We will assume that only one ray of light bounces off of the cornea at the unknown point , reflects around a normal vector, , and travels through the camera lens to the film plane. So we know the starting point, , and the final point, , and would like to determine the unknown reflection point, .

The location of the reflection point, , is dependent on the source of light, , so . Thus the resulting film data point is also a function of since . Since we know , we can construct a unit vector that points in the same direction as and can thus consider our reflection point as the vector .

Since is a function of , both and are functions of , as is the normal vector . But since , we also observe that , , , , and are all functions of x.

One important observation about this system is that we can describe everything as a function of one variable. A second important observation is that and are known quantities. Since we know the starting point of one ray of light (the source grid ) and we have the distorted image of this ray of light (on the film plane at ), we also know the unit vector . Since we are trying to find the unknown point , what we really need to find is the distance from the origin to , which is . Our problem resolves to determining .