Equiresolution Mirrors

R. Andrew Hicks
Ronald K. Perline
Department of Mathematics
Drexel University
ahicks@drexel.edu

As you may have observed, the size and shape of an object viewed in a curved mirror depends on the shape of the mirror and the position of the object. If the object moves, then it usually gets distorted into different shapes as it moves. In particular, for most catadioptric sensors, the area of the object can change drastically, even if the distance from the object to the sensor remains constant.To quantify this, consider a ring of spheres, as depicted on the right. (you can click on the image if you want a bigger version.)

Now suppose in the center of the ring of spheres, you place a catadioptric sensor, consisting of a parabolic mirror viewed with an orthographic projection. A simulated image from such a sensor appears to the right. (One caveat - the "observer", i.e., the camera, does not appear in the simulations.) A striking feature of this image is that the size of the spheres changes greatly, depending on their elevation. Actually the fact that they are circular is an interesting fact unto itself, and was proved by Geyer & Daniilidis. The botton line is though, that the number of pixels allocated to each sphere varies greatly - by 75% in fact. The reason for this is that the determinant of the Jacobian of the projection map is non-constant, i.e., the number of pixels per solid angle varies. This quantity is sometimes referred to as the resolution of the sensor.

What are the mirror shapes that do give constant resolution, when combined with an orthographic projection ? That is, what are equiresolution sensors ? To find out, one has to solve the equation
det(dp) = a constant
where p is the sensor's projection, from the sphere at infinity to the image plane. (Actually in practice it is easier to compute with the inverse of p.)

The sphere is one such surface. Here is what the above ring of spheres looks like imaged with a spherical mirror. The images get longer in one direction and shorter in the other, but the area stays constant.

It turns out that there are other solutions to the equation det(dp)=constant. In fact the solutions turn out to be exactly the surfaces of revolution of constant Gaussian curvature. Here are two representitives, for positive and negative constants, which have positive and negative curvature, resp.:

All such surfaces can be nicely represented in terms of elliptic integrals. The surface on the right has negative Gaussian curvature, and is called the pseudosphere - it is a realization of Lobachevsky's hyperbolic geometry! The qualitative difference between the two cases is that the positively curved surface preserves orientation, while the negatively curved surface reverses it. It is also important to realize that the pseudosphere can be extended indefinitely in the "skinny" direction, but not in the broader direction, i.e. it is not a complete surface. Of course, the positive curved football shape is not complete either.

In the case where the camera model used is perspective, not orthographic, the ode can be solved numerically (although it is more fussy about the numerics), but we don't have a nice interpretation of what these surfaces are.

The problem of designing equiresolution mirrors has another interpretation, namely if you think of them as surfaces that a beam is being scattered off of, then these shapes are the surfaces that will give uniform scattering.

To find out more about equiresolution sensors see our paper Equiresolution Catadioptric Sensors .

Other designs by Andy Hicks
The Page of Catadioptric Sensor Design

Last modified Wed Jan 14 12:54:28 EST 2004