Rectifying Mirrors

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




( What are catadioptric sensors ? )

This page is for gently explaining the idea of a rectifying mirror. If you would like a more technical introduction, then you might look at this paper(pdf) , or this html version of the paper.


We have all seen curved mirrors at one time or another. Often they are employed in stores, to look for shoplifters. Some trucks have small curved mirrors attached to their the rearview mirrors. There are numerous famous instances in art. Among the most famous is the mirror that appears in Jan van Eycks "The Arnolfini Marriage", the image to the left.






You may have trouble seeing it, but behind the couple is a curved mirror on the wall. Here is a close up:







Another work of art featuring a curved mirror is "Hand with Reflecting Sphere" by M.C. Escher:






A related phenomenon in art is anamophosis . This applies to two different things really. One is the method of painting something that is meant to be viewed from some funny angle. For example in Holbein's "The Ambassadors" , the funny smudge on the bottom of the painting appears as a skull when viewed from the side. The other meaning to anamorphosis is an image that is distorted but appears "correct" in a curved mirror, usually a vase that is placed upon the painting. To the left is an 18th century example (I don't know who the artist is, but the image comes from the book "Mathematics" by David Bergamini, Life Science Library). Here is a large version.


Curved mirrors are known as catoptrics and conventional lenses systems are known as dioptrics. Devices consisting of both are know as catadoptrics. There are several nice things about catadioptric sensors.

First of all, they can give a wide field of view. This is the primary reason that there has been a big increase in interest in the past few years in the computer vision and robotics communities (not to mention the military etc.) The second interesting thing about them is that they perform some sort of tranformation on an image (as illustrated above). The nature of this transformation depends on the shape of the mirror. And of course, the transformation occurs very quickly. So in some sense catadioptrics are analog computers, albeit not very programmable ones.

Now, to get a wide field of view, generally one uses a convex surface of revolution. So the geometry of the surface is determined by the profile curve that is revolved. One way to get an idea of how such a mirror distorts the environment is to see what it does to a checkerboard pattern. Here is an scene of such a pattern set up in the GRASP lab. Amidst the pattern is a sensor consisting of a spherical mirror and a normal CCD camera. The mirror is suspended above the camera, which is pointed towards the ceiling.



Here is what images of the pattern taken with the spherical (left) mirror and a parabolic mirror (right):


Notice how the spherical mirror causes more distortion, i.e. the checkers are bigger (than the ones in the parabolic image) near the camera and shrink very quickly. So if the paraboloid distorts less than the sphere, is there a mirror shape where there is not distortion, i.e. the checker board appears as if you were just looking at it from above ? Certainly a flat mirror has this property, but a flat mirror does not provide a wide field a view. It turns out that there is a family of mirrors with a field of view arbitrarily close to 180 degrees which leave the checkerboard unwarped. Here is what the checkerboard looks like using such a mirror:



And here is a picture of the mirror:



To find out more about catadioptric sensors/omnidirectional vision, look at:

The Page of Omnidirectional Vision
The Page of Catadioptric Sensor Design
Catadioptric Sensors Designs by R. Andrew Hicks



Last modified Mon Dec 29 11:43:15 EST 2003