Integrability in optical design

TLDR: In this paper, the problem of designing a catadioptric sensor that will induce a given transformation does not have a solution and the error associated with these approximations is measured.

Abstract: Catadioptric sensors can be designed to induce certain transformations between an object surface and an image plane. We discuss techniques for designing these sensors and examine the error involved when dealing with non-integrable distributions. Rotationally symmetric mirrors can be used to produce panoramic images. Two methods for producing rotationally symmetric mirrors that induce cylindrical projections are described. The first has the advantage of imaging without distorting vertical lines and the second has the advantage of producing uniform resolution panoramic images. If we do not required a surface to be rotationally symmetric we will have more flexibility in the design process. However, in general, when dealing with the non-rotationally symmetric case, the problem of designing a catadioptric sensor that will induce a given transformation does not have a solution. If a distribution is integrable thensolutions to this problem will exist. Frobenius’ Integrability Theorem can be use to check for integrability. When a distribution is non-integrable we use methods that will produce approximate solutions to the problem. We then introduce ways to measure the error associated with these approximations. We show that on an open set with compact closure we can find a lower bound on our error metric. We also describe a Cartesian slice method for constructing a surface to approximate a given distribution and show that when this method is used, we can measure the error associated with the resulting surface. These error measurements are important because they give us a way to measure integrability and determine when a distribution is “badly” behaved with respect to integrability.%%%%Ph.D., Mathematics  – Drexel University, 2009