Title : Designing Mirrors for Catadioptric Systems that Minimize Image Errors

Authors : Rahul Swaminathan
GRASP Lab, Univ. of Pennsylvania.

and
Michael D. Grossberg and Shree K. Nayar
Columbia University.

Abstract:
A critical step in designing any catadioptric system is determining
the shape of its component mirrors. Many mirror shapes have been
designed in the past for numerous applications. Invariably these
mirrors were designed using case specific tools and required
considerable effort on the part of the designer. Recently, some new
general methods have been developed to automate the design process.
However, all previous approaches determined the mirror by minimizing
errors in the mirror shape rather than the image formed. A more
principled formulation is to instead determine a mirror that reduces
geometric errors in the images that the system produces. In general,
such a formulation makes the problem non-linear and computationally
prohibitive.

In this paper we present a linear method to determine the mirror shape
that meets a designer's requirements. The requirements are specified
as a map from pixels to scene points which we call the image-to-scene
map. For the case when an exact solution for the mirror shape does
exist, our method finds it. A more interesting and common case is
when no mirror shape exists that implements the required map. In this
case, our method finds the mirror shape that minimizes geometric
errors in image formation. The method achieves this by relating the
errors in the mirror normals to a first-order approximation of image
errors. We show how the constraints of a previous shape based methods
can be modified so as to minimize image errors. Our proposed method is
also directly applicable to the design of projection systems. We
demonstrate the effectiveness of our approach on various catadioptric
imaging and projection systems showing up to an 80 percent reduction
in errors with respect to previous shape based methods.

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