Time-band limiting operators, corresponding to classical discrete orthogonal families, admit commuting second order difference operators. A new proof is presented.

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We present a general method of catadioptric sensor design for realizing prescribed projections. Our method makes use of geometric distributions in 3-dimensional space, which are generalizations of vectorfields. The main idea is this: ifone desires a rejective surface that will image the world in a certain way, then this condition determines the orientation of the tangent planes to the surface. Analytically, this means that the surface will then be determined by a pair of partial differential equations, which may or may not have a common solution. We show how to check i f a common solution exists. If no common solution exists, we describe a method for obtaining optimal approximate solutions in a least-squares sense.As an example application, we construct a mirror that will give a panoramic view of a scene without any digital unwarping.

We consider the problem of designing a passenger side mirror for an automobile that does not have a blind spot and that does not distort the image. Our model consists of a coupled pair of partial differential equations that do not have a common solution. Using a best mean-square-error functional, we find approximate solutions using nonlinear optimization. In one case a local minimum provides a mirror that solves the problem, but it does not reverse the image.

A prominent characteristic of most catadioptric sensors is their lack of uniformity of resolution. We describe catadioptric sensors whose associated projections from the viewing sphere to the image plane have constant Jacobian determinants and so are equiresolution in the sense that any two equal solid angles are allocated the same number of pixels in the image plane. We show that in the orthographic case the catoptric component must be a surface of revolution of constant Gaussian curvature. We compare these equiresolution sensors in both the perspective and orthographic cases with other sensors that were proposed earlier for treating the uniformity-of-resolution problem.