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We present two families of reflective surfaces that are capable of providing a wide field of view, and yet still approximate a perspective projection to a high degree. These surfaces are derived by considering a plane perpendicular to the axis of a surface of revolution and finding the equations governing the distortion of the image of the plane in this surface. We then view this relation as a differential equation and prescribe the distortion term to be linear. By choosing appropriate initial conditions for the differential equation and solving it numerically, we derive the surface shape and obtain a precise estimate as to what degree the resulting sensor can approximate a perspective projection. Thus these surfaces act as computational sensors, allowing for a wide-angle perspective view of a scene without processing the image in software. The applications of such a sensor should be numerous, including surveillance, robotics and traditional photography.Recently, many researchers in the robotics and vision community have begun to consider visual sensors that are able to obtain wide fields of view. Such devices are the natural solution to various difficulties encountered with conventional imaging systems.The two most common means of obtaining wide fields of view are fish-eye lenses and reflective surfaces, also known as catoptrics. When catoptrics are combined with conventional lens systems, known as dioptrics, the resulting sensors are known as catadioptrics. The possible uses of these systems include applications such as robot control and surveillance. In this paper we will consider only catadioptric based sensors. Often such systems consist of a camera pointing at a convex mirror, as in figure (1).How to interpret and make use of the visual information obtained by such systems, e.g. how they should be used to control robots, is not at all obvious. There are infinitely many different shapes that a mirror can have, and at least two different camera models (perspective and orthographic projection) with which to combine each mirror. Convex mirror Camera Figure 1. The generic setup of the type of sensor that we consider in this paper.

The problem of controlling a single ray bundle with a single reflector is not generally solvable, but approximate solutions may often be found that are acceptable for applications. We introduce a new technique for finding such approximations and apply it to the design of a driver-side mirror for an automobile that has no blind spot and minimal distortion.

I present a design technique for realizing given projections as catadioptric sensors. In general, these problems do not have solutions, but approximate solutions may often be found that are visually acceptable. The method described reduces the problem to solving a linear system. A given transformation from the image plane to an object surface is shown to determine a vector field that is normal to the surface in the case where the vector field is a gradient. For the case when the vector field is not a gradient, several functionals are presented that may be minimized to give approximate solutions. As an application several new designs are described, including a mirror that directly gives a full 360-deg cylindrical projection without the need for any digital processing.

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