Catadioptric sensors that approximate wide-angle perspective projections

Abstract

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.

Camera Models and Fundamental Concepts Used in Geometric Computer Vision

“…Hicks and Bajcsy as well as Kweon et al showed how to compute mirror shapes such that a particular scene plane that is "frontoparallel" to the mirror is imaged without distortion, while still providing a very wide field of view [222,223,297]. This was done for both, perspective and orthographic cameras looking at the mirror.…”

Panorama Imaging System Using Hyperbolic Cylinder Reflector and Image Reconstruction Algorithm for Automative AVM System

“…한편 카메라와 부가적인 반사체를 혼합하여 영상을 획득하는 것을 catadioptric(catoptric + dioptric) 방법이라고 한다 [10]. Catadioptric 시스템은 한 대의 카메라를 통해 여러 시점의 영상을 획득하거나 [7,8] …”

Design of Focusing Catadioptric Systems using Differential Geometry

“…However, a big problem of today's catadioptric systems is their focusing as this is not considered during system design. Instead common designs only deal with the geometric mapping that describes the relation between image points and corresponding lines of sight (Hicks and Bajcsy, 2000;Stürzl and Srinivasan, 2010). Neglecting wave effects, the light transport through an optical system can be described using light rays (Malacara and Malacara, 2003).…”