Equidistant (f θ) fish-eye perspective with application in distortion centre estimation

Radial distortion in an image is a geometric distortion that causes a non-linear variation in resolution across the image, with a higher spatial resolution in the central areas of the image, and lower resolution in the peripheral areas of the image. This is particularly evident in fish-eye cameras, with very wide fields-of-view. Equidistant fish-eye cameras are designed such that the distance between a projected point and the distortion centre of the image is proportional to the incident angle of the projected ray, scaled only by the focal length. The perspective of the projection of a given scene in an equidistant fish-eye camera differs greatly from the projection of the same scene in a rectilinear pin-hole camera. For example, while the field-of-view is significantly larger for a fish-eye camera, the non-linear radial distortion of the scene results in straight lines mapping to curves of a particular shape in the equidistant fish-eye image. In this paper, we describe equidistant fish-eye perspective in terms of the projection of sets of parallel lines to the equidistant fish-eye plane, and derive an equation that describes the projection of a straight line. We also demonstrate how the shape of a projected straight line can be accurately described by arcs of circles on the distorted image plane. We also describe an application of the equidistant perspective properties, by showing that the distortion centre of an equidistant fish-eye camera can be estimated by the extraction of the vanishing points. Additionally, we examine the accuracy of this estimation procedure on a large set of synthetically created images and a smaller set of real images from fish-eye cameras.