Geometric Properties of Multiple Reflections in Catadioptric Camera with Two Planar Mirrors

Abstract

A catadioptric system consisting of a pinhole camera and two planar mirrors is deeply investigated in this paper. The two mirrors combine to form a corner and face-to-face with the pinhole. Their relative pose is unknown. An object will be reflected in the mirror corner one-time or multiple-times. Using the pinhole, we may take an image containing the object and its reflections, i.e., simultaneously imaging multiple views of an object by a single camera. We discovered that each 3D point and its reflections lie on a circle. We call the point set composed of a 3D point and its reflections, a Reflection Point Group (RPG), and the circle related to a RPG is called a RPG circle. All RPG circles are parallel to one another. Furthermore, each RPG can be partitioned into two separate subgroups. Shape formed by all points in a subgroup is invariant with respect to location of the 3D point. From these geometric properties, two calibration approaches can be utilized: One is based on parallel circles; the other is using 2D homographies among invariant shapes. Experiments validate our approaches.