"How many planar viewing surfaces are there in noncentral catadioptric cameras?" Towards singe-image localization of space lines

In-door environments often contain several straight line segments. The 3D reconstruction of such environments can thus reduce to the localization of lines in the 3D space. Multi-view reconstruction requires the solution of the correspondence problem. The use of a single image to localize space lines is attractive, since the correspondence problem can be avoided. However, using a perspective camera (or a central one), a line can not be localized, since its viewing surface is planar, and hence it can contain infinite lines other than the correct one. In this paper we study the number of planar viewing surfaces for a general class of catadioptric cameras, constituted by an axial symmetric mirror and a perspective camera placed at generic relative position. We show that, under broad conditions, there is only a discrete set of planar viewing surfaces for the considered class of cameras. This result establishes a qualitative difference with respect to axial-symmetric cameras (e.g., catadioptric cameras constituted by an axial-symmetric mirror plus a perspective camera, whose viewpoint is constrained to be on the mirror axis), where an infinite set of planar viewing surfaces exists. Then, some conditions are derived for the localization of lines in the 3D space from single images. Preliminary experiments are also reported.