Solutions and ambiguities of the structure and motion problem for 1D retinal vision

In this paper we investigate the geometry and algebra of multiple one-dimensional projections of a two-dimensional environment. This is relevant for one-dimensional cameras, e.g. as used in certain autonomous guided vehicles. It is also relevant for understanding the projection of lines in ordinary vision. A third application is on ordinary vision of vehicles undergoing so called planar motion. The structure and motion problem for such cameras is studied and the two possible minimal cases is solved. The technique of solving these questions reveal interesting ambiguities. It is shown that for each solution with three images there is an ambiguous solution. It is also shown that for each solution for four points there is an ambiguous solution. The connection between these two different types of ambiguities is also given. Although the paper deals with calibrated cameras, it is shown that similar results exist for uncalibrated cameras.