Photogrammetric Theory in Stereo Vision

This article addresses the photogrammetric theory in stereo vision, particular with non-metric camera. The discussion is embedded in projective geometry, which can be regarded as an extension to conventional photogrammetry with metric camera in Euclidean geometry. The cross-ratio of volume elements is shown as projective invariant after the spatial projective transformation is properly introduced. The volume cross-ratio is thus defined as projective independent measure for object description. The fundamental matrix in stereo vision is decomposed into a production of two matrices, projective base matrix and projective-rotation matrix, as defined in the article. This concept inherits the one of the essential matrix for metric camera, and leads a concise soultion to projective coordinates and homogeneous corrdinates of the projective model. In order to reconstruct the object from its projective model, a three dimensional direct linear transformation (3-D DLT) is proposed, with which the object reconstruction can be performed linearly with minimum five conjugate known object points. Tests results and analyses verify the theory and methodology.