Inverse perspective projection of a line segment pair

Abstract

This paper discusses how a line segment and segment pair in a 3-dimensional space are constrained by an image obtained by their perspective projection. Also, it introduces the concept of polarity in the interpretation of a line-segment image obtained under perspective projection and gives a definition of a spherical PAT curve. In addition, the relationship is explained between a PAT constraint [5], which is of an angle formed by two line segments on a plane, and the spherical PAT curve [7], which is a constraint of two line segments at an arbitrary position in a 3-dimensional space (an extended idea of the former). Although the spherical PAT curve an be obtained formally by projecting a plane PAT curve onto a sphere, and this is justified for a semisphere, the meaning of this on the opposite semisphere has not been clarified with a justification. This paper justifies the sphere PAT curve by discussing the polarity of the segment interpretation, and explains its accurate meaning. This paper also describes the characteristics of a spherical PAT curve. The idea will have many applications to image understanding.