Perspective angle transform: Principle of shape from angles

Abstract

This paper introduces a new relation, called the perspective angle transform (PAT), to deal with shape-from-angle problems, together with its application to 3D configuration recovery from an image. Three main aspects of PAT are presented in this paper. The first is the derivation of PAT which holds between the apparent and real angles under perspective projection. A concept is proposed of a virtual image plane and a new coordinate system, named the first perspective moving coordinate (FPMC) system, for analysis of the shape-from-range problems. Characteristics of FPMC are discussed briefly. The second point is the analysis of PAT properties, for which the general PAT form is introduced on another new coordinate system, named the second perspective moving coordinate (SPMC) system. Using this general form, the gradient of the plane including the real angle is constrained on a curve (PAT curve) of the fourth degree in the virtual image plane. The characteristics of PAT curves and relations between the general PAT form and skewed symmetry are summarized briefly. The last point concerns an application of PAT. As an example, we treat 3D configuration recovery from three arbitrary line segments in the image plane. This recovery corresponds to a generalization of the right-angled interpretation problem proposed and discussed by S.T. Barnard. A solution to the problem is shown using the general PAT form in conjunction with the concept of a virtual crossing point. The right-angled interpretation problem is ascribed to a quadratic equation. A simplified solution is also provided for the case where three line segments have a common crossing point in the image plane. This solution is based on the primary PAT form and is applicable to interpretation of a trihedral vertex with at least two right angles.