Constraints on quadratic curves under perspective projection

The authors address the problem of three-dimensional (3-D) location estimation based on quadratic-curved features. They derive the mathematical relations or constraints on the 3-D position and orientation of quadratic-curved features using the standard rotation and the standard transformation concepts introduced by K.I. Kanatani, (1988), and assuming that the true size and shape of a given quadratic feature are known a priori, with its projection image given. In this context, an analytical method is introduced for estimation of the standard rotation and determination of the shape of a quadratic-curved feature at its canonical position. It is shown that, in general, knowledge of the true shape and size of a quadratic-curved feature does not yield a sufficient number of constraints to determine the 3-D position and orientation uniquely. As a result, extra constraints must be acquired from various sources of information and fused with these constraints to obtain unique 3-D position and orientation estimates