Using line correspondences in invariant signatures for curve recognition

Abstract

Plane curves and space curves distorted by an affine or projective transformation may be recognized if Invariant descriptions of them are available. Recent research in this area has shown that it is possible to identify transformed curves through the use of various combinations of differential invariants and point correspondences. Purely differential invariants usually require very high order derivatives of the space curves, though taking advantage of point correspondences sharply reduces the order of derivatives necessary. In cases where point correspondences are not available but line correspondences are, it is still possible to construct invariant signatures of the curves without increasing the order of derivatives necessary. Using just first order derivatives, invariant signature functions can be established for plane curves using one line correspondence for curves subjected to affine transformations and using two line correspondences for curves subjected to projective transformations. Still with only first order derivatives, invariant signatures can be found for space curves using two line correspondences for curves subjected to affine transformations, and using three line correspondences for curves subjected to projective transformations. In each of the four cases, these invariant signatures are graphs of one invariant quantity versus another. Determining the equivalence of objects then requires identification of a pair of two-dimensional graphs. Planar objects and surfaces In space may be recognized by matching their boundaries using these variants. Furthermore a group of partially occluded curves may be resolved Into Its individual components.