Using Affine Invariants on Perspective Projections of Plane Curves

Abstract

One method for determining whether two curves may be different perspective views of the same curve is to use invariant signature functions. A significant difficulty with this approach is that projective invariants often require unwieldy combinations of point correspondences and high order derivatives. A special case of the projective viewing transformation is the affine transformation. Affine invariants require fewer point correspondences and/or lower order derivatives than projective invariants, and as such are faster to compute and less sensitive to noise. In this paper, affine invariants requiring one point correspondence and second order derivatives, and requiring two point correspondences and first order derivatives, are investigated. In both of these cases, the affine invariant is useful when the ratio of the sine of the angle of inclination of the plane of the transformed curve to the image plane to the distance of the transformed curve from the image plane is less than a curve-dependent constant. Examples are worked out in detail to illustrate our results.