Registration of a Curve on a Surface Using Differential Properties

Abstract

This article presents a new method to find the best rigid registration between a curve and a surface. It is possible to write a compatibility equation between a curve point and a surface point, which constrains completely the 6 parameters of the sought rigid displacement. This requires the local computation of third order differential quantities and leads to an algebraic equation of degree 16. A second approach consists in considering pairs of curve and surface points. Then only first order differential are necessary to compute locally the parameters of the rigid displacement. Although computationally more expensive, the second approach is more robust, and can be accelerated with a preprocessing of the surface data. To our knowledge, it is the first method which takes full advantage of local differential computations to register a curve on a surface.