Registration of a Curve on a Surface Using Differential Properties

This article presents a new method to nd the best spatial registration between a rigid curve and a rigid surface. We show how to locally exploit the knowledge of diierential properties computed on both the curve and the surface to constrain the rigid matching problem. It is in fact 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 diierential quantities and leads to an algebraic equation of degree 16. A second approach consists in considering pairs of curve and surface points. It is then possible to use only rst order diierential constraints to compute locally the parameters of the rigid displacement. Each approach leads to a diierent matching algorithm. Although computationally more expensive, the second approach is more robust, and can be accelerated with a preprocessing of the surface data. The paper presents the mathematical details of both approaches, algorithms , and a preliminary experimental study on both synthetic and real 3D medical data. To our knowledge, it is the rst method which takes full advantage of local diierential computations to register a curve on a surface. R esum e : Cet article pr esente une nouvelle approche pour eeectuer le recalage rigide d'une courbe sur une surface. Les caract eristiques dii eren-tielles calcul ees sur la courbe et la surface permettent de contraindre les 6 param etres du recalage rigide. Il est en eeet possible d' ecrire une equation de compatibilit e entre un point d'une courbe et le point homologue de la surface. Cela n ecessite le calcul de d eriv ees jusqu'' a l'ordre 3 et conduit a une equation alg ebrique de degr e 16. Une deuxi eme approche consiste a utiliser des couples de points de la courbe et de la surface. Alors les caract eristiques dii erentielles du premier ordre suusent a estimer localement le d eplacement rigide. Bien que plus gourmande en temps de calcul, cette approche est plus robuste, et peut ^ etre acc eler ee par un pr etraitement de la surface. Apr es avoir d emontr e les contraintes g eom etriques du probl eme, nous pr esenterons les deux types d'algorithmes ainsi que des r esultats obtenus sur des donn ees synth etiques et r eelles. A notre connaissance, il s'agit de la premi ere …