Modelling Objects having Quadric Surfaces Incorporating Geometric cCnstraints

Naoufel Werghi, Robert B. Fisher, Craig Robertson, Anthony Ashbrook

Abstract: This paper deals with the constrained shape reconstruction of objects having quadric patches. The incorporation of geometric constraints in object reconstruction was used first by Porrill [10]. His approach combined the Kalman filter equations with linearized constraint equations. This technique was improved by De Geeter et al [5] to reduce the effects of linearization error. The nature and the specificity of this technique make it limited in scope and application. In their approach for 3-D object pose estimation, Bolle et al [2] constrained some quadrics to have a certain shape (circular cylinder and sphere) by using a specific representation for these particular surfaces. Our work uses a new approach to global shape improvement based on feature coincidence, position and shape constraints. The key idea is to incorporate user specific geometric constraints into the reconstruction process. The constraints are designed to fix some feature relationships (such as parallel surface separations, or cylindrical surface axis relationships) and then use least squares fitting to fix the remaining parameters. An optimization procedure is used to solve the reconstruction problem. In this paper, constraints for planar and general quadric surface classes are given. Results with quadric surfaces show much improvement in shape reconstruction for both constrained and unconstrained relationships. The proposed approach avoids the drawbacks of linearization and allows a larger category of geometric constraints. To our knowledge this work is the first to give such a large framework for the integration of geometric relationships in object modelling. The technique is expected to have a great impact in reverse engineering applications and manufactured object modelling where the majority of parts are designed with intended feature relationships.