Homeomorphic Geometrical Transform for Collision Response in Surgical Simulation

Abstract

A new technique for multi-point collision response among 3D objects is presented. The method is applied to the interpenetration between deformable and non-deformable models. Accuracy in the response to collided points is a critical premise that determines the fidelity and the stability of the subsequent deformations. Previous approaches assume a set of simplifications in the collision models that usually derives in inadequate model behaviors. The method pursues a twofold objective: (1) to consider the 3D character of the surgical instruments (2) to cope with the complex nature, concavities and convexities, of the interpenetrated volume. For that purpose, an overall approximation of a local deformation is performed. First, the fidelity of each local deformation is modeled as a combination of penetration and sliding forces inferred from a feedback fuzzy logic system, and, second, the stability of the global deformation is ensured by the definition of an homeomorphic transformation which enforces the preservation of the topology of deformable objects. A linear combination of a set of Compact Supported Radial Basis Functions (CS-RBF) provides a customary mathematical description for finding a smooth displacement field. The resulting deformation field is thus expressed as an optimization problem. Preliminary results show its suitability and benefits for surgical simulation.