Verified distance computation between non-convex superquadrics using hierarchical space decomposition structures

Abstract

Verified distance computation is an important task in various application domains. In some domains a proof of correctness is crucial. In this paper, we show how we can apply the methods provided by our uniform framework for verified geometric computations to derive verified bounds on the distance between non-convex objects. The framework features a layered structure enabling the algorithm to run independently whether the objects are described by implicit functions or parametric ones or by polyhedrons. The approach is based on the use of adaptively constructed hierarchical decompositions of the models. As a practical example we use various scenarios occurring in automatic surgery assistance systems for total hip replacement (THR). To ensure that an implant selected by the system fits into the patient's femoral shaft, we have to derive verified bounds on the distance between them. In this case, the models are either superquadrics or polyhedrons, both of which can be non-convex We first show how to increase the enclosure quality of implicit objects by incorporating interval contractors into the hierarchical space decomposition. Next, we describe the construction of a decomposition structure for parametric objects. After that, we present an improvement of the case selector for computing the distance between interval tree nodes, yielding tighter results. We also show how to integrate surface normals into the algorithm if first-order information is available and how to accelerate the solving process by incorporating information gained by non-verified floating-point solvers. Finally, we provide numerical results for all distance query types occurring during the THR procedure and examine whether it is advisable to perform the computation on the implicit model or on the parametric one if both are available. Further numerical results are presented for test cases involving contractors in the decomposition structures.