Eva Dyllong

The GJK Distance Algorithm: An Interval Version for Incremental Motions

The tracking of distance between two convex polyhedra is commonly used in the field of robotics, including collision detection or path planning. One of the well-known algorithms in this area is the distance algorithm developed by Gilbert, Johnson and Keerthi. Although this algorithm is widely-used in robotics, up till now, there has been no verification of the computed results. This paper will present an interval version for tracking the distance between convex polyhedra using the C++ library PROFIL/BIAS.

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

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.

A Comparison of verified distance computation between implicit objects using different arithmetics for range enclosure

This paper describes a new algorithm for computing verified bounds on the distance between two arbitrary fat implicit objects. The algorithm dissects the objects into axis-aligned boxes by constructing an adaptive hierarchical decomposition during runtime. Actual distance computation is performed on the cubes independently of the original object’s complexity. As the whole decomposition process and the distance computation are carried out using verified techniques like interval arithmetic, the calculated bounds are rigorous. In the second part of the paper, we test our algorithm using 18 different test cases, split up into 5 groups. Each group represents a different level of complexity, ranging from simple surfaces like the sphere to more complex surfaces like the Kleins bottle. The algorithm is independent of the actual technique for range bounding, which allows us to compare different verified arithmetics. Using our newly developed uniform framework for verified computations, we perform tests with interval arithmetic, centered forms, affine arithmetic and Taylor models. Finally, we compare them based on the time needed for deriving verified bounds with a user defined accuracy.

A reliable extended octree representation of CSG objects with an adaptive subdivision depth

Octrees are among the most widely used representations in geometric modeling systems, apart from Constructive Solid Geometry and Boundary Representations. An octree model is based on recursive cell decompositions of the space and does not depend greatly on the nature of the object but much more on the chosen maximum subdivision level. Unfortunately, an octree may require a large amount of memory when it uses a set of very small cubic nodes to approximate an object. This paper is concerned with a novel generalization of the octree model that uses interval arithmetic and allows us to extend the tests for classifying points in space as inside, on or outside a CSG object to whole sections of the space at once. Tree nodes with additional information about relevant parts of the CSG object are introduced in order to reduce the depth of the required subdivision. The proposed extended octrees are compared with the common octree representation.

Proximity Queries between Interval‐Based CSG Octrees

This short paper is concerned with a new algorithm for collision and distance calculation between CSG octrees, a generalization of an octree model created from a Constructive Solid Geometry (CSG) object. The data structure uses interval arithmetic and allows us to extend the tests for classifying points in space as inside, on the boundary, or outside a CSG object to entire sections of the space at once. Tree nodes with additional information about relevant parts of the CSG object are introduced in order to reduce the depth of the required subdivision. The new data structure reduces the input complexity and enables us to reconstruct the CSG object. We present an efficient algorithm for computing the distance between CSG objects encoded by the new data structure. The distance algorithm is based on a distance algorithm for classical octrees but, additionally, it utilizes an elaborated sort sequence and differentiated handling of pairs of octree nodes to enhance its efficiency. Experimental results indicate that, in comparison to common octrees, the new representation has advantages in the field of proximity query.

A Reliable Convex-Hull Algorithm for Interval-Based Hierarchical Structures

This paper presents a new approach for constructing the convex polyhedral enclosure of an interval-based hierarchical structure of any dimension. To reduce the number of points in the hull construction considered, only relevant vertices on the boundary-called presumable extreme points- are involved. Additionally, a suitable update of the presumable extreme points enhances the performance whenever the maximum level of the hierarchical structure is changed. This method utilizes interval arithmetic and combines adaptation of the concept of presumable extreme points to higher dimensions with a convex-hull algorithm based on an interval linear solver.

Some Applications of Interval Arithmetic in Hierarchical Solid Modeling

Reliable computing techniques, like interval arithmetic, can be used to guarantee reliable solutions even in the presence of numerical round-off errors. The use of such techniques can eliminate the need to trace bounds for the error function separately. In this paper, we show how the techniques and algorithms of reliable computing can be applied to the construction and further processing of hierarchical solid representations, using the octree model as an example.

Convex Polyhedral Enclosures of Interval-Based Hierarchical Object Representations

In this paper, we discuss approaches to constructing convex polyhedral enclosures of interval-based hierarchical structures. Hierarchical object representations are the data structures most frequently used for reconstructing real scenes. This object modelling does not depend on the nature of a real solid but only on the chosen maximum level of the hierarchical structure. This is a useful property for objects with complex shapes that are difficult to describe via exact mathematical formulas. We focus on reliable object modeling using an interval-based octree data structure. To obtain a convex polyhedral enclosure of an octree, we seek feasible ways to limit the number of considered points. For this purpose, we use the concept of extreme vertices of the tree nodes. Accurate algorithms for constructing the convex hull of these vertices yield a convex polyhedron as an adaptive and reliable object enclosure at each level of the tree.