The weighted region problem: finding shortest paths through a weighted planar subdivision
Journal of the ACM (JACM)
Delaunay triangulation and 3D adaptive mesh generation
Finite Elements in Analysis and Design - Special issue: adaptive meshing part 2
A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract)
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximating weighted shortest paths on polyhedral surfaces
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Approximation algorithms for geometric shortest path problems
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Mesh generation for domains with small angles
Proceedings of the sixteenth annual symposium on Computational geometry
The Quadtree and Related Hierarchical Data Structures
ACM Computing Surveys (CSUR)
Eighteenth national conference on Artificial intelligence
Automatic extraction of Irregular Network digital terrain models
SIGGRAPH '79 Proceedings of the 6th annual conference on Computer graphics and interactive techniques
Geometry and Topology for Mesh Generation (Cambridge Monographs on Applied and Computational Mathematics)
A theorem on polygon cutting with applications
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
BUSHWHACK: An Approximation Algorithm for Minimal Paths through Pseudo-Euclidean Spaces
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
The focussed D* algorithm for real-time replanning
IJCAI'95 Proceedings of the 14th international joint conference on Artificial intelligence - Volume 2
Querying Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
Theta*: any-angle path planning on grids
Journal of Artificial Intelligence Research
Approximate shortest path queries on weighted polyhedral surfaces
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
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Classic shortest path algorithms operate on graphs, which are suitable for problems that can be represented by weighted nodes or edges. Finding a shortest path through a set of weighted regions is more difficult and only approximate solutions tend to scale well. The Field D* algorithm efficiently calculates an approximate, interpolated shortest path through a set of weighted regions and was designed for navigating robots through terrains with varying characteristics. Field D* operates on unit grid or quad-tree data structures, which require high resolutions to accurately model the boundaries of irregular world structures. In this paper, we extend the Field D* cost functions to 2D triangulations and 3D tetrahedral meshes: structures which model polygonal world structures more accurately. Since robots typically have limited resources available for computation and storage, we pay particular attention to computation and storage overheads when detailing our extensions. We begin by providing analytic solutions to the minimum of each cost function for 2D triangles and 3D tetrahedra. Our triangle implementation provides a 50 % improvement in performance over an existing triangle implementation. While our 3D extension to tetrahedra is the first full analytic extension of Field D* to 3D, previous work only provided an approximate minimization for a single cost function on a 3D cube with unit lengths. Each cost function is expressed in terms of a general function whose characteristics can be exploited to reduce the calculations required to find a minimum. These characteristics can also be exploited to cache the majority of cost functions, producing a speedup of up to 28 % in the 3D tetrahedral case. We demonstrate that, in environments composed of non-grid aligned data, Multi-resolution quad-tree Field D* requires an order of magnitude more faces and between 15 and 20 times more node expansions, to produce a path of similar cost to one produced by a triangle implementation of Field D* on a lower resolution triangulation. We provide examples of 3D pathing through models of complex topology, including pathing through anatomical structures extracted from a medical data set. To summarise, this paper details a robust and efficient extension of Field D* pathing to data sets represented by weighted triangles and tetrahedra, and also provides empirical data which demonstrates the reduction in storage and computation costs that accrue when one chooses such a representation over the more commonly used quad-tree and grid-based alternatives.