Micro-Aerial Vehicles (MAVs) have the advantage of moving freely in 3D space. However, creating compact and sparse map representations that can be efficiently used for planning for such robots is still an open problem. In this paper, we take maps built from noisy sensor data and construct a sparse graph containing topological information that can be used for 3D planning. We use a Euclidean Signed Distance Field, extract a 3D Generalized Voronoi Diagram (GVD), and obtain a thin skeleton diagram representing the topological structure of the environment. We then convert this skeleton diagram into a sparse graph, which we show is resistant to noise and changes in resolution. We demonstrate global planning over this graph, and the orders of magnitude speed-up it offers over other common planning methods. We validate our planning algorithm in real maps built onboard an MAV, using RGB-D sensing.
Other methods for building topologically-aware graphs include SPARTAN, where a distance field is computed up to a certain distance from obstacles @cite_0 , and connected in straight lines between distance field boundaries. This representation is well-suited for very sparse environments, as it allows shortcuts through large open-space regions, but does bug-algorithm-style planning around obstacles. While their method is based on visibility graphs, we use Voronoi diagrams as our basis.
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2020141201 2089950862 2111994433 2014635219
We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the "hyperbolic tree browser" demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space (H2).In the new approach, H-MDS maps proximity data directly into the H2. This removes the restriction to "quasihierarchical", graph-based data--a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general.We review important properties of the hyperbolic space and, in particular, the circular Poincare model of the H2. It enables effective human-computer interaction: by mouse dragging the "focus", the user can navigate in the data without loosing the context. In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce "ring"-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H2--creating a "space of movies" for interactive exploration. This paper presents novel, highly efficient approaches for updating Euclidean distance maps and Voronoi diagrams represented on grid maps. Our methods employ a dynamic variant of the brushfire algorithm to update only those cells that are actually affected by changes in the environment. In experiments in different environments we show that our update strategies for distance maps and Voronoi diagrams require substantially fewer cell visits and significantly less computation time compared to previous approaches. Furthermore, the dynamic Voronoi diagram also improves on previous work by correctly dealing with non-convex obstacles such as building walls. We also present a dynamic variant of a skeletonization-based approach to Voronoi diagrams that is especially robust to noise. All of our algorithms consider actual Euclidean distances rather than grid steps. An open source implementation is available online [1]. We present techniques for fast motion planning by using discrete approximations of generalized Voronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a 3D distance mesh for each Voronoi site. The sites can be points, line segments, polygons, polyhedra, curves and surfaces. The computation of the generalized Voronoi diagram provides fast proximity query toolkits for motion planning. The tools provide the distance to the nearest obstacle stored in the Z-buffer, as well as the Voronoi boundaries, Voronoi vertices and weighted Voronoi graphs extracted from the frame buffer using continuation methods. We have implemented these algorithms and demonstrated their performance for path planning in a complex dynamic environment composed of more than 140,000 polygons. In the real world, there are many phenomena that occur on a network or alongside a network; for example, traffic accidents on highways and retail stores along streets in an urbanized area. In the literature, these phenomena are analysed under the assumption that distance is measured with Euclidean distance on a plane. This paper first examines this assumption and shows an empirical finding that Euclidean distance is significantly different from the shortest path distance in an urbanized area if the distance is less than 500 m. This implies that service areas in urbanized areas cannot be well represented by Voronoi diagrams defined on a plane with Euclidean distance, termed generalized planar Voronoi diagrams. To overcome this limitation, second, this paper formulates six types of Voronoi diagrams defined on a network, termed generalized network Voronoi diagrams, whose generators are given by points, sets of points, lines and polygons embedded in a network, and whose distances are given by inward outward distances, and additively multiplicatively weighted shortest path distances. Third, in comparison with the generalized planar Voronoi diagrams, the paper empirically shows that the generalized network Voronoi diagrams can more precisely represent the service areas in urbanized areas than the corresponding planar Voronoi diagrams. Fourth, because the computational methods for constructing the generalized planar Voronoi diagrams in the literature cannot be applied to constructing the generalized network Voronoi diagrams, the paper provides newly developed efficient algorithms using the 'extended' shortest path trees. Last, the paper develops user-friendly tools (that are included in SANET, a toolbox for spatial analysis on a network) for executing these computational methods in a GIS environment.