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R. Marfil and A. Bandera

Ingeniería de Sistemas Integrados Departamento de Tecnología Electrónica Universidad de Málaga (Spain). Graph abstraction preserving the topology: Application to environment mapping for mobile robotics. R. Marfil and A. Bandera.

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R. Marfil and A. Bandera

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  1. Ingeniería de Sistemas Integrados Departamento de Tecnología Electrónica Universidad de Málaga (Spain) Graph abstraction preserving the topology: Application to environment mapping for mobile robotics R. Marfil and A. Bandera International Workshop on Computational Algebraic Topology within Image Context

  2. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  3. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  4. Introduction • Metric paradigm • Topological paradigm

  5. Introduction Metric Map A metric map tries to represent accurately the real features of the world. Advantage: It is easy to build and to maintain. Disadvantages: It has associated a huge data load and time complexity.

  6. Introduction Topological Map • It usually represents the environment using a simple graph: • Nodes are different regions of the environment. • Arcs represent the spatial relationships among regions. • Advantage: they reduce the data to be stored allowing fast path planning.

  7. Introduction Topological Map • Problems of using a simple graph: • It only takes into account adjacency relationships. Then…

  8. Introduction Topological Map Problems of using a simple graph: It is unable to distinguish from the graph an adjacency relation from an inclusion relation between two regions. If there is two non-connected boundaries which will allow a robot to cross from one region to another one, the simple graph only joins these nodes by one arc.

  9. Introduction Goal of the proposed approach • Integrating into the same framework the topological and metric maps using a dual graph pyramid. • That is, a dual graph pyramid is built over the metric map. Each level of the pyramid can be seen as a topological map with different level of resolution. Each node of the topological map has associated a region of the metric map.

  10. Introduction Simple graph-based topological map Dual graph-based topological map

  11. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  12. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  13. Dual Graph Pyramid • Data structure • Decimation process

  14. Dual Graph Pyramid Gl Gl Original image Each level consists of a dual pair (Gl ,Gl) of planar graphs Gland Gl. If level l defines a partition of the image into a connected subsets of pixels, then the vertices of Glare the representatives of these subsets and the edges of Glrepresent their neighbourhood relationships. The edges of Glrepresent the boundaries of these connected subsets in level l and the vertices of Gldefine meeting points of boundary segments of Gl. Data structure D. Willersinn, W.G. Kropatsch, Dual graph contraction for irregular pyramids, in: 12th IAPR International Conference on Pattern Recognition, vol. 3, 1994, pp. 251–256.

  15. Dual Graph Pyramid Decimation process • In general, the process to build a level from the level below in a pyramid is the following: • Selection of the nodes Vl+1 among Vl. These nodes are the surviving nodes. • Allocation of each non-surviving node of level l to a survivor, which generates the son–parent edges. • Creation of edges El+1 by defining the adjacency relationships among the surviving nodes of level l. • The rules used to perform points 1 and 2 define a decimation process.

  16. Dual Graph Pyramid 1. 2. 3. Decimation process: MIES • The MIES algorithm consists of three steps: • Find a maximal matching M from Gl. A matching is a set of edges in which no pair of edges has a common end vertex. • Enlarge M to forest M+ by connecting isolated vertices of Glto the maximal matching M. • M+ is reduced by breaking up trees of diameter three into trees of depth one. A tree is a set of edges connected at their ends containing no closed loops (cycles). Y. Haxhimusa, R. Glantz, M. Saib, G. Langs, W.G. Kropatsch, Logarithmic tapering graph pyramid, Proc. of the 24th German Association for Pattern Recognition Symposium, LNCS vol. 2449, 2002, pp. 117–124.

  17. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  18. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  19. Hybrid mapping Sonar Scan Local map Sonar model The metric map is based on a two dimensional occupancy grid: each grid cell (x,y) in the map yields the occupancy probability O(x,y) of the corresponding region ofthe environment. Metric map generation

  20. Hybrid mapping Proposed approach • Metric map thresholding. Each occupancy value in the metric map is thresholded: • If O(x,y) < U1 (x,y) is free-space. • If U1 < O(x,y) < U2(x,y) is nonexplored. • Others (x,y) is occupied. • Free-space and non-explored cells are the nodes of the base level of the graph pyramid. • Two base nodes are related by an arc if the two corresponding cells are neighbors. • Nodes are attributed with a discrete name wn(n) which can only take two different values: free-space or non-explored. • Arcs are attributed with the Euclidean distance wa(a) between the two metric cells they link.

  21. Hybrid mapping Proposed approach • Dual graph pyramid generation. A node n is linked with a surviving node s if wn(n) == wn(s) and wa((n,s)) is below a threshold Ud. • Set arc attributes of Gl+1. The atributte of the arc al+1 between two surviving nodes (n,m) in Gl+1 is set to the maximum value of the atributtes of the arcs in the path which link n and m in Gl.

  22. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  23. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  24. Results Thresholded metric map Ud=4 m

  25. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  26. Index • Introduction • Dual Graph Pyramid • Proposed approach for hybrid mapping • Results • Conclusions and future work

  27. Conclusions and future work • This paper proposes an integrated method for indoor robot environment mapping. It combines the metric and topological paradigms. • Topological maps are generated using a hierarchy of dual graphs that divides the metric map into homogeneous compact regions. The use of the dual graph made it possible to preserve the topology of the metric map and to correctly code the relation of adjacency and inclusion between topological regions. • Future work will be focused on implementing high-level navigation tasks in the proposed map representation.

  28. Ingeniería de Sistemas Integrados Departamento de Tecnología Electrónica Universidad de Málaga (Spain) Graph abstraction preserving the topology: Application to environment mapping for mobile robotics R. Marfil and A. Bandera International Workshop on Computational Algebraic Topology within Image Context

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