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Characterizing Obstacle-Avoiding Paths Using Cohomology Theory

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Computer Analysis of Images and Patterns (CAIP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6854))

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Abstract

In this paper, we investigate the problem of analyzing the shape of obstacle-avoiding paths in a space. Given a d-dimensional space with holes, representing obstacles, we ask if certain paths are equivalent, informally if one path can be continuously deformed into another, within this space. Algebraic topology is used to distinguish between topologically different paths. A compact yet complete signature of a path is constructed, based on cohomology theory. Possible applications include assisted living, residential, security and environmental monitoring. Numerical results will be presented in the final version of this paper.

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References

  1. Tovar, B., Cohen, F., LaValle, S.M.: Sensor Beams, Obstacles, and Possible Paths. In: Proc. Workshop on the Algorithmic Foundations of Robotics, WAFR (2008)

    Google Scholar 

  2. de Silva, V., Ghrist, R.: Coverage in sensor networks via persistent homology. Alg. & Geom. Topology 7, 339–358 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Soro, S., Heinzelman, W.: A Survey of Visual Sensor Networks. Advances in Multimedia 2009 (2009)

    Google Scholar 

  4. Edelsbrunner, H.: Geometry and Topology for Mesh Generation. Cambridge University Press, Cambridge (2001) ISBN 9780521793094

    Book  MATH  Google Scholar 

  5. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  6. Dłotko, P., Specogna, R.: Efficient cohomology computation for electromagnetic modeling. CMES: Computer Modeling in Engineering & Sciences 60(3), 247–278 (2010)

    MathSciNet  Google Scholar 

  7. Dłotko, P., Specogna, R.: Critical analysis of the spanning tree techniques. SIAM Journal of Numerical Analysis (SINUM) 48(4), 1601–1624 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cromwell, P.R., Beltrami, E., Rampichini, M.: The Borromean Rings. Mathematical Intelligencer 20(1), 53–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Jagiellonian University, Poland

    Paweł Dłotko & Hubert Wagner

  2. Pattern Recognition and Image Processing Group, Vienna University of Technology, Austria

    Walter G. Kropatsch

Authors
  1. Paweł Dłotko
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  2. Walter G. Kropatsch
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  3. Hubert Wagner
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Editor information

Editors and Affiliations

  1. Dpto. Matematica Aplicada I, Escuaela Técnica Superior de Ingeniería Informática, Universite de Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain

    Pedro Real

  2. Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingeniería Informática, University of Seville, Avenida Reina Mercedes s/n, 41012, Sevilla, Spain

    Daniel Diaz-Pernil  & Helena Molina-Abril  & 

  3. Departamento de Didáctica de la Mathemática y de las CC.Experimentales, Universidad del País Vasco-Esukal Herriko Unibertsitatea, Escuela Universitaria de Magisterio, Ramón y Cajal, 72, 48014, Bilbao (Bizcaia), Spain

    Ainhoa Berciano

  4. Institute of Computer Graphics and Algorithms, Pattern Recognition and Image Processing Group, Vienna University of Technology, Favoritenstraße 9/186-3, 1040, Vienna, Austria

    Walter Kropatsch

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© 2011 Springer-Verlag Berlin Heidelberg

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Dłotko, P., Kropatsch, W.G., Wagner, H. (2011). Characterizing Obstacle-Avoiding Paths Using Cohomology Theory. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds) Computer Analysis of Images and Patterns. CAIP 2011. Lecture Notes in Computer Science, vol 6854. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23672-3_38

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  • DOI: https://doi.org/10.1007/978-3-642-23672-3_38

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23671-6

  • Online ISBN: 978-3-642-23672-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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