Document Open Access Logo

Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance

Authors Quirijn W. Bouts, Irina Irina Kostitsyna, Marc van Kreveld, Wouter Meulemans, Willem Sonke, Kevin Verbeek

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2016.22.pdf
  • Filesize: 0.65 MB
  • 16 pages

Document Identifiers

Subject Classification

Keywords
  • grid mapping
  • Hausdorff distance
  • Fréchet distance
  • digital geometry

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.

Cite As Get BibTex

Quirijn W. Bouts, Irina Irina Kostitsyna, Marc van Kreveld, Wouter Meulemans, Willem Sonke, and Kevin Verbeek. Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ESA.2016.22

Author Details

Quirijn W. Bouts
Irina Irina Kostitsyna
Marc van Kreveld
Wouter Meulemans
Willem Sonke
Kevin Verbeek

References

  1. Helmut Alt, Bernd Behrends, and Johannes Blömer. Approximate matching of polygonal shapes. Annals of Mathematics & Artificial Intelligence, 13(3-4):251-265, 1995. Google Scholar
  2. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5:75-91, 1995. Google Scholar
  3. Helmut Alt, Christian Knauer, and Carola Wenk. Comparison of distance measures for planar curves. Algorithmica, 38(1):45-58, 2003. Google Scholar
  4. Ernst Althaus, Friedrich Eisenbrand, Stefan Funke, and Kurt Mehlhorn. Point containment in the integer hull of a polyhedron. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 929-933, 2004. Google Scholar
  5. K. Joost Batenburg, Sjoerd Henstra, Walter A. Kosters, and Willem Jan Palenstijn. Constructing simple nonograms of varying difficulty. Pure Mathematics and Applications (Pu. MA), 20:1-15, 2009. Google Scholar
  6. Daniel Berend, Dolev Pomeranz, Ronen Rabani, and Ben Raziel. Nonograms: Combinatorial questions and algorithms. Discrete Applied Mathematics, 169:30-42, 2014. Google Scholar
  7. Quirijn W. Bouts, Irina Kostitsyna, Marc van Kreveld, Wouter Meulemans, Willem Sonke, and Kevin Verbeek. Mapping polygons to the grid with small hausdorff and fréchet distance. Computing Research Repository (arXiv), abs/1606.06660, 2016. Google Scholar
  8. Rafael G. Cano, Kevin Buchin, Thom Castermans, Astrid Pieterse, Willem Sonke, and Bettina Speckmann. Mosaic drawings and cartograms. Computer Graphics Forum, 34(3):361-370, 2015. Google Scholar
  9. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6:485-524, 1991. Google Scholar
  10. Jinhee Chun, Matias Korman, Martin Nöllenburg, and Takeshi Tokuyama. Consistent digital rays. Discrete & Computational Geometry, 42(3):359-378, 2009. Google Scholar
  11. Mark de Berg, Dan Halperin, and Mark H. Overmars. An intersection-sensitive algorithm for snap rounding. Computational Geometry, 36(3):159-165, 2007. Google Scholar
  12. Olivier Devillers and Philippe Guigue. Inner and outer rounding of boolean operations on lattice polygonal regions. Computational Geometry, 33(1-2):3-17, 2006. Google Scholar
  13. Michael T. Goodrich, Leonidas J. Guibas, John Hershberger, and Paul J. Tanenbaum. Snap rounding line segments efficiently in two and three dimensions. In Proceedings of the 13th Annual Symposium on Computational Geometry (SoCG), pages 284-293, 1997. Google Scholar
  14. Daniel H. Greene and F. Frances Yao. Finite-resolution computational geometry. In Proceedings of the 27th Annual Symposium on Foundations of Computer Science (FOCS), pages 143-152, 1986. Google Scholar
  15. Warwick Harvey. Computing two-dimensional integer hulls. SIAM Journal on Computing, 28(6):2285-2299, 1999. Google Scholar
  16. John Hershberger. Stable snap rounding. Computational Geometry, 46(4):403-416, 2013. Google Scholar
  17. Reinhard Klette and Azriel Rosenfeld. Digital Geometry - geometric methods for digital picture analysis. Morgan Kaufmann, 2004. Google Scholar
  18. Reinhard Klette and Azriel Rosenfeld. Digital straightness - a review. Discrete Applied Mathematics, 139(1-3):197-230, 2004. Google Scholar
  19. Johannes Kopf, Ariel Shamir, and Pieter Peers. Content-adaptive image downscaling. ACM Transactions on Graphics, 32(6):Article No. 173, 2013. Google Scholar
  20. Wouter Meulemans. Similarity Measures and Algorithms for Cartographic Schematization. PhD thesis, Technische Universiteit Eindhoven, 2014. Google Scholar
  21. Wouter Meulemans. Discretized approaches to schematization. Computing Research Repository (arXiv), abs/1606.06488, 2016. Google Scholar
  22. Emilio G. Ortíz-García, Sancho Salcedo-Sanz, José M. Leiva-Murillo, Ángel M. Pérez-Bellido, and José Antonio Portilla-Figueras. Automated generation and visualization of picture-logic puzzles. Computers & Graphics, 31(5):750-760, 2007. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact