Skip to main content
SpringerLink
Log in

Neighborhood Sequences in the Diamond Grid – Algorithms with Four Neighbors

  • Conference paper
Combinatorial Image Analysis (IWCIA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5852))

Included in the following conference series:

Abstract

In digital image processing digital distances are useful; distances based on neighborhood sequences are widely used. In this paper the diamond grid is considered, that is the three-dimensional grid of Carbon atoms in the diamond crystal. This grid can be described by four coordinate values using axes of the directions of atomic bonds. In this way the sum of the coordinate values can be either zero or one. An algorithm to compute a shortest path defined by a neighborhood sequence between any two points in the diamond grid is presented. The metric and non-metric properties of some distances based on neighborhood sequences are also discussed. The constrained distance transformation and digital balls obtained by some distance functions are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Borgefors, G.: Distance Transformations on Hexagonal Grids. Pattern Recognition Letters 9, 97–105 (1989)

    Article  Google Scholar 

  2. Brimkov, V.E., Barneva, R.P.: Analytical Honeycomb Geometry for Raster and Volume Graphics. The Computer Journal 48(2), 180–199 (2005)

    Article  MathSciNet  Google Scholar 

  3. Conway, J.H., Sloane, N.J.A., Bannai, E.: Sphere-packings, lattices, and groups. Springer, New York (1988)

    MATH  Google Scholar 

  4. Csébfalvi, B.: Prefiltered Gaussian reconstruction for high-quality rendering of volumetric data sampled on a body-centered cubic grid. IEEE Visualization, 40 (2005)

    Google Scholar 

  5. Das, P.P., Chakrabarti, P.P., Chatterji, B.N.: Distance functions in digital geometry. Information Sciences 42, 113–136 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal and triangular arrays. Communications of the ACM 15, 827–837 (1972)

    Article  Google Scholar 

  7. Her, I.: A symmetrical coordinate frame on the hexagonal grid for computer graphics and vision. ASME J. Mech. Design 115, 447–449 (1993)

    Article  Google Scholar 

  8. Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. on Image Processing 4(9), 1213–1222 (1995)

    Article  Google Scholar 

  9. Her, I.: Description of the F.C.C. lattice geometry through a four-dimensional hypercube. Acta Cryst. A 51, 659–662 (1995)

    Article  MathSciNet  Google Scholar 

  10. Klette, R., Rosenfeld, A.: Digital geometry - Geometric methods for digital picture analysis. Morgan Kaufmann Publ., San Francisco (2004)

    MATH  Google Scholar 

  11. Middleton, L., Sivaswamy, J.: Framework for practical hexagonal-image processing. Journal of Electronic Imaging 11, 104–114 (2002)

    Article  Google Scholar 

  12. Nagy, B.: Shortest Paths in Triangular Grids with Neighbourhood Sequences. Journal of Computing and Information Technology 11(2), 111–122 (2003)

    Article  Google Scholar 

  13. Nagy, B.: Distance functions based on neighbourhood sequences. Publicationes Mathematicae Debrecen 63(3), 483–493 (2003)

    MATH  MathSciNet  Google Scholar 

  14. Nagy, B.: A Family of Triangular Grids in Digital Geometry. In: Proc. of the 3rd ISPA, International Symposium on Image and Signal Processing and Analysis, Rome, Italy, pp. 101–106 (2003)

    Google Scholar 

  15. Nagy, B.: A symmetric coordinate frame for hexagonal networks. In: Proc. of IS-TCS 2004, Theoretical Computer Science - Information Society, Ljubljana, Slovenia, pp. 193–196 (2004)

    Google Scholar 

  16. Nagy, B.: Generalized triangular grids in digital geometry. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 20, 63–78 (2004)

    MATH  Google Scholar 

  17. Nagy, B.: Characterization of digital circles in triangular grid. Pattern Recognition Letters 25(11), 1231–1242 (2004)

    Article  Google Scholar 

  18. Nagy, B.: Distances with Neighbourhood Sequences in Cubic and Triangular Grids. Pattern Recognition Letters 28, 99–109 (2007)

    Article  Google Scholar 

  19. Nagy, B., Strand, R.: Neighborhood sequences in the diamond grid. In: Barneva, R.P., Brimkov, V.E. (eds.) Image Analysis - From Theory to Applications, Research Publishing, Singapore, Chennai, pp. 187–195 (2008)

    Google Scholar 

  20. Nagy, B., Strand, R.: A connection between \(\mathbb Z^n\) and generalized triangular grids. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Porikli, F., Peters, J., Klosowski, J., Arns, L., Chun, Y.K., Rhyne, T.-M., Monroe, L., et al. (eds.) ISVC 2008, Part II. LNCS, vol. 5359, pp. 1157–1166. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Nagy, B., Strand, R.: Non-traditional grids embedded in \(\mathbb Z^n\). Int. Journal of Shape Modeling (2009) (accepted for publication)

    Google Scholar 

  22. Nagy, B., Strand, R.: Neighborhood sequences in the diamond grid: algorithms with 2 and 3 neighbors. Int. Journal of Imaging Systems and Technology 19(2), 146–157 (2009)

    Article  Google Scholar 

  23. Piper, J., Granum, E.: Computing distance transformations in convex and non-convex domains. Pattern Recognition 20(6), 599–615

    Google Scholar 

  24. Rosenfeld, A., Pfaltz, J.L.: Sequential Operations in Digital Picture Processing. Journal of the ACM 13(4), 471–494 (1966)

    Article  MATH  Google Scholar 

  25. Rosenfeld, A., Pfaltz, J.L.: Distance Functions on Digital Pictures. Pattern Recognition 1, 33–61 (1968)

    Article  MathSciNet  Google Scholar 

  26. Rosenfeld, A.: Digital Geometry: Introduction and Bibliography. Techn. Rep. CS-TR-140/CITR-TR-1 (Digital Geometry Day 1997), Computer Science Dept., The University of Auckland, CITR at Tamaki Campus (1997)

    Google Scholar 

  27. Snyder, W.E., Qi, H., Sander, W.A.: A coordinate system for hexagonal pixels. Proc. of SPIE, Medical Imaging Pt.1-2, 716–727 (1999)

    Google Scholar 

  28. Strand, R., Nagy, B.: Distances Based on Neighbourhood Sequences in Non-Standard Three-Dimensional Grids. Discrete Applied Mathematics 155(4), 548–557 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Takahashi, T., Yonekura, T.: Isosurface construction from a data set sampled on a face-centered-cubic lattice. In: Proc. of ICCVG 2002, vol. 2, pp. 754–763 (2002)

    Google Scholar 

  30. Yamashita, M., Honda, N.: Distance Functions Defined by Variable Neighborhood Sequences. Pattern Recognition 5(17), 509–513 (1984)

    Article  MathSciNet  Google Scholar 

  31. Yamashita, M., Ibaraki, T.: Distances Defined by Neighborhood Sequences. Pattern Recognition 19(3), 237–246 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Faculty of Informatics, University of Debrecen, PO Box 12, 4010, Debrecen, Hungary

    Benedek Nagy

  2. Centre for Image Analysis, Uppsala University, Box 337, SE-75105, Uppsala, Sweden

    Robin Strand

Authors
  1. Benedek Nagy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robin Strand
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Automatic Control, Centro de Investigación y de Estudios Avanzados (CINVESTAV- IPN), Av. I.P.N. 2508, Col. San Pedro Zacatenco, 07000 D.F., Mexico

    Petra Wiederhold

  2. Department of Computer Science, State University of New York at Fredonia, NY 14063, Fredonia,, USA

    Reneta P. Barneva

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nagy, B., Strand, R. (2009). Neighborhood Sequences in the Diamond Grid – Algorithms with Four Neighbors. In: Wiederhold, P., Barneva, R.P. (eds) Combinatorial Image Analysis. IWCIA 2009. Lecture Notes in Computer Science, vol 5852. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10210-3_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10210-3_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10208-0

  • Online ISBN: 978-3-642-10210-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation