IWCIA 2004: Combinatorial Image Analysis pp 98-109

# Calculating Distance with Neighborhood Sequences in the Hexagonal Grid

• Benedek Nagy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)

## Abstract

The theory of neighborhood sequences is applicable in many image-processing algorithms. The theory is well examined for the square and the cubic grids. In this paper we consider another regular grid, the hexagonal one, and the distances based on neighborhood sequences are investigated. The points of the hexagonal grid can be embedded into the cubic grid. With this injection we modify the formula which calculates the distances between points in the cubic space to the hexagonal plane. Our result is a theoretical one, which is very helpful. It makes the distances based on neighborhood sequences in the hexagonal grid applicable. Some interesting properties of these distances are presented, such as the non-symmetric distances. It is possible that the distance depends on the ordering of the elements of the initial part of the neighborhood sequence. We show that these two properties are dependent.

## Keywords

Digital geometry Hexagonal grid Distance Neighborhood sequences

## References

1. 1.
Danielsson, P.E.: 3D Octagonal Metrics. In: Eighth Scandinavian Conference on Image Processing, pp. 727–736 (1993)Google Scholar
2. 2.
Das, P.P., Chakrabarti, P.P., Chatterji, B.N.: Distance functions in digital geometry. Information Sciences 42, 113–136 (1987)
3. 3.
Deutsch, E.S.: Thinning algorithms on rectangular, hexagonal and triangular arrays. Communications of the ACM 15(3), 827–837 (1972)
4. 4.
Fazekas, A., Hajdu, A., Hajdu, L.: Lattice of generalized neighborhood sequences in nD and ∞D. Publicationes Mathematicae Debrecen 60, 405–427 (2002)
5. 5.
Freeman, H.: Algorithm for generating a Digital Straight Line on a Triangular Grid. IEEE Transactions on Computers C-28, 150–152 (1979)
6. 6.
Hajdu, A., Nagy, B., Zörgő, Z.: Indexing and segmenting colour images using neighborhood sequences. In: IEEE International Conference on Image Processing, ICIP 2003, Barcelona, September 2003, pp. I/957–960 (2003)Google Scholar
7. 7.
Kong, T.Y., Rosenfeld, A.: Digital Topology: Introduction and Survey. Computer Vision, Graphics and Image Processing 48, 357–393 (1989)
8. 8.
Nagy, B.: Distance functions based on neighbourhood sequences. Publicationes Mathematicae Debrecen 63, 483–493 (2003)
9. 9.
Nagy, B.: Shortest Path in Triangular Grids with Neighborhood Sequences. Journal of Computing and Information Technology 11, 111–122 (2003)
10. 10.
Nagy, B.: A family of triangular grids in digital geometry. In: 3rd International Symposium on Image and Signal Processing and Analysis (ISPA 2003), Rome, Italy, September 2003, pp. 101–106 (2003)Google Scholar
11. 11.
Nagy, B.: A symmetric coordinate system for the hexagonal networks. In: Information Society 2004 – Theoretical Computer Science (IS 2004-TCS), ACM Slovenija conference, Ljubljana, Slovenia (October 2004) (accepted paper)Google Scholar
12. 12.
Nagy, B.: Non-metrical distances on the hexagonal plane. In: 7th International Conference on Pattern Recognition and Image Analysis: New Information Technologies (PRIA-7-2004), St. Petersburg, Russian Federation (October 2004) (accepted paper)Google Scholar
13. 13.
Nagy, B.: Distance with generalised neighborhood sequences in nD and ∞D. Discrete Applied Mathematics (submitted)Google Scholar
14. 14.
Rosenfeld, A., Pfaltz, J.L.: Distance functions on digital pictures. Pattern Recognition 1, 33–61 (1968)
15. 15.
Yamashita, M., Ibaraki, T.: Distances defined by neighborhood sequences. Pattern Recognition 19, 237–246 (1986)