Calculating Distance with Neighborhood Sequences in the Hexagonal Grid
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.
KeywordsDigital geometry Hexagonal grid Distance Neighborhood sequences
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- 1.Danielsson, P.E.: 3D Octagonal Metrics. In: Eighth Scandinavian Conference on Image Processing, pp. 727–736 (1993)Google Scholar
- 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
- 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.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.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.Nagy, B.: Distance with generalised neighborhood sequences in nD and ∞D. Discrete Applied Mathematics (submitted)Google Scholar