Hyperspheres of N-sequence distances

P. P. Das

doi:10.1117/12.142186

9 April 1993 Hyperspheres of N-sequence distances

P. P. Das

Proceedings Volume 1832, Vision Geometry; (1993) https://doi.org/10.1117/12.142186
Event: Applications in Optical Science and Engineering, 1992, Boston, MA, United States

Abstract

Let d be a metric on the n-D digital space Zⁿ. The hypersphere H_d(r) of radius r, r integer, and center at origin is defined as H_d(r) equals {x : x (epsilon) Zⁿ & d(x) ≤ r}. For example, Das and Chatterji studied the structure, volume and surface of such digital hyperspheres for the m-neighbor distance dⁿ_m. On generalization to dⁿ_m the N-sequence distance d(B) was proposed with the contention that these will define hyperoctagons in n-D. However, the hyperoctagonality of d(B)'s has not been established so far except for the special cases of 2- and 3-D and for dⁿ_m's in n- D. In this paper we explore the structure of the hyperspheres of d(B)'s in n-D to show that they truly are hyperoctagons. In particular we derive a formula to compute the corners of such hyperoctagons given a B and a r.

Citation Download Citation

P. P. Das "Hyperspheres of N-sequence distances", Proc. SPIE 1832, Vision Geometry, (9 April 1993); https://doi.org/10.1117/12.142186

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