Lattice of generalized neighbourhood sequences in nD

Properties for distances based on neighbourhood sequences on the face-centred cubic (fcc) and the body-centred cubic (bcc) grids are presented. Formulas to both compute the distances and assure that the distances satisfy the conditions for being metrics are presented and proved to be correct. The formulas are used to calculate the neighbourhood sequences that generates distances with lowest deviation from the Euclidean distance.

Section: Introductionmentioning

confidence: 99%

Distances based on neighbourhood sequences in non-standard three-dimensional grids

Strand¹,

Nagy²

2007

“…In this section we recall some definitions and notation from [2] and [6] concerning neighbourhood sequences. In what follows, n denotes a positive integer, N denotes an element of N ∪ {∞}.…”

Section: Basic Notation and Conceptsmentioning

confidence: 99%

“…The so-called 'faster' relation has been introduced among neighbourhood sequences (see [6] for N D with arbitrary n.s.-es) in the following way.…”

Section: Lemma 10 the Finite Path Exists (Ie The B-distance Is Finmentioning

confidence: 99%

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Distance with generalized neighbourhood sequences innDand∞D

Nagy¹

2008

In this paper we generalize some former results of Das et al. about distances with n-dimensional periodic neighbourhood sequences. We use a more general definition of neighbourhood sequences, which does not require periodicity. As an extension of the earlier results, we give a formula to calculate the distance between two arbitrary points with general neighbourhood sequences in an arbitrary finite dimension. Moreover we extend the result to the infinite-dimensional digital space.

“…The theory for ns-distances with periodic neighborhood sequences was further developed in [12][13][14][15], where for example formulas for point-to-point distance and conditions for metricity were given. In [16][17][18][19][20], ns-distances for the non-periodic case were considered for standard and non-standard grids.…”

mentioning

confidence: 99%

Weighted distances based on neighborhood sequences for point-lattices

Strand

2009

a b s t r a c tA path-based distance is defined as the minimal cost-path between two points. One such distance function is the weighted distance based on a neighborhood sequence. It can be defined using any number of neighborhood relations and weights in conjunction with a neighborhood sequence. The neighborhood sequence restricts some steps in the path to a smaller neighborhood. We give formulas for computing the point-to-point distance and conditions for metricity for weighted distances based on neighborhood sequences with two neighborhood relations for the general case of point-lattices.