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

Abstract

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.

Weighted Distances Based on Neighbourhood Sequences in Non-standard Three-Dimensional Grids

“…We remark that by translation-invariance and symmetry, the distance between any two grid points is given by the formula below. The formulas in Lemma 1 are proved (as Theorem 2 and 5) in [4]. …”

“…The corresponding algorithm using a path-based approach is simple, fast, and easy to generalize to higher dimensions [7]. Examples of path-based distances are weighted distances, where weights define the cost (distance) between neighbouring grid points [8,3,2], and distances based on neighbourhood sequences (ns-distances), where the cost is fixed but the adjacency relation is allowed to vary along the path [9,4]. These path-based distance functions are generalizations of the well-known city-block and chessboard distance function defined for the square grid in [10].…”

“…These path-based distance functions are generalizations of the well-known city-block and chessboard distance function defined for the square grid in [10]. The rotational dependency of weighted distances and ns-distances can be minimized by optimizing the weights [8,3,2] and neighbourhood sequences [9,4], respectively.…”

“…In this paper, weighted distances and ns-distances for the fcc and bcc grids, presented in [2,3] and [4] respectively, are combined by defining the distance as the shortest path between two grid points using both weights for the local distance between neighbouring grid points and allowing the adjacency relation to vary along the path. Functional forms are given of the distance functions for grid points in the fcc and bcc grids.…”

“…This is one reason for the increasing interest in using these grids in e.g. image acquisition [1], image processing [2,3,4], and image visualization [5].…”