Generalized distances in digital geometry

Neighborhood sequences in the diamond grid: Algorithms with two and three neighbors

“…The literature on distances based on neighborhood sequences is rich; a theory for periodic neighborhood sequences not connected to any specific neighborhood relations in Z n is presented in the article by Yamashita and Honda (1984) and Yamashita and Ibaraki (1986) and further developed for the natural neighborhood structure, by the so-called octagonal neighborhood sequences in studies by Das and Chakrabarti (1987a) and Das et al (1987b). Results for general (not necessarily periodic) neighborhood sequences are presented in the article by Nagy (2003).…”

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

“…This was first noted in [25]; the authors state that the approximation obtained when the ratio between the number of steps using the different neighbourhood relations in Z 2 is equal to 1 : √ 2 is optimal. The literature on distances based on neighbourhood sequences is rich; a theory for periodic neighbourhood sequences not connected to any specific neighbourhood relations in Z n is presented in [29,30] and further developed for the natural neighbourhood structure, by the so-called octagonal neighbourhood sequences in [7,6]. Results for general (not necessarily periodic) neighbourhood sequences are presented in [9,21].…”

Distance with generalized neighbourhood sequences innDand∞D

“…By allowing arbitrary mixture of the basic motions, Das, Chakrabarti and Chatterji introduced the concept of periodic neighbourhood sequences in [2] and in [5]. The concept of neighbourhood sequences was extended to the infinite dimension also in [6].…”

“…In [5] Das and his co-authors gave a formula to determine the distance between two points in Z n using any fixed neighbourhood criterion. In [2] they presented a complex formula (we detail it later) which determines the distance between two points in nD using an arbitrary periodic neighbourhood sequence.…”