Generalized distances in digital geometry
This paper proposes a generalized distance measure called m-neighbour distance in quantized n-dimensional space. Given two points \(P=\{x_ i\}\) and \(Q=\{y_ i\}\) for \(1\leq i\leq n\), the m-neighbour distance is defined as: \[ d^ n_ m(P,Q)=\max (\max^{n}_{k=1}X_ k,\quad \lceil \sum^{n}_{k=1}X_ k/m\rceil)\quad, \] where \(X_ k=| x_ k-y_ k|\), \(1\leq k\leq n\) and m,n\(\in {\mathbb{N}}.\) Given a point \(P=\{x_ i\}\); an r-neighbour of P is defined as \(Q=\{y_ i\}:\) \(0\leq | x_ i-y_ i| \leq 1\) for \(1\leq i\leq n\) and \(\sum^{n}_{1}| x_ i-y_ i| =r\). An r-neighbour of P has `n-r' vertices in common with P. It is shown that \(d^ n_ m(P,Q)\) is a distance metric, and an algorithm is given to trace the shortest path composed of successive r- neighbours between points P and Q. It is shown that \(d^ n_ r(P,Q)\) is the length of the path.
scientific article; zbMATH DE number 4028923