The neighborhood relation for a given pixel $p(x,y)$ describes a [[Set|set]] of adjacent pixels. The $N_{4}$ neighbor relates to the vertical and horizontal direct neighbors. $\huge \begin{align} N_{4}(p) &= \set{(x-1,y),(x+1,y),(x,y-1),(x,y+1)} \\ \end{align} $ $ \huge q \in N_{4}(p) \implies p \in N_{4}(q) $ Each member of $N_4(p)$ is a [[Unit Vector|unit]] [[Distance|distance]] from $p$. $N_D(p)$ refers to the diagonal [[Pixel|Pixels]], with each having a distance of $\sqrt{ 2 }$ from $p$. $N_8(p)$ refers to all adjacent neighbors with is a [[Union]] of the previous two. $\huge N_{8}(p) = N_{4}(p) \cup N_{D}(p) $