Pixel connectivity

Consider solving for ${\displaystyle G|{\vec {q}}=(0,1,1)}$

In this scenario, ${\displaystyle N=3}$ since the vector is 3-dimensional. ${\displaystyle n_{0}=1}$ since there is one ${\displaystyle q_{i}=0}$. Likewise, ${\displaystyle n_{1}=2}$. ${\displaystyle k=1,n=3}$ since ${\displaystyle \max q_{i}=1}$. ${\displaystyle d={\sqrt {0^{2}+1^{2}+1^{2}}}={\sqrt {2}}}$. The neighborhood is ${\displaystyle M_{3}^{3}}$ and the hypersphere is ${\displaystyle S_{3}^{\sqrt {2}}}$

${\displaystyle E={\frac {3!}{1!*2!*0!}}2^{3-1}={\frac {6}{2}}4=12}$

The basic ${\displaystyle {\vec {q}}}$ in the neighborhood ${\displaystyle N_{3}^{3}}$, ${\displaystyle {\vec {q_{1}}}=(0,0,0)}$. The Manhattan Distance between our vector and the basic vector is ${\displaystyle \left\Vert {\vec {q}}-{\vec {q_{0}}}\right\Vert _{1}=2}$, so ${\displaystyle {\vec {q}}={\vec {q_{3}}}}$. Therefore,

${\displaystyle G_{\vec {q_{3}}}=V_{\vec {q_{3}}}-1=E_{\vec {q_{1}}}+E_{\vec {q_{2}}}+E_{\vec {q_{3}}}-1=E_{{\vec {q}}=(0,0,0)}+E_{{\vec {q}}=(0,0,1)}+E_{{\vec {q}}=(0,1,1)}}$

${\displaystyle E_{{\vec {q}}=(0,0,0)}={\frac {3!}{3!*0!*0!}}2^{3-3}={\frac {6}{6}}1=1}$

${\displaystyle E_{{\vec {q}}=(0,0,1)}={\frac {3!}{2!*1!}}2^{3-2}={\frac {6}{2}}2=6}$

${\displaystyle G=1+6+12-1=18}$

Which matches the supplied table

The assumption that all ${\displaystyle \left\Vert {\vec {q_{p}}}\right\Vert =r}$ are unique does not hold for higher values of k & N. Consider ${\displaystyle N=2,k=5}$, and the vectors ${\displaystyle {\vec {q_{A}}}=(0,5),{\vec {q_{B}}}=(3,4)}$. Although ${\displaystyle {\vec {q_{A}}}}$ is located in ${\displaystyle M_{2}^{5}}$, the value for ${\displaystyle r=25}$, whereas ${\displaystyle {\vec {q_{B}}}}$ is in the smaller space ${\displaystyle M_{2}^{4}}$ but has an equivalent value ${\displaystyle r=25}$. ${\displaystyle {\vec {q_{C}}}=(4,4)\in M_{2}^{4}}$ but has a higher value of ${\displaystyle r=32}$ than the minimum vector in ${\displaystyle M_{2}^{5}}$.

For the this assumption to hold, ${\displaystyle {\begin{cases}N=2,k\leq 4\\N=3,k\leq 2\\N=4,k\leq 1\end{cases}}}$

At higher values of ${\displaystyle k}$ & ${\displaystyle N}$, Values of ${\displaystyle d}$ will become ambiguous. This means that specification of a given ${\displaystyle d}$ could refer to multiple ${\displaystyle {\vec {q_{p}}}\in M_{n}^{N}}$.

4-connected pixels are neighbors to every pixel that touches one of their edges. These pixels are connected horizontally and vertically. In terms of pixel coordinates, every pixel that has the coordinates

${\displaystyle \textstyle (x\pm 1,y)}$ or ${\displaystyle \textstyle (x,y\pm 1)}$

is connected to the pixel at ${\displaystyle \textstyle (x,y)}$.

6-connected pixels are neighbors to every pixel that touches one of their corners (which includes pixels that touch one of their edges) in a hexagonal grid or stretcher bond rectangular grid.

There are several ways to map hexagonal tiles to integer pixel coordinates. With one method, in addition to the 4-connected pixels, the two pixels at coordinates ${\displaystyle \textstyle (x+1,y+1)}$ and ${\displaystyle \textstyle (x-1,y-1)}$ are connected to the pixel at ${\displaystyle \textstyle (x,y)}$.

8-connected pixels are neighbors to every pixel that touches one of their edges or corners. These pixels are connected horizontally, vertically, and diagonally. In addition to 4-connected pixels, each pixel with coordinates ${\displaystyle \textstyle (x\pm 1,y\pm 1)}$ is connected to the pixel at ${\displaystyle \textstyle (x,y)}$.

6-connected pixels are neighbors to every pixel that touches one of their faces. These pixels are connected along one of the primary axes. Each pixel with coordinates ${\displaystyle \textstyle (x\pm 1,y,z)}$, ${\displaystyle \textstyle (x,y\pm 1,z)}$, or ${\displaystyle \textstyle (x,y,z\pm 1)}$ is connected to the pixel at ${\displaystyle \textstyle (x,y,z)}$.

18-connected pixels are neighbors to every pixel that touches one of their faces or edges. These pixels are connected along either one or two of the primary axes. In addition to 6-connected pixels, each pixel with coordinates ${\displaystyle \textstyle (x\pm 1,y\pm 1,z)}$, ${\displaystyle \textstyle (x\pm 1,y\mp 1,z)}$, ${\displaystyle \textstyle (x\pm 1,y,z\pm 1)}$, ${\displaystyle \textstyle (x\pm 1,y,z\mp 1)}$, ${\displaystyle \textstyle (x,y\pm 1,z\pm 1)}$, or ${\displaystyle \textstyle (x,y\pm 1,z\mp 1)}$ is connected to the pixel at ${\displaystyle \textstyle (x,y,z)}$.

26-connected pixels are neighbors to every pixel that touches one of their faces, edges, or corners. These pixels are connected along either one, two, or all three of the primary axes. In addition to 18-connected pixels, each pixel with coordinates ${\displaystyle \textstyle (x\pm 1,y\pm 1,z\pm 1)}$, ${\displaystyle \textstyle (x\pm 1,y\pm 1,z\mp 1)}$, ${\displaystyle \textstyle (x\pm 1,y\mp 1,z\pm 1)}$, or ${\displaystyle \textstyle (x\mp 1,y\pm 1,z\pm 1)}$ is connected to the pixel at ${\displaystyle \textstyle (x,y,z)}$.