In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors.

Formulation

The 9 possible connectivities in a 5x5x5 neighborhood

In order to specify a set of connectivities, the dimension ${\displaystyle N}$ and the width of the neighborhood ${\displaystyle n}$, must be specified. The dimension of a neighborhood is valid for any dimension ${\displaystyle n\geq 1}$. A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions.

Let ${\displaystyle M_{N}^{n}}$ represent a N-dimensional hypercubic neighborhood with size on each dimension of ${\displaystyle n=2k+1,k\in \mathbb {Z} }$

Let ${\displaystyle {\vec {q}}}$ represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of ${\displaystyle M_{N}^{n}}$. This implies that each element ${\displaystyle q_{i}\in \{0,1,...,k\},\forall i\in \{1,2,...,N\}}$ and that at least one component ${\displaystyle q_{i}=k}$

Let ${\displaystyle S_{N}^{d}}$ represent a N-dimensional hypersphere with radius of ${\displaystyle d=\left\Vert {\vec {q}}\right\Vert }$.

Define the amount of elements on the hypersphere ${\displaystyle S_{N}^{d}}$ within the neighborhood ${\displaystyle M_{N}^{n}}$ as ${\displaystyle E}$. For a given ${\displaystyle {\vec {q}}}$, ${\displaystyle E}$ will be equal to the amount of permutations of ${\displaystyle {\vec {q}}}$ multiplied by the number of orthants.

Let ${\displaystyle n_{j}}$ represent the amount of elements in vector ${\displaystyle {\vec {q}}}$ which take the value ${\displaystyle j}$. ${\displaystyle n_{j}=\sum _{i=1}^{N}(q_{i}=j)}$

The total number of permutation of ${\displaystyle {\vec {q}}}$ can be represented by a multinomial as ${\displaystyle {\frac {N!}{\prod _{j=0}^{k}n_{j}!}}}$

If any ${\displaystyle q_{i}=0}$, then the vector ${\displaystyle {\vec {q}}}$ is shared in common between orthants. Because of this, the multiplying factor on the permutation must be adjusted from ${\displaystyle 2^{N}}$ to be ${\displaystyle 2^{N-n_{0}}}$

Multiplying the number of amount of permutations by the adjusted amount of orthants yields,

${\displaystyle E={\frac {N!}{\prod _{j=0}^{k}n_{j}!}}2^{N-n_{0}}}$

Let ${\displaystyle V}$ represent the number of elements inside of the hypersphere ${\displaystyle S_{N}^{d}}$ within the neighborhood ${\displaystyle M_{N}^{n}}$. ${\displaystyle V}$ will be equal to the number of elements on the hypersphere plus all of the elements on the inner shells. The shells must be ordered by increasing order of ${\displaystyle \left\Vert {\vec {q}}\right\Vert =r}$. Assume the ordered vectors ${\displaystyle {\vec {q}}}$ are assigned a coefficient ${\displaystyle p}$ representing its place in order. Then an ordered vector ${\displaystyle {\vec {q_{p}}},p\in \{1,2,...,\sum _{x=1}^{k}(x+1)\}}$ if all ${\displaystyle r}$ are unique. Therefore ${\displaystyle V}$ can be defined iteratively as

${\displaystyle V_{\vec {q_{p}}}=V_{\vec {q_{p-1}}}+E_{\vec {q_{p}}},V_{\vec {q_{0}}}=0}$,

or

${\displaystyle V_{\vec {q_{p}}}=\sum _{x=1}^{p}E_{\vec {q_{x}}}}$

If some ${\displaystyle \left\Vert {\vec {q_{x}}}\right\Vert =\left\Vert {\vec {q_{y}}}\right\Vert }$, then both vectors should be considered as the same ${\displaystyle p}$ such that

${\displaystyle V_{\vec {q_{p}}}=V_{\vec {q_{p-1}}}+E_{\vec {q_{p,1}}}+E_{\vec {q_{p,2}}},V_{\vec {q_{0}}}=0}$

Note that each neighborhood will need to have the values from the next smallest neighborhood added. Ex. ${\displaystyle V_{{\vec {q}}=(0,2)}=V_{{\vec {q}}=(1,1)}+E_{{\vec {q}}=(0,2)}}$

${\displaystyle V}$ includes the center hypervoxel, which is not included in the connectivity. Subtracting 1 yields the neighborhood connectivity, ${\displaystyle G}$

${\displaystyle G=V-1}$^[1]

Types of connectivity

2-dimensional

Example of neighborhood of pixels - association of eight and four pixels

4-connected

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 ${\displaystyle \textstyle (x,y)}$.

See also: Von Neumann neighborhood

6-connected

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

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 ${\displaystyle \textstyle (x,y)}$.

See also: Moore neighborhood

See also

• Grid cell topology
• Moore neighborhood

References

• A. Rosenfeld , A. C. Kak (1982), Digital Picture Processing, Academic Press, Inc., ISBN 0-12-597302-0
• Cheng, CC; Peng, GJ; Hwang, WL (2009), "Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding", IEEE Transactions on Image Processing, 18 (1): 52–62, doi:10.1109/TIP.2008.2007067, PMID 19095518, retrieved 2009-02-16
• Cheng, CC; Peng, GJ; Hwang, WL (2009), "Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding", IEEE Transactions on Image Processing, 18 (1): 52–62, doi:10.1109/TIP.2008.2007067, PMID 19095518, retrieved 2009-02-16