Pixel connectivity

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

In order to specify a set of connectivities, the dimension N and the width of the neighborhood n, must be specified. The dimension of a neighborhood is valid for any dimension

n

≥

1

{\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

M

N

n

{\displaystyle M_{N}^{n}}

represent a N-dimensional hypercubic neighborhood with size on each dimension of

n

=

2

k

+

1

,

k

∈

Z

{\displaystyle n=2k+1,k\in \mathbb {Z} }

Let

q

→

{\displaystyle {\vec {q}}}

represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of

M

N

n

{\displaystyle M_{N}^{n}}

. This implies that each element

q

i

∈

{

0

,

1

,

.

.

.

,

k

}

,

∀

i

∈

{

1

,

2

,

.

.