Connectivity

In a binary (black and white) image, two neighboring pixels (as defined above) are connected if their values are the same, i.e., both equal to 0 (black) or 255 (white).

In a gray level image, two neighboring pixels are connected if their values are close to each other, i.e., they both belong to the same subset of similar gray levels: $p \in V$ and $q \in V$, where $V$ is a subset of all gray levels in the image.

Specifically, the connectivity can be defined as one of the following:

• 4-connected Two pixels $p$ and $q$ are 4-connected if they are 4-neighbors and $p \in V$ and $q \in V$;

• 8-connected Two pixels $p$ and $q$ are 8-connected if they are 8-neighbors and $p \in V$ and $q \in V$;

• mixed-connected Two pixels $p$ and $q$ are mix-connected if
  • $p$ and $q$ are 4-connected, or
  • $p$ and $q$ are 8-connected and not 4-connected through a third pixel ( $N_4(p) \cap N_4(q) \not\subset V$)

  The second condition states that if $p$ and $q$ are 8-connected and they are also 4-connected through a third pixel, the tighter 4-connectivity through a third pixel is preferred and therefore $p$ and $q$ are no longer considered as 8-connected.

Two pixels at $p$ at $(x,y)$ and $q$ at $(u,v)$ not 4, 8, or mix-connected can still be connected through a path composed of a sequence (chain) of pixels

$$(x_0,y_0)=(x,y), (x_1,y_1), \cdots, (x_{n-1},y_{n-1}),(x_n,y_n)=(u,v)$$ (10)

with all neighboring pixels $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$ 4, 8, or mix-connected.

Example:

The upper-right pixel and the lower-left pixel are 8 and mix-connected, but they are not 4-connected:

0	0	1
0	1	0
1	0	0

The upper-right pixel and the lower-left pixel are 4, 8 and mix-connected:

0	1	1
0	1	0
1	1	0