Image Processing Reference
In-Depth Information
p
FIGURE 1.2: Partitioning by an 8-connected curve in a digital grid.
as well. This violates the notion of partitioning as discussed in Jordan's curve
theorem for a 2-D plane. On the other hand, if we use the definition of 4-
adjacency, though there are two 4-connected components in the background,
the foreground (of black pixels) consists of 4 isolated points, which do not form
a curve. To resolve this paradox of connectivity, Rosenfeld [177] proposed to
use two different types of connectivity for foreground and background, i.e.,
either 4-connectivity for foreground and 8-connectivity for background, or the
reverse. With this notion of connectivity, the grid topology is distinguished.
1.3 Grid Topology
Following the notations of [118], we denote a grid topology for 2-D and
3-D binary images by a quadruple (U,m,n,S), where U could be either G
2
or G
3
; (m,n) ∈ {(4,8),(8,4)} if U is G
2
, else for G
3
it is an element of
{(6,18),(18,6),(6,26),(26,6)}; and S is the set of foreground points in U.
The type of adjacency used for defining neighbors in S is denoted by m, and
n is the corresponding type for the background points, i.e., for S = U−S. The
quadruple grid topology is also referred to here as a digital picture. We also
denote the border of U as B(U), and the connected components of background
