International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
n - connected object O are
equivalence classes of spels according to this relation. Using
this equivalence relation on the complement of an object we can
components of an
define a background component of O as an
n -connected component of Ô .
(b) 12-adjacent
Figure 7. The definition of 3-adjacent and 12-adjacent
In 2D spherical surface digital space, we consider spherical
surface triangle mesh to express spherical surface digital image.
In this paper, points refer to grid points in spherical surface
digital space unless stated otherwise. Two nonempty sets of
points S, and S, are said to be 3 - adjacent or
12-adjacent if at least one point of Sys
3-adjacent or 12- adjacent to at least one point of
So: The adjacency definition is important not only in the
computation of raster distance between two spels but also in
topological analysis (LI et al. 2000). Let S be a nonempty set
of points. An 3 - path between two points D, in S
means a sequence of distinct points
D = Par Pise P, = Got S such that p, is 3-adjacent
tons 254 5 0s i«m o Twe:poins-pDge Se «dae
3-connected in S if there exists an. 3- path from p to
q in S. An 3 - component of S is a maximal subset
of S where each pair of points is 3 - connected .
A 2D spherical surface digital image Ç can be treated as a subset
of Z2 together with some fixed neighborhood structure. It is
defined as a quadruple (7312. £3. Here / is the image
space, which is a set of all grid points. E is defined as the set of
black points that is spatial entity in ] and I — E is the set
of white points. 3-adjacency or 12-adjacency are the adjacencies
used for finding 3 - components and
12.- components in E and / — E , respectively. Note
that / — E denotes the set of white points in £ . In this
paper, we use 12 - adjacency for black points and
3- adjacency for white points and call
12 - components of E black components and
3 - components of I — E white component. The basic
white spels are defined as being
topological components of a spatial entity in spherical surface
digital space are still interior, boundary and exterior. A point
PIE E is called an interior point of. Æ if N(p) eX,
otherwise p is called a border point of E . The set of all
interior points of Æ is called the interior of E and is
denoted as E .The set of all border points of. Æ is called the
border of E and is denoted as QE. The closure of. E is
denoted as E. The relationship between interior, closure and
boundary is as follows:
E"I 0E -
E'YóE-E
E=EI (BE
3.2 Spherical surface digital Jordan Therom
3.3 Topological paradox associated with definition of
adjacency in T
The classical Jordan curve theorem says that the complement of
a Jordan curve in the Euclidean in the Euclidean plan
R° consists of exactly two connectivity components. This
theorem is the basic topological property in vector space and it
?
would be preferable to keep it in the Z' raster space. So a
— 2
topological paradox in Z~ has arisen (figure 8). Kong and
2 . gn . en .
Rosenfeld has solved this problem in. Z^. if solved if the white
spels are defined as being 4-connected and black spels
8 - connected , or vice versa.
Figure 8. Topological paradox in raster space (from LI et al.
2000)
>
However, no one has discussed the topological paradox in T5.
In Figure 9, there are six black spels, one gray spel and some
white spels. The gray spel is surrounded by the six black spels.
If 12-adjacency is defined, the black spels are connected and
should form a closed line; however, this black line cannot
separate the central gray spel from the white spels. If
3-adjacency is defined, the black spels do separate the central
gray spel from the white spels; however, these black spels are
totally disconnected and thus no closed line has been formed by
the black spels in this case. So this leads to the topological
2
paradox in raster space I^, To deal with this paradox, the
3 - connected and black
spels 12 - connected
digital space, background and object have the different
connectedness. That is to say, the spatial entity in spherical
surface is defined as being 12-connected, but the background is
vice versa. In spherical surface
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