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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