International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 Inter 
  
  
(b) The relation between raster model and real world 
Figure 1. Relation between real world and discrete computation 
This paper is concerned with spherical surface digital topology. 
Digital topology provides a sound mathematical basis for 
various image-processing applications including surface 
detection, border tracking, and thinning in 2D Euclidean space. 
We often use voxel representation to describe objects on a 
computer. Specifically, spherical surface digital space is 
partitioned into.unit triangles. In this representation, an object in 
spherical surface is described by an array of bits. In this way, a 
spherical surface digital object can be defined as an array 
augmented by a neighborhood structure. The emphasis of this 
paper is on the differences between planar and spherical surface 
digital topology. It is specific to the basic topology model on the 
surface of an earth, and thus, the ellipsoidal nature of the earth 
and its vertical dimension are not considered. 
The paper is organized as follows. Next section presents the 
definitions of spherical surface digital space based on manifold. 
[n Section 3, the basic topology model of spherical surface 
digital space is discussed. In the end, the discussions and the 
future works are given. 
2. THE DEFINITION OF SPHERICAL SURFACE 
DIGITAL SPACE BASED ON MANIFOLD 
Regular grid sampling structures in the plane are a common 
spatial framework for many applications. Constructing grids 
with desirable properties such as equality of area and shape is 
more difficult on a sphere (White et al. 1998). To deal with the 
problems on the Earth conveniently, it is necessary to construct 
a similar regular mesh structure as a common spatial framework 
for spherical surface just as planar. Such similar regular mesh 
system is named as spherical surface digital space, which is the 
digitization of the spherical surface. That is, spherical surface 
can be described with discrete point sample in spherical surface 
digital space. Therefore, it is necessary to subdivide the 
spherical surface according to its characteristics. There are three 
steps to get the sphere digital space just as follows. 
2.1 Initial partition of the spherical surface 
The Platonic solids are reasonable starting points for a spherical 
subdivision (shown in Figure 2). Three of the five polyhedrons 
have triangular faces, such as the tetrahedron (four faces), the 
octahedron (eight faces), and the icosahedron (20 faces). The 
other Platonic solids are the cube (six faces) and the pentagonal 
dodecahedron (12 faces) The icosahedron has the greatest 
number of initial faces, and would therefore show the least 
distortion in the subdivision. However, the larger number of 
faces makes it somewhat harder to deal with the problems 
through the borders of the initial faces. In a word, the sphere is 
more easily covered by triangles and the triangles of the initial 
partition need not be equilateral. Distortion could be decreased 
considerably by dividing each equilateral triangular side of an 
initial Platonic figure into equivalent scalene triangles ( White et 
al. 1998). 
  
n3 
  
  
If th 
base: 
8 x 
discr 
Figure 2. Platonic solids and it’s spherical subdivision diii 
(White et al. 1992) Syste 
coor 
The octahedron has more distortion, but it has the advantage Space 
that its faces and vertices map to the important global features: fe 
meridians, the equator, and the poles (Goodchild and Shiren sphet 
1992). Therefore, in this paper octahedron is selected as - 
common initial partition in which eight base triangles are Gold 
produced. | 
2.2 Subdivision of triangular cells 
There are several ways to hierarchically subdivide an equilateral 
triangle such as quaternary subdivision and binary subdivision 
(shown in Figure 3). All of these are subject to distortion when 
transferred to the spherical surface. Different decisions will 
have different effects on the uniformity of shape and size of 
cells within a given level of the hierarchy, as well as on the ease 
of calculation. Here, the quaternary subdivision is selected, in 
which a triangle is subdivided by joining the midpoints of each 
side with a new edge, to create four sub-triangles. ba 
= - To ev 
curve 
sub-tr 
betwe 
surfac 
resolu 
index: 
(8) ©) Barth: 
1991) 
contin 
Figure 3. Quaternary subdivision (a) and binary subdivision space 
(b) multi- 
the ini 
The quaternary subdivision is a good compromise. It is equal 
relatively easy to work with, and non-distortifg on the plane: a 
planar equilateral triangle is: divided into four equilateral 
triangles. But a spherical base triangle may be divided into four A 
equivalent triangles. The result of subdivision based on f S 
octahedron with quaternary subdivision is as follows in Figure 4 / 
(Dutton 1996). / 
/ 
2 jet 
E Ei. 
(a) lev 
Figui 
Planar 
surface 
Figure 4. The result of subdivision based on octahedron surface 
(Dutton 1996) space. 
surface 
2.3 The definition of sphere digital space based on manifold no hon 
can be 
Manifold is the extension of Euclidean just because every point Althou 
in manifold has a homeomorphism of an open set in Euclidean. approx 
So local coordinates system can be set up for every point in continu 
manifold. It seems that manifold is a result plastered with many 2001). 
Euclidean spaces. It can be proved that sphere is a 2-dimension 
smooth manifold (Evidence omitted).