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).