SPATIAL SUBDIVISION AND CODING OF A GLOBAL THREE-DIMENSIONAL GRID: SPHEOID DEGENERATED-OCTREE GRID YU Jie-qing
WU Li-xin
Academy of Disaster Reduction and Emergency Management, Ministry of Civil Affairs, Ministry of Education (Beijing Normal University), Beijing 100875, P.R. China,
[email protected]
avoided or ignored if the scale of objects and interesting
ABSTRACT Map projections had brought a lot of problems to regional and global spatial data organization and representation,
area is small, but they are inevitable and unelectable to the large or global scale objects representation.
such as distortion, geometric fissure and space metric error.
Fortunately, Global Discrete Grid (GDG) gives a feasible
Fortunately, Discrete Global Grid (DGG) gave a feasible
way to solve the problems. It takes recursively subdivision
solution, but it was only confined to the Earth surface, not
on the earth surface so that the metric space transformation
reached to the Earth inside and outside, which was also
is avoided and large scale objects can be represented in
very important to human activity. In this paper, a global
global multiresolution discrete grids. However, GDG is
three-dimensional
Spheoid
confined to the surface, not reaches to the interior and
Degenerated Octree Grid (SDOG), was presented, and two
exterior of the surface. Actually, human activity had
kinds of coding methods for SDOG, called as SDZ and
extended to the whole earth space, including the inside and
MDZ, were proposed. The properties of SDOG were
the outside of earth surface. This paper is to present a
discussed and the performance of SDZ and MDZ was
global 3D grid, called as Spheoid Degenerated-Octree
compared with QuaPA experimentally. It revealed that
Grid(SDOG), which can be served as a 3D grid framework
SDOG is a 3D grid with properties of global continuity,
for global spatial data organization and representation.
(3D)
grid,
called
as
multi-hierarchy, similar shape and approximate size, and
SDOG SUBDIVISION
can serve well for global spatial data organizing and
Goodchild(1992) and Kimerling(1999) had proposed some
representing; SDZ and MDZ are high performance coding
principles that general GDGs required. The most important
methods, and MDZ is a nicer choice for SDOG coding.
ones were global continuity, multi-hierarchy at different
Key Words: Spheoid Degenerated Octree Grid(SDOG),
levels, equal grid shape at any level, equal grid area at the
three-dimensional grid, global grid, grid coding
same level.
Whereas, some requirements had been
proved being difficult to meet at the same time. In fact, an excellent grid system was a balance between them(Zhao,
INTRODUCTION
2007), and global continuity, multi-hierarchy, similar
Early theories and methodologies for geographical objects
shape and approximate size had actually been the key
representation are generally based on map projection, and
principles for a good grid system.
objects on the Earth surface need to be projected onto the Cartesian plane before modeling and analyzing. However, map projection brings with many problems. One of them is projection distortion in that area, distance and angle can not be all controlled at the same time; another is geometric fissure that appeared on the joint of projection zoning; A third is space metric error, as distance between two points on the spherical surface is a great circle arc, while a straight line on Cartesian plane. These problems may be
978-1-4244-3395-7/09/$25.00 ©2009 IEEE
SDOG is a global 3D grid, which takes virtual spheoid as the division target. A virtual spheoid is a normal spheoid with a variable radius, whose space may include sphere, aerosphere or even outer space. The core idea of subdivision is to divide virtual spheoid in octree and merge the
degenerated
degenerated-octree
grids
into
one,
subdivision.
which
Details
is on
called SDOG
subdivision may be separated into four steps.
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IGARSS 2009
Step 1: Choosing a suitable radius for virtual spheoid
other octants subdivision.
according to our interesting areas. If interesting area is
Fig.2 gave the visualization of SDOG on one octant from
confined to the earth surface or under the surface,
level 1 to level 4.
approximate spheoid of reference ellipsoid defined by
SDOG PROPERTIES
IUGG is better. But if interesting area contains aerosphere, SDOG
a spheoid including aerosphere is required. Step 2: Dividing virtual spheoid into eight octants equally in equator plane, 0 o meridian plane and 90 o meridian plane.
on
the
surface
is
Degenerated-Quadtree Grid(DQG).
actually
acted
as
Cui (2007) gave the
maximum-minimum area radio of it. As showed in Fig.3, the radio increases as subdivision level grown, but the
Step 3: Taking one octant as the basic object and
increasing speed is slowed down, at last it converges at
subdividing it in degenerated octree. As showed in Fig.1a,
2.23, while QTM converges at 1.8.
octant was divided by three curved faces, red cone plane, green plane, and blue sphere. Each curved face was defined by spheoid core and midpoints of longitude line, latitude line or radius line. In other words, each intersecting point on the new curved face was a midpoint. It should be pointed out that grid “3” don’t need to be divided by red cone plane and green plane, and the same to grid “2”. After the first level division, octant was divided into four grids.
Fig.3 The maximum-minimum area ratio of SDOG
We called grid “0” and grid “1” as Normal Grid(NG), grid “2”
as
Latitude-degenerated
Grid(LG),
grid
“3”
as
Volume of SDOG can be calculated by the following
Sphere-degenerated Grid(SG). SG, LG and NG would be
formula. Fig.4 was the result of
maximum-minimum
three basic grids in further SDOG subdivision.
volume radio of SDOG in forms of graph. Just like the area, the ratio of maximum-minimum volume is also grown with subdivision level, but it finally converges at 8.89.
V
( r23 r13 )(sin M 2 sin M1 ) ˄O 2 O1˅ 3
(1)
V ü volume of SDOG; Ȝ 2 ü maximal longitude; Ȝ 1 ü b) LG
c) NG
minimal longitude; r 2 ü maximal radius; r 1 ü minimal
Fig. 1 The subdivision on three basic grids SG
is
actually
a
smaller
sized
octant,
its
radius; ij 1 ü maximal latitude; ij ü minimal latitude. further
Max Volume / Min Volume
a) Octant or SG
subdivision can follow that of octant. Further subdivision of LG and SG was showed in Fig.1b and Fig.1c. Similar to octant, it was divided by three curved faces too, with each intersecting point being a midpoint. In LG subdivision, green plane did not divide the two upper grids.
Fig.4 The maximum-minimum ratio of SDOG volume Both
maximum-minimum
area
ratio
and
maximum-minimum volume ratio of SDOG converge at a small value, so the size of SDOG is approximate. Except a) level 1
b) level 2
c) level 3
for the grids near the pole and the spheoid core(LGs or
d) level 4
SGs), they are all NGs, and have the similar shape. Beside,
Fig.2 The visualization of SDOG Step 4: Repeating step 3 to finish the higher level grid and
SDOG covers the whole spheoid space without any seam, and its father grid can cover their child grids exactly. It is
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global continuity and multi-hierarchy. Therefore, SDOG is an excellent global 3D grid, with properties of global continuity, multi-hierarchy, similar shape and approximate size. Fig.6 Coding on the 1 st level SDOG
SDOG CODING Grid code is the coordinate of grid and used to local the gird position through code inversion. It is essentially a procedure of traversal with Space Filling Curve(SFC). Z curve is a high performance SFC that uesd frequently. However, normal Z curve requires that parent grid own eight children exactly. But once subdivision on the parent grid of SDOG will produce 4 to 8 sub-grids. Obviously, Z curve should be improved if it was used to fill the SDOG. A feasible way is to follow the way of subdivision. Fig.5a was a normal Z curve filling, while F ig.5b was an improved Z curve filling, called as Degenerated Z (DZ) curve. Numbers near by the nodes were their codes. When grid “2” and grid “3” were merged into one grid, the corresponding codes were merged into one code too. Here we preserved the small value code and abandoned the big one.
SDZ and MDZ has a same result on the first level grid coding, as showed in Fig.6. Fig.7 and Fig.8 were the results of SDZ and MDZ coding on the second level SDOG. Each subfigure in them was a part of the whole level 2 subdivision grids, for example, the subfigure of part “2” was a further subdivision on grid “2” in Fig.6. If all DZ curves in every part are joined together, they will be a complete DZ curve in SDZ coding. However, MDZ takes the different ways. Only grids on the same level share the same DZ curve. Fig.6 was a DZ curve, while Fig.8 was another DZ curve. Code of child grid is a code of parent grid added by a new code element in the back. For example, grid “2” in Fig.6 was the parent of grids “20”~ “26” in Fig.8c, so new code elements “0”~ “6” were added to code “2”.as children grid codes.. Both SDZ and MDZ were the grid code, and they needed to combine with quadrant in the front to become a complete SDOG code.
a) normal z curve Fig.5
b) degenerated z curve Z curve filling
a) part “0”
SDOG code is composed of quadrant and grid code.
b) part “1”
c) part “2”
d) part “4”
Fig.7 SDZ coding on the 2 nd level SDOG
Quadrant is the quadrant code of octant. Grid code is the filling codes of grids in an octant with DZ curve. According to the filling way, two different coding methods were
offered
here:
Single
hierarchical
Degenerated
Z-curve Filling (SDZ) coding and Multiply hierarchical Degenerated Z-curve Filling(MDZ) coding. SDZ took grids
a) part “0”
on fixed level as objects, filling the objects with DZ curve
b) part “1”
c) part “2”
Fig.8 MDZ coding on the 2
directly without any recursion. However, MDZ took grids
nd
d) part “4”
level SDOG
on multiply levels as objects. High level grid code was
SDZ code is valid only on the specific subdivision level.
based on low level grid, with adding a new code element to
Because codes of different level grids may fall into
it. Code element was either a number from 0 to 7, which
collision, such as gird “0” in Fig.6 and grid “0” in Fig.7.
was decided by the filling of grids at the same level with
Grid can not be located definitely through SDZ code
DZ curve.
except subdivision level is given, which indicated the size or resolution of coding grid. It is a static and single resolution code. However, MDZ takes the opposite. Codes of different level grids can coexist without any collision,
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and we can obtain the information of grid level (or size)
MDZ is a dynamic and multiresolution coding method,
from code length, as the value of code length is the value
while SDZ is a static and single resolution one. From the
of grid level. It is a dynamic and multiresolution code.
view
of
multi-scale
spatial
data
organization
and
representation, MDZ will be no doubt more excellent than
CODE COMPARISION QuaPA primary code (WU, 1999), shortened as QuaPA, is
SDZ for SDOG coding.
a high performance grid coding method. As to show the
CONCLUSIONS
performance of the grid coding, it was chosen to compare
A new global three-dimensional grid was present in this
with SDZ and MDZ on coding time, decoding time and
paper, called as SDOG, satisfied the principles that general
code length experimentally. The Results were showed in
global
Fig.9 to Fig.11.
multi-hierarchy, similar shape and approximate size. It can
grid
required,
i.e.,
global
continuity,
serve as 3D grid framework for global spatial data storage, indexing and representing, and can be applied in Digital Earth and Earth System Science areas. Two kinds of coding methods were advanced on SDOG coding, called as SDZ and MDZ. Both of them are high performance on coding and decoding time. SDZ is static Fig.9 Comparison on coding time
and single resolution code while MDZ is dynamic and multiresolution one. From the view of multi-scale spatial data organization and representation, MDZ is more excellent than SDZ and is a nicer choice for SDOG coding. REFERENCES [1]Cui MJ, Zhao XS. Tessellation and distortion analysis based
Fig.10 Comparision on decoding time
on
spherical
DQG
[J].
Geography
and
Geo-Information Science. vol. 23, no.6, pp.23-25,2007. [2]Goodchild M.F., Yang Shiren. A hierarchical data structure for global geographic information systems[J]. Computer vision and geographic image processing. vol.54, no.1, pp.31-44, 1992. [3] Kimerling A, Sahr K, White D, et al. Comparing Fig.11 Comparision on code length
geometrical properties of global grids[J]. Cartography
As showed in the figures, they all are a nearly linear
and Geographic Information Science. vol.26,
function
271-87,1999.
of
subdivision
level.
On
coding
time,
no.4, pp.
QuaPA
