[GAGIS]A Hierarchical Representation and Computation Scheme of Arbitrary-dimensional Geometrical Primitives Based on CGA

VGE
Key Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal University
Faculty of Geography
Department of Cartography and geography
information system
A Hierarchical Representation and
Computation Scheme of Arbitrary-dimensional
Geometrical Primitives Based on CGA
Wen Luo, Yong Hu, Zhaoyuan Yu∗, Linwang Yuan and Guonian Lü
Key Laboratory of Virtual Geographic Environment, Ministry of Education (VGE)
Nanjing Normal University, Faculty of Geography, Nanjing, China
GACSE 2016 - Heraklion, Crete, Greece
yuzhaoyuan@njnu.edu.cn
Doc. Wen Luo
Jun 28th, 2016

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information system
Index:
• Background
- Increasingly richer in geographical data source
- GIS data structures
- Geographical objects modeling
- Development of spatial data structures
• Theoretical basis
• MVTree structure
• Case Study
• Conclusions
1

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Complex
geographic scene
Increasingly richer in geographical data source
Remote-sensing
image
spatial-temporal
trajectoryVideo image Point cloud
Field-sequence
data
Big data

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GIS data structures
More unified data structures are needed for GIS data
representation
Higher
dimension
Space-time
expression

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A B
C
D
H
L
G
K
A'E F
I
J
B'
C'D'
E' F'
G'H'
I' J'
K'L'
L1
L2
L3
L4 L1'
L2'
L3'
L4'
K
S1'
S1
Points
Segments
Polygons
Polyhedron
Geographical object
Geographical objects modeling
The algebraic expressions of geometry hierarchy are needed.

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Family Tree of spatial data structures
Therefore, a well defined algebra system which can combined the
geometric and algebra is needed for GIS data representation

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Summary:
 High dimension representation
 Unified representing hierarchy
 Supporting of GIS computation
For the GIS data representation, several requires are needed:

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Summary:
This paper will proposed a GA-based data structure for GIS
primitives representation and computations, which is the basic
elements of GIS systems development.
GA is a potential tools for GIS data modeling. Based on the GA ,
a data structure used for data representation and computation
of GIS primitives is discussed in this contribution.

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Index:
• Background
• Theoretical basis
- Outer product-based geometric representation
- Grassmann structure of representation
• Case Study
• Conclusions
8

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Outer product-based geometric representation
For the given conformal 3-dimentional space n+1,1 , the 3-
dimentional points can be expressed with vector:
Given conformal points: pi, pj, pk, pl, pm , the 3D primitives can be
expressed by outer product:

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Grassmann structure of representation
The hierarchical structure of Outer Product-based representation
can be expressed as:
As shown in the figure, the dimensions are in accordance with
the Grassmann structures of objects. E.g. the line representation
can be generated by two points.
Point
Point pair
Circle
Sphere
Infinite point
Flat point
Line
Plane
Definition

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Index:
• Background
• Outer product and Grassmann structure
- Primitives representation of GIS data
- Multivector and MVTree structure
- Operations of MVTree structure
- Meet operation based on MVTree structure
• Case Study
• Conclusions
11

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Primitives representation of GIS data
For the GIS data, the primitives (GeoPri) can be represented with
GeoCarrier and GeoBounds:
(1)GeoCarrier is defined as the container or carrier of GeoPri, which
can be generated by outer product.
(2) GeoBounds is defined as a set of k-1-dimensional CGA
objects which represent the boundaries of GeoPri.
(3) GeoPrik can be written as: GeoPrik= GeoCarrierk{GeoBoundsk-1}

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Primitives representation of GIS data
GeoPri Expression Representation
Point GeoCarrier0 Pi
Segment GeoCarrier1{GeoBounds0} Sr=PiPje{Pi, Pj}
Polygon GeoCarrier2{GeoBounds1} Pgx=PiPjPke{Sr, Ss, St}
Polyhedron GeoCarrier3{GeoBounds2} PiPjPkPl{Pgx, Pgy, Pgp, Pgq}
Therefor, the GeoPris in 3D GIS representation can be list as:
From the table, it can be seen that the GeoPris are represented with
a hierarchical structure: the polyhedron represented based on
polygons; polygon represented based on segments; and segment
represented based on points.

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Multivector and MVTree structure
The GIS primitives with different dimensions can be combined
with multivector:
ObjMv = Obj.Points  Obj.Geo
= Obj.Points  Obj.Lines
<Lines.Pointsindex>  Obj.Polygons
<Polygons.Linesindex> 
Obj.Polyhedrons<Polyhedrons.polygonindex>
s9
s2s1
l1
l4
l2
l3
l18
l15
l7l14
p8
p1
p2
p9
p10
p3
p5
p6
p7
p4
P1=x1e1+y1e2+z1e3
P2=x1e1+y1e2+z1e3
P10=x10e1+y10e2+z10e3
Points expression Geometric structure expression
𝑀 = 𝑂𝑏𝑗. 𝑃𝑜𝑖𝑛𝑡𝑠 ⊕ 𝑂𝑏𝑗. 𝐿𝑖𝑛𝑒𝑠 ⊕ 𝑂𝑏𝑗. 𝑃𝑙𝑎𝑛𝑒𝑠 ⊕
⋯ ⊕ 𝑂𝑏𝑗. 𝑛ℎ𝑦𝑝𝑒𝑟𝑠𝑝ℎ𝑒𝑟𝑒𝑠
The multivector-based representation is not suitable for large
scale GIS computation.

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Multivector and MVTree structure
A tree like data structure MVTree can be defined here. Any node of
MVTree, it meet:
(1) when dim 𝑇𝐴 = 𝑣 > 0, 𝑇𝐴. 𝐶ℎ𝑖𝑙𝑑 𝑖 = 𝐺𝑒𝑜𝑃𝑟𝑖𝑣 𝑣−1, 1 ≤ 𝑖 ≤ 𝑚,
𝑚 is the number of the GeoPris in GeoBounds;
(2) when dim 𝑇𝐴 = 0, 𝑇𝐴. 𝐶ℎ𝑖𝑙𝑑 𝑖 = 𝑁𝑈𝐿𝐿,and this node will
always be the leaf node.

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Operations of MVTree structure
(1) Accessing nodes by child index
TA0
TA21
TA.Child(i) means the ith child of
TA, it can be also written as TAi .
and TAij means the jth child of TAi .
TA

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TA
(2) Accessing nodes by level index
All the nodes in level i of TA can
be accessed by TA.Level(i).
TA.Level(1)
TA.Level(3)

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TA
TA.SubTree(2)
(3) Accessing subtrees
The subtrees of node TA were defined as the
collections of one of its child node and all the
descendants of this child node.

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TA
TA.Value = TA.Child(0)  TA.Leaf(1)
(3) Accessing subtrees
(4) Accessing of nodes value
According to the OP-based representation, the
value of nodes can be recalled by their child nodes
by the equation:

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Meet operation based on MVTree structure
According to the hierarchical structure of the MVTree, the meet
operator of two MVTree TA and TB can be defined as the hierarchical
judgment structures.
where the symbol  is a judgment mark that only when the equation
in  is meet can the equation on the right of ⊨ be calculated.

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Meet operation based on MVTree structure
For the two given triangles (ABC) and (DEF), the hierarchical
judgment structures can be expressed as:
Because all the computations in the symbol  are judged first and a
great many of computations can be omitted if the judgements are
not meet. By using this strategy, the meet computation can be
implemented with smaller complexity.
Omitt
ed

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Index:
• Background
• Case Study
- Topological Relationship Computation
- Intersection between triangles
• Conclusions
22

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Topological Relationship Computation
Based on the computability and the hierarchical computing
structures of MVTree, a deductive approach can be
proposed:
Interior boundary External
ExternalboundaryInterior
Triangles
T2
T1
9-IM model
T1 T2
Structures
analysis
Hierarchical
structures
Topological
Relationship
GA model

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According to the hierarchical judgment computing framework,
the meet of (ABC) and (DEF) can be computed as:
MVTree-based representation of triangles
Hierarchy computation of meet

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Then, the hierarchical structures of meet computing can be
abstracted as a topological JudgeTree. The meaning of the
nodes in JudgeTree is shown as below:
Construction of topology judgetree
∈ { }
∈ {Connection, 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛, 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡}

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Relation Amount
DC 3
EC 8
PO 10
EQ 1
TPP 3
TPPI 3
NTPP 1
NTPPI 1
Total 30
By analyzing the topological JudgeTree, the topological
relations can be extracted (totally 30 relations):
DC DC DC EC EC EC EC EC
EC EC EC PO PO PO PO PO
PO PO PO PO PO EQ TPP TPP
TPP TPPI TPPI TPP NTPP NTPPI

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0
5
10
15
20
25
30
3
8
10
1
3 3
1 1
30
1
5
7
1 1 1 1 1
18
Topological relations comparison
GA model
9-IM
Compared with the 9-IM model, because of the dimensional
hierarchical structure, additional topological relations (e.g. in
DC, EC and PO relations) can be distinguished.

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Intersection between triangles
For the given triangles Ti and Tj , the intersection of them can
be solved by the meet operator:
Because the sign of the square of the meet operator can be
used to determine the intersection/touch/disjoint relations.
The judgement operation  can be solved directly by:

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The DTIN (Delaunay-Triangulated Irregular Network ) data can
be represented with the collection of triangles .

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The DTIN data of a 3D ice model was used for the data source.
We chose five time points to illuminate the computing process.
The result is shown in the right figure.
Label Data
T1 33,800KaBP
T2 33,550KaBP
T3 33,300KaBP
T4 33,400KaBP
T5 33,200KaBP
DTIN data set: 3D dynamical ice
model of the Antarctica from
34,000 kaBP to 33,200 kaBP
Intersection of v1 and v2 Intersection of v2 and v3
Intersection of v3 and v4 Intersection of v4 and v5

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Take the Guigue-Devillers method and Möller method as the
comparison. The number of the intersected segments was
recorded:
Type Method v1-v2 v2-v3 V3-v4 V4-v5
Number of
intersected segments
Our method 2213 3471 3634 2583
Guigue-Devillers 5329 7289 7574 5778
Möller 5178 7155 7431 5671
Number of redundant
Our method 574 1652 1862 1046
Möller 3539 5336 5659 4134
Number of available
Our method 1639 1819 1772 1537
Möller 1639 1819 1772 1537

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Index:
• Background
• Case Study
• Conclusions and further researches
32

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Conclusions
 The geometrical primitives can be represented by outer
product in a hierarchical structure.
 The proposed MVTree structure can represent the hierarchical
structure in a unified way, and computed in a hierarchical
judgment structure.
 The MVTree structure is geometrically meaningful and has the
potential power to support complex GIS analysis.

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Further researches
 The construction of new GA-based multidimensional unified
data model of GIS.
 The introduction of some optimization methods like GA-
oriented FPGA and Gaalop.

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References
[1]D. Hildenbrand. Foundations of Geometric Algebra Computing. Springer, 2013.
[2]L. Yuan L , Z. Yu, W. Luo, et al. Multidimensional-unified topological relations computation: a
hierarchical geometric algebra-based approach[J]. International Journal of Geographical
Information Science, 2014, 28(12): 2435–2455.
[3]L. Yuan, Z. Yu, S. Chen, et al. CAUSTA: Clifford Algebra-based Unified Spatio-Temporal Analysis[J].
Transactions in GIS, 2010, 14(S1): 59–83.
[4]L. Yuan, Z. Yu, W. Luo, et al. Geometric Algebra for Multidimension-Unified Geographical
Information System. AACA, 2013, 23(2): 497–518.
[5]M. F. Goodchild. Citizens as sensors: the world of volunteered geography. GeoJournal, 2007,
69(4):211-221.
[6]R. Abdul and M. Pilouk. Spatial Data Modelling for 3D GIS. Springer-Verlag, 2007.
[7]Silvia Franchini, Antonio Gentile, Filippo Sorbello, Giorgio Vassallo, and Salvatore Vitabile. An
embedded, FPGA-based computer graphics coprocessor with native geometric algebra support.
Integration, the VLSI Journal, 2009, 42(3):346-355.
[8]Z. Yu, W. Luo, Y. Hu et al. Change detection for 3D vector data: a CGA-based Delaunay–TIN
intersection approach. International Journal of Geographical Information Science, 2015, 29(12):
2328–2347.

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The Video can be watched in YouTube by searching “GAGIS”;
The slide can be download in SlideShare by searching “GAGIS”;
yuzhaoyuan@njnu.edu.cn

[GAGIS]A Hierarchical Representation and Computation Scheme of Arbitrary-dimensional Geometrical Primitives Based on CGA

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