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 [email protected]Doc. Wen Luo Jun 28 th , 2016
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[GAGIS]A Hierarchical Representation and Computation Scheme of Arbitrary-dimensional Geometrical Primitives Based on CGA
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VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
A Hierarchical Representation and Computation Scheme of Arbitrary-dimensional
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation 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
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
2
Complex geographic scene
Increasingly richer in geographical data source
Remote-sensing image
spatial-temporal trajectoryVideo image Point cloud
Field-sequence data
Big data
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
3
GIS data structures
More unified data structures are needed for GIS data representation
Higher dimension
Space-timeexpression
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
4
A B
CD
H
LG
K
A'E F
IJ
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.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
5
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
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
6
Summary:
High dimension representation
Unified representing hierarchy
Supporting of GIS computation
For the GIS data representation, several requires are needed:
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
7
Summary:
This paper will proposed a GA-based data structure for GISprimitives representation and computations, which is the basicelements 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 computationof GIS primitives is discussed in this contribution.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
Index:
• Background
• Theoretical basis
- Outer product-based geometric representation- Grassmann structure of representation
• MVTree structure
• Case Study
• Conclusions
8
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
9
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:
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
10
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
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
Index:
• Background
• Outer product and Grassmann structure
• MVTree structure
- Primitives representation of GIS data- Multivector and MVTree structure - Operations of MVTree structure - Meet operation based on MVTree structure
• Case Study
• Conclusions
11
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
12
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}
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
The multivector-based representation is not suitable for large scale GIS computation.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
15
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 = 𝐺𝑒𝑜𝑃𝑟𝑖𝑣𝑣−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.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
16
Operations of MVTree structure
(1) Accessing nodes by child indexTA0
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
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
17
Operations of MVTree structure
(1) Accessing nodes by child indexTA
(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)
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
18
Operations of MVTree structure
(1) Accessing nodes by child indexTA
(2) Accessing nodes by level index
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.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
19
Operations of MVTree structure
(1) Accessing nodes by child indexTA
(2) Accessing nodes by level index
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:
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
20
Meet operation based on MVTree structure
According to the hierarchical structure of the MVTree, the meetoperator of two MVTree TA and TB can be defined as the hierarchicaljudgment structures.
where the symbol is a judgment mark that only when the equationin is meet can the equation on the right of ⊨ be calculated.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
21
Meet operation based on MVTree structure
For the two given triangles (ABC) and (DEF), the hierarchicaljudgment structures can be expressed as:
Because all the computations in the symbol are judged first and agreat many of computations can be omitted if the judgements arenot meet. By using this strategy, the meet computation can beimplemented with smaller complexity.
Omitted
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
Index:
• Background
• Outer product and Grassmann structure
• MVTree structure
• Case Study
- Topological Relationship Computation- Intersection between triangles
• Conclusions
22
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
23
Topological Relationship Computation
Based on the computability and the hierarchical computingstructures of MVTree, a deductive approach can beproposed:
Interior boundary External
Exte
rnal
boundary
Inte
rior
Triangles
T2
T1
9-IM model
T1 T2
Structures analysis
Hierarchical structures
Topological Relationship
GA model
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
24
Topological Relationship Computation
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
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
25
Then, the hierarchical structures of meet computing can beabstracted as a topological JudgeTree. The meaning of thenodes in JudgeTree is shown as below:
Construction of topology judgetree
∈ { }∈ {Connection, 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛, 𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡}
Topological Relationship Computation
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
26
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 topologicalrelations 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
Topological Relationship Computation
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
27
0
5
10
15
20
25
30
3
810
13 3
1 1
30
15
7
1 1 1 1 1
18
Topological relations comparison
GA model9-IM
Compared with the 9-IM model, because of the dimensionalhierarchical structure, additional topological relations (e.g. inDC, EC and PO relations) can be distinguished.
Topological Relationship Computation
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
28
Intersection between triangles
For the given triangles Ti and Tj , the intersection of them canbe solved by the meet operator:
Because the sign of the square of the meet operator can beused to determine the intersection/touch/disjoint relations.The judgement operation can be solved directly by:
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
29
The DTIN (Delaunay-Triangulated Irregular Network ) data canbe represented with the collection of triangles .
Intersection between triangles
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
30
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
Intersection between triangles
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
31
Take the Guigue-Devillers method and Möller method as thecomparison. The number of the intersected segments wasrecorded:
Intersection between triangles
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 intersected segments
Our method 574 1652 1862 1046
Guigue-Devillers 3690 5470 5802 4241
Möller 3539 5336 5659 4134
Number of available intersected segments
Our method 1639 1819 1772 1537
Guigue-Devillers 1639 1819 1772 1537
Möller 1639 1819 1772 1537
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
Index:
• Background
• Outer product and Grassmann structure
• MVTree structure
• Case Study
• Conclusions and further researches
32
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
33
Conclusions
The geometrical primitives can be represented by outerproduct in a hierarchical structure.
The proposed MVTree structure can represent the hierarchicalstructure in a unified way, and computed in a hierarchicaljudgment structure.
The MVTree structure is geometrically meaningful and has thepotential power to support complex GIS analysis.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
34
Further researches
The construction of new GA-based multidimensional unifieddata model of GIS.
The introduction of some optimization methods like GA-oriented FPGA and Gaalop.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
35
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.
VGEKey Laboratory of Virtual Geographic Environment
Ministry of Education
Nanjing Normal UniversityFaculty of GeographyDepartment of Cartography and geographyinformation system
36
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