[gagis]a hierarchical representation and computation scheme of arbitrary-dimensional geometrical...

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

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 LuoJun 28th, 2016




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




2

Complex geographic scene

Increasingly richer in geographical data source

Remote-sensing image

spatial-temporal trajectoryVideo image Point cloud

Field-sequence data

Big data




3

GIS data structures

More unified data structures are needed for GIS data representation

Higher dimension

Space-timeexpression




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.




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




6

Summary:

High dimension representation

Unified representing hierarchy

Supporting of GIS computation

For the GIS data representation, several requires are needed:




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.




Index:

• Background

• Theoretical basis

- Outer product-based geometric representation- Grassmann structure of representation


• Case Study

• Conclusions

8




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:




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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




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




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}




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Primitives representation of GIS data

GeoPri Expression Representation

Point GeoCarrier0 Pi

Segment GeoCarrier1{GeoBounds0} Sr=PiPje{Pi, Pj}

Polygon GeoCarrier2{GeoBounds1} Pgx=PiPjPke{Sr, Ss, St}

Polyhedron GeoCarrier3{GeoBounds2} PiPjPkPl{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

l4l2

l3

l18l15

l7l14

p8

p1p2

p9

p10

p3

p5p6

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.




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.




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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




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(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)




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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.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:




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.




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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




Index:

• Background



• Case Study

- Topological Relationship Computation- Intersection between triangles

• Conclusions

22




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




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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 beabstracted as a topological JudgeTree. The meaning of thenodes 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 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





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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.





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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:




29

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





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





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Take the Guigue-Devillers method and Möller method as thecomparison. The number of the intersected segments wasrecorded:


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


Möller 3539 5336 5659 4134

Number of available intersected segments

Our method 1639 1819 1772 1537


Möller 1639 1819 1772 1537




Index:

• Background



• Case Study

• Conclusions and further researches

32




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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.




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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.




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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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