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Representation of spatial dataPowerPoint Presentation

Representation of spatial data

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### Representation of spatial data

GIS thematic layers, raster and vector, conversion, subdivision representation, continuous data: contours, DEMs, TINs

Thematic map layers

- Separatestorageof data according to theme: map layers
- GIS typically use tens to hundreds of map layers
- For example: municipality borders, land use, cadastral boundaries, water pipes, churches, etc.

Geometry, topologyand attributes

- Geometry: coordinates
- Topology: adjacency relations of objects
- Attributes: properties, values

Example: CountrymapofSouth America

Geometry: coordinatesof the bordersTopology: whichcountries border which Attributes: namesofcountries, population, etc.

Representationof geometry

- Twomain approaches: raster and vector
- Can also be mixed in a GIS, anymap layer
- Conversion raster-vector and vice versa possible
- Representationdepends ontype of data, way ofacquisition, desired operations, etc.

Rasterstructure

- Division of space intoequal-size cells (squares, pixels)
- Themegives cellsavalue (nominal, ordinal, interval, ratio, vector, …)
- Cellsshould not contain any furtherspatialinformation (more detail)

Data in raster form

Point object in

raster form

Line object in

raster form

Plane object in

raster form

Raster maps

Simple structure

Simple operations

Obtained after scanning, remote sensing

Less suitablefor point and line objects: representationdoes not follow intuition

Networkanalysisdifficult

Not adaptive: no difference in detail possible in different regions

Eitherexpensive in memory, orlittle precision

Not obtained after digitizing

Raster: pros and consRaster: memory reduction

- Run-length encoding: no 2-dim array but coding start pixel withvalue and lengthofrun
- Block encoding: 2-dim version
- Disadvantage: makes structureand operationsmuch morecomplex

(34,67) forest 9

(34,67) forest 4,6

Vectorstructure

- Objectsstored as points, linesandareas
- Points have coordinates; linesconnect points; areas are delimited by lines
- Attributesare stored with the objects (point, line orareal)

Elegant structure; fits withboth point, lineand areal objects

Small storage consumption

Precise

Adaptive: additional controlpointspossible

Network and clusteranalysispossible

Obtained after digitizing

Relatively complex

Map overlay and buffer computation complex

Vector: pros and consVector representation of a region

- Not necessarily simply-connected:
- NL has islands
- NL has holes(Baarle-Nassau / Baarle-Hertog); there are even regions in these holes

Subdivisions: spaghetti model

- Every chain is represented by a list with coordinate pairs
- Splitnodesare doubly stored
- Areas are not present explicitly

C1

C2

C5

C4

C3

C6

C1: (..,..), (..,..), (..,..), ...

C2: (..,..), (..,..), (..,..), ...

C3: (..,..), (..,..), (..,..), ...

Subdivisions: polygon ringstructure

- Every area is represented by a list with coordinate pairs
- Controlpointsare doublystored
- Neighbor areas aredifficult to determine
- Consistency is difficult to maintain

P1

P2

P3

P1: (..,..), (..,..), (..,..), ...

P2: (..,..), (..,..), (..,..), ...

P3: (..,..), (..,..), (..,..), ...

Subdivisions: topological structure

- Nodes are objectswith coordinates
- Edges are connectionsof nodes
- Sequences of edges along polygon boundaries form cycles
- Polygons are objects that can access their boundaries

Doubly-connected edge list

Subdivisions: topological structure

- Edges are split into directed half-edges
- Half-edges have pointers to
- Twin half-edge
- Origin vertex
- Next and Prev half-edges of incident polygon
- Incident polygon

- Polygons have pointers to half-edges, one in each bounding cycle

Origin

polygon

Twin

Prev

Next

polygon

Subdivisions: topologicalchain structure

- Splitting nodes are objectswith coordinates
- Chains are connections of splitting nodes and contain zero or morenodes with coordinates
- Sequences of chains along polygon boundaries formcycles
- Polygonsare objects that can access their boundaries

half-chains

Doubly-connected chain list

Memory Duplication Polygon Topology

retrieve retrieve

VectorstructuresSpaghetti ++ + -- -Polygon ring - -- ++ -DC edge list -- ++ - +DC chain list ++ ++ + ++

Raster-vector conversion

E.g. for data integration

- Vector-to-raster: Like in computer graphics: scan-conversionof lines, etc.

- Raster-to-vector: Consider pixel sides between pixels with different values as boundary and put in vectorrepresentation Thinning, line simplification

DP-algorithm

- Draw line segmentbetween first and last point
- If all points in between are within error: ready
- Otherwise, determine farthest point and recursively continueon the part until farthest point and the part afterfarthest point

DP-algorithm

DP-standard(i, j, )

Determine farthest point pkbetween piandpj

Ifdistance(pk, pi pj) > then DP-standard(i, k, )

DP-standard(k, j, )

Return the concatenationof

the simplifications

Properties of the DP-algorithm

- DP-algorithm does not minimizethe number of points in the simplification

DP-algorithm

Optimal

Properties of the DP-algorithm

- Determining farthest point takesO(n) time
- Whole algorithm takesT(n) = T(m) + T(n-m+1) + O(n),T(2) = O(1) time,splitting in m and n-m+1 points
- “Fair” split givesO(n log n) time
- Worst case gives quadratic time

Properties of the DP-algorithm

- DP-algorithm may giveself-intersections in the output

Solution: test output forself-intersectionsand continue adding controlpoints if necessary

Improved DP-algorithm

DP-improved(i, j, )

Simp = DP-standard(i, j, )

V = set ofintersecting segmentsofSimpRepeat

For all segments s VRefine(s) in Simp Do 1 refinementà la DP by adding the farthest pointV = set of intersecting segmentsofSimpUntilV is empty

Continuous data representation

Digital Elevation Model (DEM)

- Data on interval or ratio measurement scale
- Data values of points near by will usually be not very different
- Representation is necessarily an approximation:finite representation of information with infinite detail
- Raster (1x) or vector (2x)

Elevation models

Raster

Vector

Vector

21

20

21

20

15

19

20

25

10

10

(Elevation) grid

Contourlinemodel

Triangulation(TIN; triangulatedirregular network)

Elevation models

- Contourmodel well-suited for visualisation, not for representation or storage
- Interpretations grid:- elevation whole cel: not a continuous model- elevation middle cel: interpolation needed; how?
- Advantage grid: simple storage, operations simple too
- Advantage TIN: more efficient in storage, adaptive

= 19.5

4

Interpolation for grid20

20

18

18

22

18

18

22

18

20

Linear interpolation; saddle point problem

18

22

20

20

18

18

22

22

18

18

Linear interpolation;additional point

Non-linear

interpolation

Topological TIN structure

- With explicit vertex and triangle representation

t2

w

t3

t1

t1

t2

t

t

v

u

u

w

t3

v

x, y-coordinatesandelevation

Topological TIN structure

- With explicit vertex and triangle representation

t2

w

t3

t1

t1

t2

t

t

v

u

u

w

t3

v

Because t1 has pointers to two the same vertices as t, we can determine their shared edge, even though it is not represented explicitly

Topological TIN structure

- With explicit vertex and triangle representation

w

w

t1

t2

t2

t1

t

v

u

t3

v

u

t3

t1

e1

t

u

e3

Topological TIN structure- Alternatively, edges have an explicit representation too

w

t1

t2

t

e1

e2

e2

e3

v

u

t3

Summary representation

- Objects have geometry and attributes, at least the attributes are in a database
- Geometry can be stored in raster or vector form; each has advantages and disadvantages
- Important geometric types of representations are those for subdivisions and for elevation models
- For subdivisions, the doubly-connected chain list is the most suitable structure
- For elevation models, grids or TINs are most useful

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