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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)

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

20+18+18+22

= 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

w

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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