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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

- 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.

Census data, 1995

(U.S.A.)

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

Example: CountrymapofSouth America

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

- 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.

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

Point object in

raster form

Line object in

raster form

Plane object in

raster form

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

- 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

- 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

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

- 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: (..,..), (..,..), (..,..), ...

- 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: (..,..), (..,..), (..,..), ...

- 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

- 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

- 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

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

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

Raster-vector

conversion

Thinning

- Douglas-Peucker algorithm from 1973
- Input: chainp1, …, pnanderror

p1

pn

- 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-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

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

DP-algorithm

Optimal

- 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

- DP-algorithm may giveself-intersections in the output

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

DP-improved(i, j, )

Simp = DP-standard(i, j, )

V = set ofintersecting segmentsofSimpRepeat

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

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)

Raster

Vector

Vector

21

20

21

20

15

19

20

25

10

10

(Elevation) grid

Contourlinemodel

Triangulation(TIN; triangulatedirregular network)

- 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

20

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

- With explicit vertex and triangle representation

t2

w

t3

t1

t1

t2

t

t

v

u

u

w

t3

v

x, y-coordinatesandelevation

- 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

t

- With explicit vertex and triangle representation

w

w

t1

t2

t2

t1

t

v

u

t3

v

u

t3

w

t1

e1

t

u

e3

- Alternatively, edges have an explicit representation too

w

t1

t2

t

e1

e2

e2

e3

v

u

t3

- 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