3 Mapping of Spatial Data - TU Braunschweig · 2012-11-06 · •Mapping (visualization) of spatial...

3.1 Properties of Maps

3.2 Signatures, Text, Color

3.3 Geometric Generalization

3.4 Label and Symbol

Placement

3.5 Summary

Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 206

3 Mapping of Spatial Data

http://homepage.univie.ac.at/.../Janschitz_Text.pdf

• Mapping (visualization) of spatial objects by

transforming them into representation objects

(map objects)

• Map

– Generalized model in a reduced scale for representing

selected spatial information

http://www.xerokampos.eu/

• Topographic map

– Geometrically and positionally accurate

representation of

landscape objects drawn

to scale (topography,

water network, land

use, transportation

routes)

http://www.gps-touren.net/

• Thematic map

– Emphasis is on subject-specific information (geological

map, vegetation

map, biotope

http://www.lung.mv-regierung.de/

• Challenges

– Projection of the 3D surface on two dimensions

(paper, film, screen)

– Selection of the spatial objects and their attributes to be displayed

– Generalization of geometric and thematic properties (simplify, omit depending on scale)

– Exaggeration and displacement (e.g. river valley with roads and railway lines)

http://maps.google.de/

• Graticule

– Mapping the earth (sphere,

ellipsoid) to a plane

– Two tasks

• Conversion of geographic

coordinates (longitude

and latitude) to cartesian

coordinates (x, y; easting,

northing)

• Scaling of the map

www.klett.de

• Projections on

– Cone

– Plane

(azimuthal

projection)

– Cylinder

[HGM02]

• Desired properties

– Length preservation: with plane maps only limitedly

attainable (in certain directions or at certain points)

– Equivalent (equal area): preserve area measurements

shape, angle, and scale may be strongly distorted

– Conformal: important

for navigation in

shipping and air

transport

http://www.quadibloc.com/maps/

• Areas and angles can not be preserved at the

same time, therefore

– Compromise projections minimize overall distortion

• Example:

Conical

Projection

http://www.csiss.org/map-projections/

• Example:

Azimuthal

Projection

• Example:

Cylindrical

Projection

• The Gauss-Krüger coordinate system

– Used in Germany and Austria

– Cartesian coordinate system to represent small areas

– Divides the surface of the earth into zones

• Each 3° of longitude in width

• The zones are projected onto a cylinder with the earths

diameter

www.geogr.uni-jena.de/.../GEO142_3b.ppt http://homepage.univie.ac.at/.../Janschitz_Text.pdf

– Origin of the coordinate system:

intersection point of the central meridian and the

equator

– X coordinate from the

origin positive towards east,

y coordinate from the origin

positive towards north

– X and y values given in meters

http://www.gerhard-tropp.de/Troppo/gauss_krueger.html

www.geogr.uni-jena.de/.../GEO142_3b.ppt

– To avoid negative x coordinates the central meridian

is set to 500,000 m (false easting)

– Each projection zone (3°, 9°, ..., 177°) is identified

by a number (1, 3, …, 59)

– This number is placed prior to the x value

– Example: the Gauss-Krüger coordinates

x: 3,567,780.339 and y: 5,929,989.731

refer to the following location

x: 67,780.339 m East of the 9° meridian

y: 5,929,989.731 m North of the equator

• Structure of maps

– Body of the map presentation of the actual map

– Map margin contains name, map, scale, legend

– Map frame borders map, contains numbering of the respective coordinate system

http://www.apat.gov.it/

• Graphical design elements of maps are

– Symbols

(signatures)

• Points

• Lines

• Areas

– Text

http://lehrer.schule.at/Ecole/

• The use of point, line, area symbols depends on

– Spatial scale of a map

– Purpose of the map

– Convention

• Point signatures are (composite) symbols for the representation of spatial objects with point geometry

[HGM02]

• Line signatures represent objects with line geometry

• Area signatures represent objects with area geometry

• Texts are needed in various fonts for the labeling of maps

[HGM02]

• Color

– Important design element

– Additive mixture

• Most relevant for displaying maps

on screen

• Common color space: RGB

– Subtractive mixture

• Most relevant for printing maps

• Common color space: CMYK

– There exist various

color tables

http://i.msdn.microsoft.com/ http://us-shop-paderborn.com/

• For maps and map series the signatures on-hand,

fonts, and colors are specified in signature

catalogs and color

tables

– Example:

topographical map

1:25.000 (TK25)

http://www.lgn.niedersachsen.de/

• Signature catalog "SK25" describes how the TK25 is derived

from the "digital landscape model" (digitales

Landschaftsmodell 1:25.000, DLM25)

• The catalog consists of derivation rules and signatures

(about

forms)

• Some

derivation

rules:

• Some signatures:

– Example:

• Real estate map 1:1000

• The ALKIS project (official property cadastre information

system, Amtliches Liegenschaftskataster Informationssystem)

specifies a signature library and derivation rules

• Approximately 670 signatures

• Approximately 850 derivation rules

• A signature

specification:

Section of a real estate map

• The direct geometric derivation (coordinate

transformation) of map objects from spatial

objects works only with large-scale maps

(e.g. real estate

• For small-scale

maps there are

too many details

http://www.geobasis-bb.de/GeoPortal1/

• Generalization simplifies map content for the

preservation of readability and comprehensibility

• Replacement of scale preserving mappings

through simplified mappings, symbols, and

signatures

• Select and summarize information

• Maintain important objects, leave

out unimportant ones

http://www.kartographie.info/

• Eight elementary operations

– To simplify

– To enlarge

– To displace

– To merge

– To select

– To symbolize

– To typify

– To classify

www.ikg.uni-hannover.de

Base map 1:5000 Topographical map 1:25.000

[HGM02] [HGM02]

Topographical map 1:25.000 Topographical map 1:50.000

[HGM02] [HGM02]

Topographical map 1:50.000 Topographical map 1:100.000

[HGM02] [HGM02]

• Typical operations for generalization

(alternative view)

[SX08]

• Smoothing of polylines by means of a simple

low-pass filter:

y(n) = 1/3(x(n)+x(n−1)+x(n−2))

• Electronic filter that passes low-frequency signals

but attenuates high-frequency signals

• Further common filter types

– High-pass

– Band-pass

– Band elimination

3.3 Low-pass Filter

• Passive electronic realization

– Simple circuit consists of a resistor and a capacitor

(RC filter, first order filter)

– More advanced circuit with

additional inductor

(RLC filter,

second order

filter)

3.3 Low-pass Filter

http://upload.wikimedia.org/

http://ecee.colorado.edu/~mathys/

• Active electronic realization

– For example RC filter with an operational amplifier

– An operational amplifier has a high input impedance

and a low output

impedance

– Voltage gain is

determined by

resistors R1, R2

3.3 Low-pass Filter

http://www.electronics-tutorials.ws/

• Digital filters are realized as

– Finite impulse response

filter (FIR)

• Impulse response lasts for

n+1 samples and settles to zero

• Output is a weighted sum of

the current and a finite

number of previous values

of the input

– Infinite impulse response

filter (IIR)

3.3 Low-pass Filter

• A moving average filter is a very simple FIR filter

– All n filter coefficients

are set to 1/n

– Realization by means of

• Delay units

• Ring buffer

3.3 Low-pass Filter

http://blog.avangardo.com/wp-content/

• In offline mode (no real-time constraints)

a moving average filter results in a simple loop

3.3 Low-pass Filter

x[1...n] input values y[1...n] output values for (int i=2; i<n; i++) { y[i] = 0.333*(x[i-1]+x[i]+x[i+1]); }; y[1] = x[1]; y[n] = x[n];

• Adaptation of simple low-pass filter to

two-dimensional

geometry

x′i = ⅓(xi−1 +xi+ xi+1)

y′i = ⅓(yi−1 +yi + yi+1)

+ very efficiently to compute

− coordinates are changed, no reduction of points

• Simplification of polylines

– Reduction of points

– No change of coordinates

• Douglas/Peucker algorithm [DP73]

– Given: polyline L, threshold g

– g is very small: L remains unchanged

g is very large: L is changed to

one single line

• Douglas/Peucker algorithm

1. given: polyline L, threshold g

2. determine line between the start and end point of L,

3. determine the point of L that is furthest from the line segment

4. if distance > g then the point is significant, repeat procedure for both sub-lines, otherwise remove all the points between the start and end point of L

http://en.wiki.mcneel.com/

Examples:

Properties:

+ simple base operation: distance measurement

− runtime of naive implementation: O(n2)

+ runtime of optimized implementation: O(n log n)

+ good results even with a strong reduction of points

+ extendable for polygons

− "outliers" are not eliminated

• Polygon to polyline conversion

– Presentation of rivers and roads

– Text placement within polygons

http://www.almenrausch.at/bergtouren/ http://www.naturathlon2006.de/

• Schoppmeyer/Heisser procedure [SH95]

1. Given: elongate polygon P

2. Determine the longitudinal axis of P

3. Drop the perpendicular from all edge points to the

longitudinal axis

4. Determine the intersection points of the extended

perpendicular lines

5. Determine the centre of the resulting axes

6. Connect adjacent centers

7. Determine suitable start and end segment

Example

1. Given: elongate polygon P

2. Determine the longitudinal axis of P

3. Drop the perpendicular from all edge points to the

longitudinal axis

4. Determine the intersection points of the extended

perpendicular lines

5. Determine the centre of the resulting axes

6. Connect adjacent centers

7. Determine suitable start and end segment

Result

Using the result for text placement

Properties:

+ relatively short runtime

+ quite good results with "good-natured" polygons

− determining the start and end segment

− procedure fails for non-elongate polygons

• Procedure for arbitrary polygons Petzold/Plümer [PP97]

1. given: Polygon P (set of points S)

2. determine Voronoi diagram of S

3. determine intersection points of the Voronoi edges and the polygon edges

4. consider resulting Voronoi skeleton:

5. choose an appropriate sequence of edges

Voronoi diagram:

assigns each point Pi ∈ S the points of the plane that are closer to Pi , than to each Pj ∈ S, i≠j.

Example

1. given: Polygon P (set of points S)

2. determine Voronoi diagram of S

3. determine intersection points of the Voronoi edges

and the polygon edges

4. consider resulting Voronoi skeleton:

5. choose an appropriate sequence of edges

Properties:

− selection of an appropriate polyline from the

skeleton

− computation of the Voronoi diagram

+ results for all kinds

of polygons

Example (polygon with 2000 points):

visualization tool: [Me11]

• Procedure with rasterization, thinning, and re-vectorization

1. given: polygon P (set of points S)

2. rasterization of P

3. topological thinning of P (building a skeleton)

4. line following and re-vectorization

5. determine the longest path

6. line simplification

Example

1. given: polygon P (set of points S)

2. rasterization of P

visualization tool: [Bu11]

3. topological thinning of P

4. line following and re-vectorization

5. determine the longest path

6. line simplification

• Simplification of polygons

– Low-pass filter

– Douglas/Peucker (adapted)

Decompose polygon P into 2 polylines L1, L2 e.g. at the points

PL1, PL2 ∈ P, with maximum distance between each other

Simplify L1 and L2

Recombine resulting polylines at PL1 and PL2

Example

polygon with points PL1, PL2 with maximum distance

– Example

Douglas/Peucker (adapted)

city area of Braunschweig,

reduction from about 90 to about 30 points

Further example

Douglas/Peucker (adapted)

g = 0.5 g = 1

cartographically desired generalization

results with Douglas/Peucker (adapted)

– Douglas/Peucker reduces the number of polygon points

– Characteristic shapes are preserved (within certain limits)

– Good results with "natural" geometries (bogs, lakes, forests)

– Less satisfactory results with polygons with predominantly right angles (buildings), therefore

• Modification to preserve right angles and long edges

• E.g. Neumann/Selke [NeS01]

• Geometric generalization (and displacement) of

building sketches is a challenging task

• Convenient results achieved by programs

• E.g. CHANGE, PUSH, TYPIFY [Se07]

http://www.ikg.uni-hannover.de/

• Placement of all texts (labels) in the same way

results with high probability in overlappings (poor

readability)

• Therefore, it is necessary

to move, scale down,

rotate, or omit texts

• In the general case this is

a NP-hard optimization

problem

3.4 Label and Symbol Placement

http://www.laum.uni-hannover.de/ilr/lehre/Isv/

• Objectives of labeling

– Easily readable

– Unambiguity: each label must be easily identified with exactly one graphical feature

– Same facts are represented in the same way

– Different facts are represented differently

– Important facts are emphasized

– Important objects are never covered

– No uniform pattern (avoidance of raster effect)

http://www.mdc.tu-dresden.de/

• Text or label placement is divided into

– Point labeling • Positions to the right are preferred to those

on the left

• Labels above a point are preferred to those below

• E.g. cities with a horizontal label

– Line labeling

• Labels should be placed as straight as possible

• E.g. rivers with names

– Area labeling

• It must be clear what the total area is

• E.g. forest areas containing their names

http://www.sprachkurs-sprachschule.com/

• For the three classes there are many specialized

algorithms, based on

– Greedy algorithm

• Labels are placed in sequence

• Each position is chosen

according to minimal

overlapping

• Acceptable results only for

very simple problems

• Very fast

http://alitarhini.files.wordpress.com/

• Example

– Local optimization

• The labels are checked several times

• On each pass a single label is repositioned an tested

• The position is kept, if the overall result improves

• Stop, if a local optimum is reached

• Suitable only for maps which relatively few labels

– Simulated annealing

• Similar to local optimization, however yielding better results

• A placement of a label can be kept even though it (initially)

downgrades the overall result

• At first "high temperature", thus leaving local optima is

possible

• Later on ("low temperature") only

small changes are possible

• Challenges: a good evaluation function,

and a good annealing schedule

http://www.hs-augsburg.de/informatik/

• Name and inspiration from annealing in metallurgy: controlled cooling of a material until it changes from liquid to solid

• Structure of the solid depends on „the cooling schedule“

– Fast cooling results in • Unordered solid

• Internal stresses

– Slow cooling results in • Ordered solid

• Low internal energy

• Macroscopic crystal lattice

3.4 Simulated Annealing

http://webuser.hs-furtwangen.de/~neutron/

• Example: SiO2

– Short range order: tetrahedron

– Crystalline form: Cristobalit

– Without long range order: Silica glass

http://de.wikipedia.org/wiki/Glas

http://webuser.hs-furtwangen.de/~neutron/

• Elements

– Initial solution (liquid)

– Modifications (vibrations of molecules)

– Cooling schedule: Change of temperature over time

• Initial temperature

• Freezing point

– Weighting function (internal energy)

• Pseudocode:

– Usually the temperature is decreased after multiple changes

T = initialTemperature; currentSolution = InitialSolution; while (T > freezingPoint){ newSolution = CHANGE(currentSolution); if (ACCEPT){ currentSolution = newSolution; } ANNEAL(T); }

– Decision if the new solution is accepted

• A better solution is always accepted

• Probability of accepting a worse solution depends on the

temperature and its cost

• Boltzmann factor: e-E/k*T ; k = 1,3806504(24) · 10−23

ACCEPT{ Δ = EVALUATE(newSolution) – EVALUATE(currentSolution) if((Δ < 0) or RANDOM(0,1) < e-Δ/T)){ return true; } return false; }

• Example: label placement

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.2327

• Initial solution: random label placement

• Weighting function

– Number of covered (or deleted) labels

– Consideration of cartographic preferences by

weighting of possible positions for point labels

• Modifications

– Move an arbitrary or covered label to a new position

– If cartographic preferences are considered an

arbitrary label should be moved

• Cooling schedule:

– Initial temperature ca 2,47 → the probability of accepting that a solution whose cost are 1 higher is accepted is 2/3, i.e.: e-1/T = 2/3

– T = 0,1 * T

– T is decreased as soon as more than 5*n new solutions have been accepted (n is the number of objects)

– Search ends

• As soon as T has been decreased 50 times

• If none of 20*n new solutions in a row has been accepted

• Result

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.2327

• Symbol placement (point signatures) is just as

complex as text placement

• In the following two examples for special cases

– Displacement and placement of trees symbols in

TK25-like presentation graphics [NPW06]

– Placement of point signatures in polygons of buildings

for real estate maps [NKP08]

• Displacement and placement of tree row symbols

– Roads as well as tree rows are given as polylines only in the digital landscape model 1:25.000 (DLM25)

– Placement of the tree symbols on the points of the tree rows does not result in an equidistant pattern

– The visualization of roads is much wider than the actual street width (depending on the type of the road (attribute: dedication, “Widmung”))

– The tree symbols are often hidden by the line signatures of the streets

– Thus, beside a coordinate transformation an alternative placement and a displacement is needed

• One single road from the DLM25

(example, XML/GML encoding)

<AtkisMember> <Strasse> <gml:name> Badstrasse </gml:name> <AtkisOID> 86118065 </AtkisOID> <gml:centerLineOf> <gml:coord><gml:X>4437952.980</gml:X><gml:Y>5331812.550</gml:Y></gml:coord> <gml:coord><gml:X>4437960.070</gml:X><gml:Y>5331818.450</gml:Y></gml:coord> <gml:coord><gml:X>4437967.200</gml:X><gml:Y>5331825.410</gml:Y> </gml:coord> </gml:centerLineOf> <Attribute> <Zustand> in Betrieb </Zustand> <AnzahlDerFahrstreifen Bedeutung=“tatsaechliche Anzahl”> 2 </AnzahlDerFahrstreifen> <Funktion> Strassenverkehr </Funktion> <VerkehrsbedeutungInneroertlich> Anliegerverkehr </VerkehrsbedeutungInneroertlich> <Widmung> Gemeindestrasse </Widmung> </Attribute> </Strasse> </AtkisMember>

• A tree row from the DLM25

<AtkisMember> <Baumreihe> <gml:centerLineOf> <gml:coord><gml:X>3524258.170</gml:X><gml:Y>5800238.690</gml:Y> </gml:coord> <gml:coord><gml:X>3524256.190</gml:X><gml:Y>5800220.270</gml:Y> </gml:coord> <gml:coord><gml:X>3524255.240</gml:X><gml:Y>5800196.070</gml:Y> </gml:coord> ... <gml:coord><gml:X>3524581.650</gml:X><gml:Y>5799674.000</gml:Y> </gml:coord> </gml:centerLineOf> <Attribute> <Vegetationsmerkmal> Laubholz </Vegetationsmerkmal> </Attribute> </Baumreihe> </AtkisMember>

• Direct visualization

• Simple displacement procedure

for all t : treeRow do begin if exists s : street (distance(t,s) ≤ minDistance(s.dedication)) then for all p : t.coord do begin r := refSegment(s,p); move(p,r) end do end if end do;

• Visualization with displacement

• Placement procedure

for all t : treeRow do begin l := length(t.centerLineOf); n := ⌊l/distanceConst⌋ + 1; t’ : new treeRow; for i=1 to n do begin computePoint (t’.coord[i], t.centerLineOf, distanceConst) end do end do;

• Visualization with displacement and placement

• Placement of point signatures in polygons of buildings

– Derivation of the real estate map (1:1.000) from ALKIS

inventory data extracts relatively straight forward

– No generalization and no displacement is needed

(topographic planimetry, "cadastral map")

– Representation of buildings, parcels, border points, etc.,

with the given signature library and the derivation rules

– E.g. symbolisation of buildings as colored polygons with a

boundary line and a typical point signature depending on

the attributes building function (Gebäudefunktion) and

further building function (weitere Gebäudefunktion)

• A building from an ALKIS inventory data extract

<gml:featureMember> <AX_Gebaeude gml:id="DEHHSERV00001FN1"> ... <position><gml:Polygon> <gml:exterior><gml:Ring> ... <gml:pos>3567807.047 5930017.550</gml:pos> <gml:pos>3567810.850 5930024.755</gml:pos> ... <gml:pos>3567807.047 5930017.550</gml:pos> ... </gml:Ring></gml:exterior> </gml:Polygon></position> <gebaeudefunktion>2000</gebaeudefunktion> <weitereGebaeudefunktion>1170</weitereGebaeudefunktion> <bauweise>2100</bauweise> <anzahlDerOberirdischenGeschosse>1 </anzahlDerOberirdischenGeschosse> </AX_Gebaeude> </gml:featureMember>

• For many point signatures which are related to buildings so-called presentation objects are supplied in the inventory data, defining the optimal position of the respective signature (given in "world coordinates")

<gml:featureMember> <AP_PPO gml:id="DEBWL00100000fAW"> <lebenszeitintervall>... </lebenszeitintervall> <modellart>... </modellart> <anlass>000000</anlass> <position> <gml:Point><gml:pos>3540847.175 5805897.864</gml:pos></gml:Point> </position> <signaturnummer>3316</signaturnummer> <dientZurDarstellungVon xlink:href="urn:adv:oid:DEBWL00100000jwR"/> <drehwinkel>67.000</drehwinkel> </AP_PPO> </gml:featureMember>

• But some buildings lack the presentation objects

• An obvious, easily determined position for the

signature:

– Choose the center of the smallest axis parallel

rectangle, which encloses the polygon of the building

– min(x1, ..., xn)+((max(x1, ..., xn)−min(x1, ..., xn))/2),

– min(y1, ..., yn)+((max(y1, ..., yn)−min(y1, ..., yn))/2)

• Unfortunately, the results are not always satisfactory

• Therefore, heuristic procedure, based on

– Convexity

– (approximate) symmetry points

– (approximate) symmetry axis

– Various quality criteria

– Discrete increments

if polygon of building convex: choose centroid if signature frame fits completely in polygon of building: place there otherwise choose appropriate point with the smallest distance to the centroid else if symmetry point in polygon of building: place there otherwise further procedure with symmetry axes

⇒ generation of a new presentation object with "optimal" positioning coordinates

• The procedure is not suited for the placement of

signatures for churches/chapels

– Besides the determination of an appropriate position

– Also the signatures alignment to the shape of the

building’s polygon is needed

– An alignment to the north south axis is rarely optimal

– Crosses for churches as parallel as possible to the

churches naves

– Crossbrace of cross should be

placed in the transept

• Heuristic procedure

– Determine a preferable large cross that just fits in the

building’s polygon

– Proportions of the large cross and the church

signature are the same

– If the large cross is found, place the signature just in

the intersection point

• For this purpose first simplify the building’s

polygon

• In several rotation angles: look for a preferable large, well placed cross

– Restriction of the potential rotation angles and positioning points (e.g. consider minimum distance to boundaries of the building’s polygon)

– Evaluate all appropriate crosses within a rotation angle

– Evaluate all the best crosses (e.g. lengths of the crosses, alignments)

• Placement of the signature in the best cross of all

rotation angles

⇒ Generation of a new presentation object with

"optimal" positioning coordinates and

"optimal" rotation angle

• Both methods applied to inventory data extracts

• Mapping of spatial data

– Topographic map

– Thematic map

• Properties of maps

– Graticule

– Projections

– Gauß-Krüger coordinate system

– Body of map, map frame, map margin

3.5 Summary

• Signatures, text, color

– Point signatures

– Line signatures

– Area signatures

– Derivation rules

– TK25, real estate map

• Geometric generalization

– Smoothing of polylines

– Simplification of polylines

3.5 Summary

– Douglas/Peucker algorithm

– Polygon to polyline conversion

– Simplification of polygons

– Geometric generalization of building’s ground plans

• Text and symbol placement

– Methods for text placement

simulated annealing

– Displacement and placement of symbols for tree rows

– Placement of point signatures in building’s polygons

3.5 Summary

GIS graticule

text signatures

collect

manage

analyse

display

generalization

area labelling

simplifying lines

polygon→ line

design elements

placement

topographic

thematic

building signatures

tree rows as optimization problem

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