2.4 Topology 2.5 Fields

2.1 Introduction 2.2 Geometry

2.3 Conversion between Vector and Raster Models

2.4 Topology

2 Spatial Data Modelling

2.4 Topology 2.5 Fields

2.6 AAA-Project 2.7 Operations

2.8 Summary

• A geographic information system (GIS) is a computer hardware and software system designed to

– Collect – Manage

2.1 Introduction

Spatial Data Management Collection of

Spatial Data

Functional Components Structural Components

Manage – Analyze

– Display

geographically referenced data (geospatial; spatial)

• It is a specialized information system consisting of a (spatial) database and a (special) database system

Visualization, Cartography

Management Spatial Data

Analysis, Modelling

• Application of GIS for spatial decision-making in politics, economy and administration is increasing

– Main applications so far

•

Surveying, cadastre

•

Urban and regional planning

•

Environmental protection

2.1 Introduction

•

Environmental protection

•

Line documentation

– Evolving applications

•

Facility management

•

Traffic management system

•

Radio network planning

•

Perturbation management

•

Site selection, marketing

business studies

http://www.awe-communications.com/

• Example question with spatial reference:

– Which wires run across federal roads?

– How many post offices are in borough C?

– Which properties adjoin a federal road?

2.1 Introduction

http://route.web.de/

Which properties adjoin a federal road?

– Where are all my properties?

– How do I get from the university to the train station?

– Find all road segments whose slope exceeds 9%?

– Which properties are crossed by transmission lines?

http://route.web.de/

• Several common names for GIS in diverse areas

– Land information system

• Large scale data

• Digital cadastre

• Land-registry and land surveying offices

2.1 Introduction

• Land-registry and land surveying offices

• Basic data for properties and topography

– Urban information system

• Information about properties (owner, area, utilization)

• Underground cadastre (water, canalization, gas, electricity)

• Further “application modules” (greens, tree cadastre,

http://www.aabsys.com/

– Natural resource information system

• Advanced geographical information system

• Documentation of contamination and hazards

• Control of air, water, soil

• Biotop mapping

2.1 Introduction

• Biotop mapping

• Full description in chapter11

– Soil information system

• Information from soil science und geology

• Soil horizons, humus content, pH-value, parent rock material

• Water balance, erosion prone

• Hydrogeology, capacity, geochemistry

http://maps.unomaha.edu/

– Network information system

• Pipes and cables of public utilities

• Graphical representation

• Technical data

• Building measure

2.1 Introduction

• Building measure

– Branch information system

• Special applications of geographic information systems

• Related to a specific field

• Architecture, geography, geology, hydrology, avalanche and environmental protection, traffic planning, tourism, and routing

http://www.siggis.be/

• Spatial object/Geoobject: element to model real world data in geographic information system

• Are described by spatial data (geodata)

• Spatial information: custom-designed spatial data

2.1 Introduction

• Spatial information: custom-designed spatial data

• Chief difference to “conventional”

objects (“What’s so special about spatial?”):

– Geometry

– Topology

• Distinction of spatial objects on basis of their contour

– Discrete objects

• Well defined, enclosed by a visible boundary

2.1 Discreta/Continua

www.maps.google.de

visible boundary

• Object surface, dissemination area, reference surface

• Examples: brook, river, building, wood, borough, industrial real estate, lake,

parcel border point, marsh, sports field, swamp, pont, tower, forest, way,

general residential building area

2.1 Discreta/Continua

– Continua

• Exists everywhere, without boundaries

• Complete

• Collectable only on distinct points

• Examples: ground level, temperature,

www.wetteronline.de

• Examples: ground level, temperature,

precipitation, air pressure, accessibility

• Models of space

– Field-based

•

Geographic information is regarded as collections of spatial distributions

•

Each distribution may be formalized as a mathematical

2.1 Discreta/Continua

Spatial Framework

Attribute Domain

spatial field

•

Each distribution may be formalized as a mathematical function from a spatial framework to an attribute

domain

•

Particularly suitable for continua

– Object-based

•

Space is populated by discrete, identifiable entities each with a geospatial reference (point, line, surface)

•

Particularly suitable for discrete objects

Domain

Object Domain

Spatial Embedding

spatial reference

• Example: statistical population data

2.1 Discreta/Continua

NR NAME < 6 6 bis 10 10 bis 16 16 bis 20 20 bis 40 40 bis 60 60 bis 65 > 65

111 Wabe-Schunter 4,2 3 5,1 3,3 22,3 28,1 6,7 27,3

112 Bienrode-Waggum-Bevenrode 5,6 4 6,3 4,4 23,1 29,3 6,3 21

113 Hondelage 4,2 3,1 4,6 3,6 22,7 30,6 8,4 22,8

114 Volkmarode 5,9 4 6,3 4,7 20,1 31,1 5,9 22,1

120 Östliches Ringgebiet 4,9 3,1 4,2 3,4 36,6 26,8 4,1 16,9

131 Innenstadt 3,4 1,6 2,5 2,7 39,9 25 5,3 19,6

132 Viewegs Garten-Bebelhof 5 2,6 3,8 3,6 33,3 27,4 4,9 19,5

132 Viewegs Garten-Bebelhof 5 2,6 3,8 3,6 33,3 27,4 4,9 19,5

211 Stöckheim-Leiferde 5,3 4,5 7,2 4,2 20,4 31,1 6,1 21,2

212 Heidberg-Melverode 3,5 2,5 4,3 3,3 20,5 26,1 6,2 33,5

213 Südstadt-Rautheim-Mascherode 5,2 4,3 6,3 4,3 22,6 29,7 5,7 21,9

221 Weststadt 5,4 3,6 5,3 4,5 22,3 27,5 5,8 25,6

222 Timmerlah-Geitelde-Stiddien 6,3 4,6 6,7 4,9 25 30,3 5,8 16,4

223 Broitzem 5,5 4 7 5,1 23,8 30,4 4,7 19,4

224 Rüningen 4,8 2,9 5,2 4,1 25,9 27,9 6,1 23,2

310 Westliches Ringgebiet 4,9 2,7 4,2 3,5 36,9 26 4,5 17,4

321 Lehndorf-Watenbüttel 5,1 4,1 6,3 4,3 21,4 30 6,2 22,5

322 Veltenhof-Rühme 4,1 3,1 5,4 4,2 26,1 30,8 6,8 19,6

323 Wenden-Thune-Harxbüttel 5 3,9 6,7 4,2 23,1 31,3 5,4 20,4

331 Nordstadt 4,6 2,5 3,9 3,5 36,9 24 4,9 19,7

332 Schunteraue 4,4 3,3 5,2 5 30 27,6 4,1 20,3

– A map to visualize this data normally shows boroughs and one distribution diagram

per borough

• Object-based

• Objects: boroughs

2.1 Discreta/Continua

• Objects: boroughs

• Attributes:

– Share of the population per age – Visualisation of the difference

between the average of all

inhabitants and the inhabitants of each borough regarding age groups:

< 6; 6 to 10; 60 to 65; > 65

– Object-based

• Clump a relation as single or groups of tuples

2.1 Discreta/Continua

id < 6 6 bis 10 60 bis 65 > 65

a + + + +

b + + + -

b + + + -

c + + - -

d - - + +

e + - - -

f - + - -

g - - + -

h - - - -

NR ^{114 213 221 321 112 211 222 132 223 323 111 113 212 224} …

ID ^{a a a a b b b c c c d d d d} …

– Field-based

• Divide the relation into variations of single or multiple attributes (columns)

2.1 Discreta/Continua

NR 111

< 6 -

NR

111 112 113 114

6 bis 10

- + - 112 +

113 114

+ - +

M M

114 120 131

+ - -

M M

NR

111 112 113 114 120

60 bis 65

+ + + + -

NR

111 112 113 114 120

> 65

+ - + + -

• Properties of fields

– Continuous – Differentiable – Isotropic

2.1 Discreta/Continua

aus: http://ifgivor.uni-muenster.de/vorlesungen/Geoinformatik

attribute

attribute attribute

investigation area

investigation area investigation area

Isotropic

• Independent of direction

– Anisotropic

• Properties vary with direction

aus: http://ifgivor.uni-muenster.de/vorlesungen/Geoinformatik

• Spatial framework: a partition of a region of space

– Forming a finite tesselation of spatial objects

– In the plane the elements of a spatial framework will be polygons

2.1 Discreta/Continua

– Is stipulated to be finite in order to be computable – Application domain is often infinite

– Imprecision introduced by the sampling process

– Regular and irregular tesselations

• Layer

– Combination of the spatial framework and the field that assigns values for

2.1 Discreta/Continua

assigns values for each location in the framework

http://worboys.duckham.org/

• In the special case where the spatial framework is a Euclidian plane and the

attribute domain is a subset of the set of real numbers,

2.1 Discreta/Continua

of the set of real numbers, then a field may be

represented as a surface in

a natural way

• Object-based models decompose an information space into objects or entities

– An entity must be

• Identifiable

• Relevant

2.1 Discreta/Continua

• Relevant

• Describable

– Entities are spatially referenced

http://worboys.duckham.org/

• Geometry

– Describes the (absolute) spatial location of an object in a 2- or 3-dimensional (metric) space

– Information about the position and extent based on a spatial reference system (georeferencing chapter 3)

2.1 Introduction

spatial reference system (georeferencing chapter 3)

– Implemented by geometrical data types, based on

• Vector data model

• Raster data model

• Topology

– Spatial relations between spatial objects – “Geometry of the relative position”

– Independent of extent

2.1 Introduction

Independent of extent

and shape

• Topological transformations

– Invertible, bijective and continous (homeomorphism)

– Translation, rotation, stretching, reflection, distortion

2.1 Introduction

reflection, distortion

– Topological properties (invariants):

neighbourhood, connectedness, containedness

– The result of applying a topological

transformation to a point-set is a

topologically equivalent point-set

• Point-set topology (analytic topology)

– Focus on sets of points, the concepts of neighbourhood, nearness, and open set

– All topological properties are definable in terms of the single concept of neighbourhood

2.1 Introduction

single concept of neighbourhood

– All important spatial relationships (e.g. connectedness, boundary) may be

expressed in point-set

topological terms

– A topological space is a collection of subsets of a given set of points S, called neighbourhoods, that satisfy the following conditions:

• Every point in S is in some neighbourhood (T1)

• The intersection of any two neighbourhoods of any point p

2.1 Introduction

• The intersection of any two neighbourhoods of any point p in S contains a neighbourhood of p (T2)

www.geoinformation.net

– Define p to be near a subset X if every

neighbourhood of p contains some point of X – Exterior X‘: complement of X

– Boundary: consists of all points which are near to both X and X‘

2.1 Introduction

both X and X‘

– Interior: all points which belong to X and are not near points of X‘

• Point: no interior, only boundary

• Line: no interior, only boundary

• Polygon: as usual

http://en.wikipedia.org/

• Formal description of binary topological relations:

9-intersection model (Egenhofer)

• Intersections between interior, boundary and exterior of objects

2.1 Introduction

exterior of objects

– Exterior: points which don‘t belong to the object – Boundary: geometry of a lower dimension

– Interior: object without boundary

EQUAL DISJOINT MEET

OVERLAP COVERS COVEREDBY

• 512 possible, 8 reasonable matrices (for polygons)

2.1 Introduction

I B E I

B E

1 0 0 0 1 0 0 0 1

I B E I

B E

0 0 1 0 0 1 1 1 1

I B E I

B E

0 0 1 0 1 1 1 1 1

OVERLAP COVERS COVEREDBY

INSIDE CONTAINS

I B E I

B E

1 1 1 1 1 1 1 1 1

I B E I

B E

1 1 1 0 1 1 0 0 1

I B E I

B E

1 0 0 1 1 0 1 1 1 I B E

I B E

1 0 0 1 0 0 1 1 1

I B E I

B E

1 1 1 0 0 1 0 0 1

• Combinatorial topology

– Base for the concept of triangulation – Theory of simplicial complexes:

tesselation of objects in

2.1 Introduction

0-simplex

1-simplex

tesselation of objects in

structural identic primitives

www.bauinf.tu-cottbus.de/mitarbeiter/homann/DISS/K ap3.html

simplicial complex no simplicial complex

2-simplex

• Theme

– Levels of measurement

2.1 Introduction

Name Operations Remark Examples

Nominal scale = ≠ no order names, postcode,

Nominal scale = ≠ no order names, postcode,

soil type Ordinal scale = ≠ < > rank order, distance is not

defined

marks, dress sizes Interval scale = ≠ < >

+ -

metric data with arbitrary zero point

temperature in celsius, dates Ratio

measurement

= ≠ < >

+ - * ÷

metric data with non- arbitrary zero value

length, age

– Layer concept

• Different characteristics of spatial objects are separated in different layers

• Separation based on objects or single attributes

• No hierarchy

2.1 Introduction

• No hierarchy

• Layers can be analysed and presented separately

• Aggregation and overlay of layers possible

• Deduced from the principle of separating map layers for printing (classical cartography)

geometric data

theme 1 theme 2

theme n

...

– Class concept

• A class comprises objects belonging to the same theme

• Hierarchical classification with subset relation between classes

2.1 Introduction

object identifier

attributes

geometric data object model

class

identifier attributes

attribute values objectclass model

object identifier

• Raster data model

– Covering of a surface with an arrangement of non- overlapping polygons most often squares (pixel)

– Discrete space, pixel is indivisible – Areal model

– Defined by

2.2 Geometry

city block metric (4 neighbours)

– Defined by

•

Origin of the raster

•

Orientation of the raster

•

Raster width (mesh size)

– The entries of the matrix (numerical values representing the object identifier or attribute values) are interpreted as

“grey scale values”

– Euclidian distance is not defined

chessboard distance

(8 neighbours)

– Points can only be represented by approximation – Lines:

• Connected sequence of pixels

– Areal objects:

• Connected area of pixels

2.2 Geometry

• Connected area of pixels

– Basic morphological operations:

• Dilatation

• Erosion

– Points can only be represented by approximation – Lines:

• Connected sequence of pixels

– Areal objects:

• Connected area of pixels

2.2 Geometry

• Connected area of pixels

– Basic morphological operations:

• Dilatation

• Erosion

– Points can only be represented by approximation – Lines:

• Connected sequence of pixels

– Areal objects:

• Connected area of pixels

2.2 Geometry

• Connected area of pixels

– Basic morphological operations:

• Dilatation

• Erosion

– Particularly suited to describe continua and areal themes

– Refined raster:

representation of objects is more accurate but also:

2.2 Geometry

but also:

higher memory requirements and computing time – Guideline: raster width half as wide as the smallest

element/distance which should be represented

Nyquist/Shannon (~ 1948)

– Lossless compression techniques

• Chain code/Crack code

– Stores the direction in which pixels with the same value are

2.2 Geometry

e ^e

e

e

• Run length encoding

– Stores the number of adjacent pixels with the same value in a row

e e e e e e e e e e e

e

e

• Block code

– Decomposition into squares which are as big as possible – Only the position and size of the squares has to be stored

2.2 Geometry

• Vector data model

– Requisite: two or three dimensional cartesian coordinate system with euklidian metric

– Line based model (edge representation)

2.2 Geometry

– Basic element: point

• Given by a vector of coordinates

• 0-dimensional

– Line segment

• Defined by two points

– Line: adjacent line segments

• Defined by a sequence of points

• Linear interpolation

• 1-dimensional

– Surface or polygon: closed line

2.2 Geometry

– Surface or polygon: closed line

• Defined by an outer boundary (ring) and any number of inner boundaries

• Boundaries do not intersect

• 2-dimensional

– Multiple elements as one geometry

2.2 Geometry

– Geometry classes

point

line segment

ring polygon multipolygon

line multiline

multipoint 2+

2

1+

1+

1+

1+

1+

– Particularly suited to represent discrete objects – Relatively little memory requirements

– Potentially infinite amount of precision

• Discretization

2.2 Geometry

Discretization

– Move intersection point to the nearest grid point

– Split lines so as to join at the moved intersection point

– Problem: Shift of the lines is

not constricted

• Greene-Yao algorithm

– Goal: control drift of lines – Grid points which belong to

the line are never moved

2.2 Geometry

– Partition in 2 to 4 segments – Advantage

• Well-defined

• Bounded error

– Disadvantage

• High fragmentation

• Loss of information

• Point

– Pixel whose center is closest to the original point

• Line

2.3 Rasterization

• Line

– Pixels intersecting the original line – Bresenham algorithm

• Polygon

– Determine for every pixel if it is inside the polygon – Polygon based fill algorithm

http://www.hdm-stuttgart.de/

• Point-in-polygon

– Semi-line algorithm

• Draw a ray out from the point

• Count the number of times that the ray intersects with

2.3 Rasterization

1 3 2 4

1

?

that the ray intersects with the boundary of the polygon

– Winding number algorithm

• Consider the triangles which are defined by a line segment of the boundary and the point

2 1

• Sum the angles

• Angular sum = 0

→ point outside

• Angular sum = 360

→ point inside

2.3 Rasterization

→ point inside

• Polygon based fill algorithm

– For each grid row

• Calculate the intersection points between the row and the edges of the polygon

• Sort the intersection points with respect to the x-axis

2.3 Rasterization

• Sort the intersection points with respect to the x-axis

• All pixel between an intersection point with odd position

and his successor belong to the polygon

• Ambiguous, manually control necessary

• Input: binary image

• Outline extraction for polygons

– Determine all edge pixels

2.3 Vectorization

– Determine all edge pixels

– Line following over the edge pixels and transformation

of their center into a cartesian coordinate system

• Topological thinning

– Classification of all possible relations

between a pixel and his 8 neighbours leads to 51 basic patterns in 6 classes:

•

Isolated point: no black neighbour

•

Inner point: all neighbours sharing an edge

2.3 Vectorization

start point

line point

•

Inner point: all neighbours sharing an edge with the pixel are black

•

Insignificant: all black neighbours are adjacent

•

Start point: exactly one black neighbour

•

Line point: two black neighbours that are not adjacent

•

Node: more than two black neighbours which are not all adjacent

node

• Centerline extraction for lines

– Determine the distance between the pixels which be- long to the line and the closest pixel which does not – Topological thinning:

Classify all pixels ordered by this distance (pixel close to the

2.3 Vectorization

• Classify all pixels ordered by this distance (pixel close to the border first), delete insignificant pixel immediately

• Classify remaining pixels again

• Extraction of nodes: calculate the center of gravity for connected nodes

• Line following (chessboard metric)

– Deficiencies

• Bumpy lines

• Corner arcs

• Displacement of nodes

• Node bridge

2.3 Vectorization

corner arc node bridge

stubble

• Node bridge

• Isles

• Stubble

isle

• Raster topology

– Implicit contained, easy to calculate

– Problems occurring when using chessboard metric

• Lines can intersect without having an intersection point

2.4 Topology

• Polygons can overlap, although their boundaries do not

intersect

• Metric space implies topological space, i.e. it is possible to determine the topological relations between objects if their geometries are known

• Access and computations are normally more

2.4 Topology

• Access and computations are normally more efficient if the topology is given explicitly

• Basic elements of topological data models:

– Vertex (V)

– Edge (E)

– Face (F)

• Relations

– Same kind of elements: Adjacency

– Different kind of elements: Incidence

2.4 Topology

node edge incidence node adjacency

face adjacency edge adjacency

node face incidence face edge incidence

wrt nodes wrt faces

wrt faces wrt nodes

• Spaghetti data structure

– Set of lists of points

– Redundant duplication of data – Inefficient in space utilization

2.4 Topology

Inefficient in space utilization

– Difficult to guarantee consistency – Improvement:

list of nodes

F1: V1 V2 V3 V4 V6 V9 F2: V9 V6 V7 V8

L1: V10 V9 V6 V5 P1: V11

P2: V1

id x y

V7 5 2

V8 5 1

V9 1 9

V10 0 0

V11 4 1

id x y

V1 0 1

V2 1 3

V3 1 5

V4 4 5

V5 6 5

V6 3 3

• Edge list

– Topological relations between points and lines are stored explicitly

2.4 Topology

edge start end right left edge start

node

end node

right face

left face

E1 V1 V2 F1 F0

E2 V2 V3 F1 F0

E3 V3 V4 F1 F0

E4 V4 V6 F1 F0

E5 V6 V9 F1 F2

E6 V9 V1 F1 F0

• Winged edge (doubly connected edge list, DCEL)

– For every edge the successor and predecessor w.r.t the right and left face are added

– Topological relations between lines are stored explicitly

2.4 Topology

explicitly

– The sequence of arcs that bound an area is easily determined

ID left face right face

left arm

right arm

left leg

right leg

e1 F1 F2 e2 e6 e3 e5

• Integrity constraints for “maps” (US Bureau of Census)

– Every edge has two incident vertices – Every edge has two incident faces

2.4 Topology

– Every face is alternately surrounded by edges and vertices

– Every vertex is alternately surrounded by edges and faces

– Edges do not intersect

• Euler characteristic: |V|- |E| + |F| = 2

• These integrity constraints are not always appropriate to model real world phenomena

– Suited for

• Land use

• Administration units

2.4 Topology

• Administration units

– But not for

• Point-shaped objects

• E.g. sources, dead ends, branch canals

– Missing redundancy might lead to problems

• Intersections between property lines and pipes

• Border rivers

http://www.lib.utexas.edu/

• Network represented as weighted graph

– Set of edges: {(ab,20), (ag,15), (bc,8), (bd,9), (cd,6), (ce,15), (ch,10), (de,7), (ef,22), (eg,18)}

– Adjacency matrix

2.4 Topology

0 0

0 18

0 0

0 15

g

0 0

0 22

0 0

0 0

f

0 18

22 0

7 15

0 0

e

0 0

0 7

0 6

9 0

d

10 0

0 15

6 0

8 0

c

0 0

0 0

9 8

0 20

b

0 15

0 0

0 0

20 0

a

h g

f e d

c b a

– Properties of adjacency matrices

• The complexity to determine the existence of an edge between two nodes is constant

• Needs |V|

space

• Graph algorithms performing a sequential scan over all

2.4 Topology

• Graph algorithms performing a sequential scan over all edges need O(|V|

) time

• For graphs with few edges the adjacency

matrix is sparse, so that a representation with O(|E|) might be better

(e.g. adjacency list)

• Adjacency list

2.4 Topology

(b,8), (d,6), (e,15), (h,10) c

(a,20), (c,8), (d,9) b

(b,20), (g,15) a

(c,10) h

(a,15), (e,18) g

(e,22) f

(c,15), (d,7), (f,22), (g,18) e

(b,9), (c,6), (e,7) d

(b,8), (d,6), (e,15), (h,10) c

http://worboys.duckham.org/

• Field-based models

– Raw data: measurements, often irregularly distributed – Primary models

• Original data

2.5 Fields

• Usually vector data

– Derivative models

• Interpolated values

• Regular grid

• Usually raster data

www.wetteronline.de (29.07.08)

www.daserste.de/wetter/

wetterstationen.asp

– Display formats

• Scatterplot

• Wireframe

• Isoline

• 2,5d representation: functional surface in space

2.5 Fields

de.wikipedia.org/wiki/Bild:Digitales_Gel%

C3%A4ndemodell.png

• 2,5d representation: functional surface in space

• Isoline

– Lines connecting points with the same numerical values or the same properties

– Only suitable 2d-representation for (continuous) continua

2.5 Fields

(continuous) continua

– Are neither borders nor edges – Are closed

– Do not intersect or touch each other – Areas are often filled

– Examples: isobar, contour line, isochrone

• Interpolation

2.5 Fields

global:

all measurements are considered

local:

points within a certain distance, or a certain number of points are

considered considered

exact:

goes through the data points

inexact:

doesn‘t go through the data points

gradual abrupt

deterministic:

well-defined, statement about the quality not possible

stochastic:

one possible distribution function, quality statement possible

• Nearest neighbour

– Every point gets the value of the nearest measurement

– Properties

Exact

2.5 Fields

• Exact

• Local

• Deterministic

• Abrupt

• Suitable for nominal

attributes

• Which post office is closest to a residence? How do the catchment areas of the post offices look like?

2.5 Voronoi diagram

• Given a set of n points P = {p

, p

, ..., p

} in the plane

• These points are called sites

• If the plane is divided by assigning every point to its nearest site p , for every site a Voronoi cell

2.5 Voronoi diagram

its nearest site p

, for every site a Voronoi cell V(p

): V(p

) = {x: | p

– x | ≤ | p

– x | for all j ≠ i } is generated

• Some points are assigned to more than one site

→ these points construct the Voronoi diagram

V(P)

• Which post office is closest to a residence? How do the catchment areas of the post offices look like?

2.5 Voronoi diagram

• Voronoi edge

– Its two nearest sites have the same distance

– It is a segment or ray (only in one special case a straight line)

2.5 Voronoi diagram

special case a straight line)

• Voronoi points

– Its (at least) three nearest sites have the same distance

– It is the center of the circle through

the three nearest sites

• Voronoi cells

2.5 Voronoi diagram

2.5 Voronoi diagram

2.5 Voronoi diagram

2.5 Voronoi diagram

• Properties

– A bounded Voronoi cell is a convex polygon

– A Voronoi cell is unbounded if its site is located on the boundary of the convex hull

2.5 Voronoi diagram

– Usually Voronoi diagrams are connected

– Voronoi points are typically of degree three

– If two sites are „nearest neighbours“ then their voronoi cells are adjacent

– A Voronoi diagram for n sites consists of at most 2n-5

points and 3n-6 edges

2.5 Voronoi diagram

A Voronoi diagram

with one point? A Voronoi diagram without points?

A uniform grid ?

http://www.pi6.fernuni-hagen.de/GeomLab/VoroGlide/

A Voronoi cell with n-1 points?

How do you construct a point of a given

degree > 3?

A Voronoi diagram with 3n-7 edges

• Constructing Voronoi diagrams with perpendicular bisectors

– The perpendicular bisector B

of p

and p

defines the half- space H(p

, p

), that contains all points being closer to p

than to p

– The Voronoi cell V of the site p

contains all points x for

2.5 Voronoi diagram

– The Voronoi cell V of the site p

contains all points x for which holds: x closer to p

than to p

and x closer to p

than to p

and ...

and x closer to p

than to p

→ V(p

) = ∩

H(p

, p

)

– Intersection of n half-spaces takes O(n log n) time →

runtime O(n

log n)

• Example: Construction of one Voronoi cell

2.5 Voronoi diagram

• Example: Construction of one Voronoi cell

2.5 Voronoi diagram

• Example: Construction of one Voronoi cell

2.5 Voronoi diagram

• Fortune‘s algorithm

– Sweep line paradigm:

• A sweep line is moving across the plane

• The solution of the passed area is determined

• The status contains all „Objects“ being relevant in the

2.5 Voronoi diagram

• The status contains all „Objects“ being relevant in the current step

• Events are points in time where the status or the set of events might change

– Problem: Voronoi diagram

„behind“ the sweep line

depends on sites „in front of“

the sweep line

– Not every point behind the sweep line can be definitely assigned to a Voronoi cell

– But all points which are closer to a passed site than to the sweep line belong to its Voronoi cell

– This area is bounded by a parabola

2.5 Voronoi diagram

– This area is bounded by a parabola

– Intersections of parabolas generate Voronoi edges – Detection of Voronoi points

•

The center of the circle through the three nearest sites

•

These three points are known before the sweep line leaves the circle

2.5 Voronoi diagram

•

After leaving the circle the Voronoi point is determined

– Status:

•

Beach line composed of pieces of parabolas

– Events:

•

Sites

•

„Circle points“: the point of a circle defined by three sites with

adjacent parabola pieces which the sweep line passes last

– Site events

• Occur when the sweep line reaches a site

• Generate a new parabola as part of the beach line

2.5 Voronoi diagram

– Circle events

• Are generated by a site event

• Generate a Voronoi point

2.5 Voronoi diagram

– Example: post offices in Braunschweig

2.5Voronoi diagram

www.diku.dk/hjemmesider/studerende/duff/Fortune/

• Some Voronoi points are outside the shown section

2.5 Voronoi diagram

• Surface constructed of triangular faces

– Triangulation of reading points – Shading of the triangles

– Properties

2.5 Fields

Properties

• Exact

• Local

• Deterministic (depends on the triangulation)

• Gradual

• Triangulation

– Valid

• No degenerated triangles (collinear)

• No overlap

• Intersections between borders

2.5 Fields

• Intersections between borders only at common edges or points

• Covers the whole space

– Regular

• Domain is connected

• Triangulation contains no hole

– Greedy Triangulation Algorithm

2.5 Fields

Input: polygon P with n points

1. Build set D = {d

, . . . , d

} of all m=n(n−3)/2 diagonals of P

2. sort D ascending on the length 2. sort D ascending on the length

d

, . . . , d

3. triangulation T ← P 4. for i ← 1 to m

5. if d

intersects no segment in T and is in P

6. T ← T d

– Goal: all triangles as equiangular as possible – Delaunay triangulation

• Dual graph of the Voronoi diagram

• Construction:

Connect all sites whose Voronoi cells are adjacent

2.5 Fields

Connect all sites whose Voronoi cells are adjacent Runtime: O(n log n)

• Ambiguous if more than three sites are cocircular

– Construction of a Delaunay-Triangulation from an arbitrary triangulation

• Consider two triangles (p

,p

,p

and p,p

,p

) with a common edge

• Check if the circumcircle of one triangle(p

,p

,p

) contains

2.5 Fields

• Check if the circumcircle of one triangle(p

,p

,p

) contains the other point p

• In that case delete the common edge p

p

and insert pp

p₃ p₃

p₁ p₂

p

p₁ p₂

p

– Example: Greedy- and Delaunay-Triangulation

2.5 Fields

– Example: interpolation with Delaunay-Triangulation

2.5 Fields

• Inverse Distance Weighting (IDW)

– The interpolant at P is determined by the values z

of the n (nearest) neighbours P

… P

and the

normalized reciprocals of their distances d

to P

2.5 Fields

– To original data points the measured value is assigned – The distances are weighted by the exponent u

∑

∑

=

−

=

−

⋅

=

i

u i n

i

i u

i

z d

d P

1 1

)

f(

and ∀i d_i ≠ 0

– Properties

• Exact

• Global/local

• Deterministic

• Abrupt

2.5 Fields

• Abrupt

• Dependent on distance

• Fast calculation

• Direction is not considered

• Problem: „Bull Eyes“

– Influence of the exponent u

2.5 Fields

u = 1 u = 0.5 u=1

u = 0.5

u = 1

u = 5

u=2 u=1

u=0.2

– Example: wind maps

2.5 Fields

visualization tool: [Sc11]

• (Ordinary) Kriging

– Invented by Danie G. Krige (1951) – Statistical method

– Principle: utilization

2.5 Fields

Principle: utilization of spatial correlation

for the estimation of values between the

reading points

http://en.wikipedia.org/

– "BLUE": Best Linear Unbiased Estimator

• Unbiased

• Linear: value at the point x

is estimated as linear combination (weighted mean) of n reading points

• Exact estimator: the estimated values at the reading points

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• Exact estimator: the estimated values at the reading points equal the measured ones

• Strong smoothing (low pass filter)

• The Kriging error (Kriging variance)

allows the evaluation of the reliability

of the estimation for every

estimated value

– Assumptions

• Normal values, for the estimation of the Kriging variance

• Regular distribution of reading points (no cluster)

• The difference between two measurements depends only

2.5 Fields

measurements depends only

on their distance and not on

the direction (second-order

stationarity)

– Procedure

1. Develop experimental variogram

2. Choose suitable variogram model

3. Setting up the Kriging equation

2.5 Fields

3. Setting up the Kriging equation 4. Solve equation

5. Calculate the estimation

– Variogram

• Describes the spatial correlation between location

dependent random variables with respect to their distance to each other

http://www.climate4you.com/

– Experimental variogram

• For each pair of reading points the difference of their values is plotted over their distance

• Example:

Reading points Variogram

2.5 Fields

y

Reading points Variogram

C: 4

E: 3 D: 2

A: 5

B: 7 differenceof thevalues

CE CD

DE AD

AC BD

AE AB

BE BC

0.8 1.0 γ(h)

– Choice of a variogram model

γ(h)

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0.0 0.2 0.4 0.6

0 10 20 30 40 50

Distance h

spherical model

exponential model

Gaussian model

linear model

– Setting up the Kriging system

• The value z

of x

is estimated as weighted average of the values of the circumjacent reading points

2.5 Fields

∑

j=

j

jz(x )

= λ z

1 0

• Wanted: weights λ

,so that the estimator is unbiased and the variance is minimized

→ Constraint optimization problem

→ Solution by Lagrange multiplier ν

( ) ⁽

⁾

1

=

−

= +

∑

= n

i n

j

j i

jγ x x ν γ x x i n

λ ^K

– The solution of the system of equations supplies the weights

– Insertion of these weights into the estimator supplies the value of x

– Properties

2.5 Fields

– Properties

• Exact

• Local/global

• Stochastic

• Gradual

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1d.gif

• Splines

– Goal: generate a surface with minimal curvature – Utilization of a sequence of different polynomials

(usually ≤ order 3) between the data points

2.5 Fields

– Properties

• Local

• Inexact

• G2 continuous

(curvature is continuous)

• Deterministic

• German Federal States are legally bound to acquire and provide spatial base data for

administration, economy and private users

2.6 AAA-Project

• A consistent structure is necessary for supra-regional

deployment of spatial data

• Working Committee of the Surveying Authorities of the States of the Federal Republic of Germany (AdV: Arbeitsgemeinschaft der

Vermessungsverwaltungen der Bundesländer)

2.6 AAA-Project

– Members

• The Cadastral and Surveying Authorities of the 16 German Federal States

• Federal ministry of the Interior, of

Defense, and of Transport, Building

and Urban Affairs