2 Spatial Data Modelling - TU Braunschweig · NR NAME < 6 6 bis 10 10 bis 16 16 bis 20 20 bis 40 ... –In the plane the elements of a spatial ... •The intersection of any two neighbourhoods

2.1 Introduction

2.2 Geometry

2.3 Conversion between Vector and Raster Models

2.4 Topology

2.5 Fields

2.6 AAA-Project

2.7 Operations

2.8 Summary

Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 48

2 Spatial Data Modelling

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1a.gif

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

– Collect

– Manage

– Analyze

– Display

geographically referenced data (geospatial; spatial)

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


2.1 Introduction

Visualization, Cartography

Spatial Data Management

Collection of Spatial Data

Analysis, Modelling

Functional Components Structural Components

• 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

• Line documentation


2.1 Introduction

http://www.energiekontor-oceanwind.de/

– Evolving applications

• Facility management

• Traffic management

system

• Radio network

planning

• Perturbation

management

• Site selection,

marketing →

business studies


2.1 Introduction

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

• Example questions with spatial reference:

– Which wires run across federal roads?

– Are there post offices

in borough C?

– Which properties

adjoin a waste deposit?

– How do I get from the

university to the train

station?


2.1 Introduction

http://route.web.de/

– Find all road segments

whose slope exceeds

9%?

– Which properties are

crossed by

transmission lines?

– Find all potential

fracking areas

intersecting ground

water bodies?


2.1 Introduction

• 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

• Chief difference to "conventional"

objects

("What’s so special about spatial?"):

– Geometry

– Topology


2.1 Introduction

http://speedymole.com/Tubes/Paris/

• Distinction of spatial objects on basis of their contour

– Discrete objects

• Well defined, enclosed by a 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

http://www.bing.com/maps/

http://www.webbaviation.de/



www.wetteronline.de http://www3.imperial.ac.uk/.../18619712.PDF

www.wetteronline.de

http://magicseaweed.com/.../pressure/in/

– Continua

• Exists everywhere, without boundaries

• Complete

• Collectable only on distinct points

• 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

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



Spatial Framework

Attribute Domain

spatial field

Object Domain

Spatial Embedding

spatial reference

• Example: statistical population data



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

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

Stadt Braunschweig 4,8 3,2 5 3,9 28,8 27,7 5,4 21,3http://www.braunschweig.de/rat_verwaltung/verwaltung/ref0120/statistik/jahrbuch/02_05d.pdf

– A map to visualize this data normally shows boroughs

and one distribution diagram

per borough

• Object-based

• 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



id < 6 6 to 10 60 to 65 65

a + + + +

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)



NR

111

112

113

114

< 6

-

+

-

+

NR

111

112

113

114

120

131

6 to 10

-

+

-

+

-

-

NR

111

112

113

114

120

131

60 to 65

+

+

+

+

-

+

NR

111

112

113

114

120

131

> 65

+

-

+

+

-

-

• Properties of fields

– Continuous

– Differentiable

– Isotropic

• Independent of direction

– Anisotropic

• Properties vary with direction



attribute

investigation area

• 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

– Regular and

irregular

tesselations



visualization tool: [Lu13]

• Layer

– Combination of the

spatial framework

and the field that

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

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

• Describable

– Entities are spatially

referenced




• 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)

– Implemented by geometrical data

types, based on

• Vector data model

• Raster data model


2.1 Introduction

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• Topology

– Spatial relations between spatial objects

– "Geometry of the relative position"

– Independent of extent

and shape


2.1 Introduction

• Topological transformations

– Invertible, bijective, and continous (homeomorphism, "elastic deformation")

• Translation

• Rotation

• Stretching

• Reflection

• Distortion


2.1 Introduction

– Topological properties

(invariants):

neighbourhood,

connectedness,

containedness

– The result of applying a

topological transformation

to a point-set is a

topologically equivalent

point-set


2.1 Introduction

• 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 neigh-

bourhood


2.1 Introduction

– 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 (N1)

• The intersection of any two neighbourhoods of any point p

in S contains a neighbourhood of p (N2)


2.1 Introduction

– 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‘

– 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


2.1 Introduction

• Formal description of

binary topological

relations: 9-intersection model

• Intersections between

interior, boundary and exterior of objects

– Exterior: points which don‘t belong to the object

– Boundary: geometry of a lower dimension

– Interior: object without boundary


2.1 Introduction

EQUAL DISJOINT MEET

OVERLAP COVERS COVEREDBY

INSIDE CONTAINS

• 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

I B E

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1 1 1 1 1 1 1 1 1

I B E

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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

• Theme

– Levels of measurement


2.1 Introduction

Name Operations Remark Examples

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

• 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)


2.1 Introduction

geometric data

theme 1

theme 2

theme n

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– Class concept

• A class comprises objects belonging to the same theme

• Hierarchical classification with subset relation between

classes


2.1 Introduction

http://theses.ulaval.ca/

• 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

• Origin of the raster

• Orientation of the raster

• Cell width

• Raster width and height


2.2 Geometry

– 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

– City block metric (4 neighbours)

– Chessboard distance (8 neighbours)


2.2 Geometry

– Points can only be represented by approximation

– Lines

• Connected sequence of pixels

– Areal objects

• Connected area of pixels

– Basic morphological

operations

• Dilatation

• Erosion


2.2 Geometry

– Particularly suited to describe continua and areal

themes

– Refined raster:

representation of objects is more accurate

but also:

higher memory requirements and computing time

– Guideline: raster width half as wide as the smallest

element/distance which should be represented


2.2 Geometry

– Lossless compression techniques

• Chain code/Crack code

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

• Run length encoding

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


2.2 Geometry

e e

e e e e e e e e e e e

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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

• Block code

– Three examples (Greedy approach)


2.2 Geometry

visualization tool: [Da12]

• Vector data model

– Requisite: two or three dimensional cartesian

coordinate system with euklidian metric

– Line based model (edge representation)

– Basic element: point

• Given by a vector of coordinates

• 0-dimensional

– Line segment

• Defined by two points


2.2 Geometry

– Line: adjacent line segments

• Defined by a sequence of points

• Linear interpolation

• 1-dimensional

– Surface or polygon: closed line

• Defined by an outer boundary (ring)

and any number of inner boundaries

• Boundaries do not intersect

• 2-dimensional


2.2 Geometry

https://www.wien.gv.at/

– Multiple elements as one geometry

– Geometry classes


2.2 Geometry

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

– 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


2.2 Geometry

• Greene-Yao algorithm (1986)

– Goal: control drift of lines

– Grid points which belong to

the line are never moved

– Partition in 2 to 4 segments

– Advantage

• Well-defined

• Bounded error

– Disadvantage

• High fragmentation


2.2 Geometry

• Loss of information

• Point

– Pixel whose center is closest to the original point

• Line

– Pixels intersecting the original line

– Bresenham algorithm (1962)

• Polygon

– Determine for every pixel if it is inside the polygon

– Polygon based fill algorithm


2.3 Rasterization

• Point-in-polygon

– Semi-line algorithm

• Draw a ray out from the point

• Count the number of times

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


2.3 Rasterization

1

3

2

2 4

1

1

?

• Sum the angles

• Angular sum = 0

→ point outside

• Angular sum = 360

→ point inside


2.3 Rasterization

• 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

• All pixel between an intersection point with odd position

and his successor belong to the polygon


2.3 Rasterization

• Ambiguous, manually control necessary

• Input: binary image

• Outline extraction for polygons

– Determine all edge pixels

– Line following over the edge pixels and transformation

of their center into a cartesian coordinate system


2.3 Vectorization

• Topological

thinning

– Consider all

256 relations

between a

pixel and his 8

neighbours


2.3 Vectorization

– Dropping all

symmetric

configurations

leads to 51

basic patterns


2.3 Vectorization

– Classification of basic patterns results in 6 classes

• Isolated point:

no black neighbour

• Inner point:

all neighbours

sharing an edge

with the pixel are black

• Insignificant: all black neighbours are adjacent


2.3 Vectorization

• 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


2.3 Vectorization

• 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

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)


2.3 Vectorization

http://www.fmepedia.com/

– Deficiencies

• Bumpy lines

• Corner arcs

• Displacement of nodes

• Node bridge

• Isles

• Stubble


2.3 Vectorization

[La13]

corner arc

node bridge

stubble

isle

• Raster topology

– Implicit contained, easy to calculate

– Problems occurring when using chessboard metric

• Lines can intersect without having an intersection point

• Polygons can overlap, although their boundaries do not

intersect


2.4 Topology

• 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

efficient if the topology is given explicitly

• Basic elements of topological data models:

– Vertex (V)

– Edge (E)

– Face (F)


2.4 Topology

• Relations

– Same kind of elements:

Adjacency

– Different kind of elements:

Incidence


2.4 Topology

• Spaghetti data structure

– Set of lists of points

– Redundant duplication of data

– Inefficient in space utilization

– Difficult to guarantee consistency

– Improvement:

list of nodes


2.4 Topology

http://www.ikg.uni-hannover.de/lehre/katalog/gis/gisII_uebung

F1: (0/1),(1/3),(1/5),(4/5), (3/3),(1/0) F2: (1/9),(3/3),(5/2),(5/1) L1: (0/0),(1/0),(3/3),(6/5) P1: (4/1) P2: (0/1)

• Spaghetti data structure

– Set of lists of points

– Redundant duplication of data

– Inefficient in space utilization

– Difficult to guarantee consistency

– Improvement:

list of nodes


2.4 Topology


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 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

F0 external

• 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

– The sequence of arcs that bound

an area is easily determined


2.4 Topology

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

– 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


2.4 Topology

• These integrity constraints are not always

appropriate to model real world phenomena

– Suited for

• Land use

• Administration units

– But not for

• Point-shaped objects

• E.g. sources,

dead ends,

branch canals


2.4 Topology

• 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 0 0 10 0 0 h 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|2 space

• Graph algorithms performing a sequential scan over all

edges need O(|V|2) 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)


2.4 Topology

• Adjacency list


2.4 Topology

(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

(a,20), (c,8), (d,9)‏ b

(b,20), (g,15)‏ a

• Field-based models

– Raw data: measurements, often irregularly distributed

– Primary models

• Original data

• Usually vector data

– Derivative models

• Interpolated values

• Regular grid

• Usually raster data


2.5 Fields

www.wetteronline.de www.daserste.de/wetter/ wetterstationen.asp

– Display formats

• Scatterplot

• Wireframe

• Isoline

• 2,5d representation: functional surface in space


2.5 Fields

parks.ca.gov/pages/468/files/AngelIsland2007reprint.pdf www.visualizationsoftware.com/3dem/gallery.html

de.wikipedia.org/wiki/Bild:Digitales_Gel% C3%A4ndemodell.png

• Isoline

– Lines connecting points with the same numerical

values or the same properties

– Are neither borders nor edges

– Are closed

– Do not intersect or

touch each other

– Areas are often filled

– Examples: isobar,

contour line, isochron


2.5 Fields

http://www.bbc.co.uk/

• Interpolation


2.5 Fields

global: all measurements are considered

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

• Local

• Deterministic

• Abrupt

• Suitable for nominal

attributes


2.5 Fields

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1a.gif

• Which post office is closest to a residence?

How do the

catchment

areas of the

post offices

look like?


2.5 Voronoi Diagram

www.meinestadt.de

• Given a set of n points P = {p1, p2, ..., pn} in the plane

• These points are called sites

• If the plane is divided by assigning every point to its nearest site pi , for every site a Voronoi cell V(pi) = {x: | pi – x | ≤ | pj – x | for all j ≠ i } is generated

• Some points are assigned to more than one site → these points construct the Voronoi diagram V(P)


2.5 Voronoi Diagram

• Which post office is closest to a residence?

How do the

catchment

areas of the

post offices

look like?


2.5 Voronoi Diagram

www.meinestadt.de

• Voronoi edge

– Its two nearest sites have the same

distance

– It is a segment or ray (only in one

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


2.5 Voronoi Diagram

www.informatik.uni-trier.de/ .../Voronoi-Diagramme.ppt

• Voronoi cells


2.5 Voronoi Diagram

http://www.olympusmicro.com/primer/techniques/fluorescence/gallery/cells/


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

– 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


2.5 Voronoi Diagram

A Voronoi diagram with one point?

A Voronoi cell with n-1 points?

A Voronoi diagram without points?

A uniform grid ?

How do you construct a point of a given degree > 3?

A Voronoi diagram with 3n-6 edges

• Constructing Voronoi diagrams with perpendicular bisectors – The perpendicular bisector Bij of pi and pj defines the half-

space H(pi, pj), that contains all points being closer to pi than to pj

– The Voronoi cell V of the site pi contains all points x for which holds: x closer to pi than to p1 and x closer to pi than to p2 and ... and x closer to pi than to pn → V(pi) = ∩i≠j H(pi, pj)

– Intersection of n half-spaces takes O(n log n) time → runtime O(n2 log n)


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 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


2.5 Voronoi Diagram

– 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

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

– 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

• 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


2.5 Voronoi Diagram

– 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


– Circle events

• Are generated by a site event

• Generate a Voronoi point


2.5 Voronoi Diagram


– Example: post offices in Braunschweig


2.5 Voronoi Diagram


– Example: post offices in Braunschweig


2.5 Voronoi Diagram


• Some Voronoi points are outside the shown

section


2.5 Voronoi Diagram


• Surface constructed of triangular faces

– Triangulation of reading points

– Rendering of the triangles

– Properties

• Exact

• Local

• Deterministic (depends

on the triangulation)

• Gradual


2.5 Fields

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1b.gif

• Triangulation

– Valid

• No degenerated triangles (collinear)

• No overlap

• Intersections between borders

only at common edges or points

• Covers the whole space

– Regular

• Domain is connected

• Triangulation contains no hole


2.5 Fields

http://www.vermessungsseiten.de/

– Greedy Triangulation Algorithm


2.5 Fields

Input: polygon P with n points

1. build set D = {d1, . . . , dm} of all m=n(n−3)/2 diagonals of P

2. sort D ascending on the length d1, . . . , dm

3. triangulation T ← P

4. for i ← 1 to m

5. if di intersects no segment in T and is in P

6. T ← T di

Output: triangulation T of P

– Triangulation of an arbitrary point set S


2.5 Fields

1. Find the convex hull of S

2. Compute the triangulation of convex hull 3. Choose an interior point and draw edges to

the three vertices of the triangle that contains it

4. Continue this process until all interior points are exhausted

– Goal: all triangles as equiangular as possible

– Delaunay triangulation

• Dual graph of the Voronoi diagram

• Construction:

Connect all sites whose Voronoi cells are adjacent

Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields




• Construction:


Runtime: O(n log n)


2.5 Fields

– Construction of a Delaunay-Triangulation from an

arbitrary triangulation

• Consider two triangles (p1,p2,p3 and p,p1,p3) with a common

edge

• Check if the circumcircle of one triangle(p1,p2,p3) contains

the other point p

• In that case delete the common edge p1p3 and insert pp2


2.5 Fields

p3 p3

p1 p2

p

p1 p2

p

– Example: interpolation with Delaunay-Triangulation


2.5 Fields

visualization tool: [Ra10]

– Interpolation of the values of a single triangle

• Arithmetic average va = ⅓(v1 +v2+ v3)


2.5 Fields

• Barycentric Interpolation

value at point p is computed on the basis of the area of

three sub-triangles vp = (A1v1+ A2v2 + A3v3) / A


2.5 Fields

• Inverse Distance Weighting (IDW)

– The interpolant at P is determined by the values zn of

the n (nearest) neighbours P1… Pn and the

normalized reciprocals of their distances dn to P

– To original data points the measured value is assigned

– The distances are weighted by the exponent u


2.5 Fields

n

i

u

i

n

i

i

u

i dzdP11

)f(0uwith

and 0 idi

– Properties

• Exact

• Global/local

• Deterministic

• Abrupt

• Dependent on distance

• Fast calculation

• Direction is not considered

• Problem: "Bull Eyes"


2.5 Fields

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1c.gif

– Influence of the exponent u


2.5 Fields

www.gitta.info/ContiSpatVar/de/html/Interpolatio_learningObject2.html

u = 0.5

u = 0.5

u = 1

u = 5

u=2 u=1

u=0.2

– Influence of the exponent u

(140 reading points, n=14)


2.5 Fields

u=9 u=3

u=1 u=0.5

– Influence of the

exponent u

(50 reading

points, n=20)


2.5 Fields

u=9 u=3

u=0.5 u=1

• (Ordinary) Kriging

– Statistical method

– Principle: utilization

of spatial correlation

for the estimation of

values between the

reading points


2.5 Fields

http://en.wikipedia.org/

– "BLUE": Best Linear Unbiased Estimator

• Unbiased

• Linear: value at the point x0 is estimated as linear

combination (weighted mean) of n reading points

• 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


2.5 Fields

– Assumptions

• Normal values, for the estimation of the Kriging variance

• Regular distribution of reading points (no cluster)

• The difference between two

measurements depends only

on their distance and not on

the direction (second-order

stationarity)


2.5 Fields

– Procedure

1. Develop experimental variogram

2. Choose suitable variogram model

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


2.5 Fields

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

x

C: 4

E: 3

D: 2

A: 5

B: 7 dif

fere

nce

of

the

val

ues

distance between points

CE

CD

DE

AD

AC

BD

AE AB

BE

BC

• A more realistic example:


2.5 Fields

visualization tool: [Zo13]

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50

γ(h)‏

– Choice of a variogram model

Distance h

γ(h)‏


2.5 Fields

www.bitoek.uni-bayreuth.de/mod/html/ss2007/geooekologie/geoinformationssysteme/GIS-Vorlesung_SS07_7.ppt

spherical model

exponential model

Gaussian model

linear model

– Setting up the Kriging system

• The value z0 of x0 is estimated as weighted average of the

values of the circumjacent reading points

• Wanted: weights λj ,so that the estimator is unbiased and

the variance is minimized

→ Constraint optimization problem

→ Solution by Lagrange multiplier ν


2.5 Fields

n

j=

jj )z(xλ=z1

0

1

,,2,1,

1

0

1

n

j

j

i

n

j

jij nixxxx

– The solution of the system of equations supplies the

weights

– Insertion of these weights into the estimator supplies

the value of x0

– Properties

• Exact

• Local/global

• Stochastic

• Gradual


2.5 Fields

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

– Example:

elevation map

based on 3300

reading points,

extracted from

Wikipedia

(e.g. "... Bonn ...

50° 44' N, 7° 6' E

... 60 m ...")


2.5 Fields

[Hu14]

• Splines

– Goal: generate a surface with minimal curvature

– Utilization of a sequence of different polynomials

(usually ≤ order 3) between the data points

– Properties

• Local

• Inexact

• G2 continuous

(curvature is continuous)

• Deterministic


2.5 Fields

http://skagit.meas.ncsu.edu/~helena/gmslab/interp/F1f.gif

• German Federal States are legally bound to

acquire and provide spatial base data for

administration, economy and

private users

• A consistent structure is

necessary for supra-regional

deployment of spatial data


2.6 AAA-Project

http://www.adv-online.de/

• Working Committee of the Surveying Authorities

of the States of the Federal Republic of Germany

(AdV: Arbeitsgemeinschaft der

Vermessungsverwaltungen der Bundesländer)

– 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


2.6 AAA-Project

http://www.adv-online.de/

– Duties and responsibilities

• Joint implementation of

projects initiated across all

federal states

• Expert statements on draft

laws

• Consulting services

• Representation of the field of

official surveying in Germany

in the EU and in international

organizations


2.6 AAA-Project

http://www.edelgrau.de/

• AFIS-ALKIS-ATKIS-Project (AAA-Project)


2.6 AAA-Project

– ISO/OGC-conform spatial data infrastructure base

component

– UML-Model

– Basic-DLM (1:25000)

• Object oriented vector data

• Complete

• Position accuracy (±3m for road

and stream network)

• No generalization

• Without graphical representation


2.6 AAA-Project

http://www.lgn.niedersachsen.de/master/C8943871_N8913975_L20_D0_I7746208.html

• Feature type catalogue contains 226 feature types

– Including definitions and descriptions of the feature

types, feature attributes and feature associations

– Modeling regulations define the way the features are

to be described and created


2.6 AAA-Project

• Nodes of topological graphs

• Multiple spatially seperated

areas

• Change in attribute values

• Change of object type


2.6 AAA-Project

– Object types:

• Simple spatial object (REO: raumbezogenes Elementarobjekt)

• Simple non-spatial object (NREO: nicht raumbezogenes

Elementarobjekt )

• Complex object (ZUSO: zusammengesetztes Objekt )

• Point-set (PMO: Punktmengenobjekt)

– Models:

• Basic-DLM, DLM50,

DLM250, DLM1000


2.6 AAA-Project

http://www.bkg.bund.de/

– Attributes


2.6 AAA-Project

ATKIS-OK Basis-DLM 6_0


2.6 AAA-Project


2.6 AAA-Project


2.6 AAA-Project


2.6 AAA-Project

– Vertical description of the earth‘s surface via overpass

references

– Further relations: formation of ZUSOs, map geometry,

generalization, technical data linkage, presentation

relation


2.6 AAA-Project

Erläuterungen zum ATKIS-OK Basis-DLM 6_0

• Metric and Euclidean algorithms

– Area

• Vector: integration

• Raster: number of cells * cell area

– Length, Circumference

• Vector: euklidian distance of points

• Raster: number of cells

– Distance

• Different distance measures for

lines and polygons e.g. minimum

or mean distance


2.7 Geometric Algorithms

yxyx iii

n

ii 11

1

12

1

http://static1.fr.de/

• Vector: minimum

distance between

a point and a line

• Raster: distance matrices for distance determination



city block metric chessboard metric Euclidean distance of centers

– Buffering

• Vector:

• Raster: building of a distance matrix, threshold



– Centre of gravity (centroid)

• Raster: average of row- and column indices

• Vector:



1

1

111 ))((6

1 n

i

iiiiiis yxyxxxF

x

1

1

111 ))((6

1 n

i

iiiiiis yxyxyyF

y

– Examples: centre of gravity



• Overlay operations

– Result consists of one or several new spatial objects

– Example: combination of parcels and soil types



visualization tool: [Lu13]

– Raster

• Logical combination of layers

• Intersection equates logical and

• Union equates logical or

– Vector

• Based on iterative application of the

line segment intersection procedure

• Topology of objects important in

deciding which line segments to

discard and which to keep



= =

• Line segment intersection

– Linear equation

• Calculate intersection

point

• Check if the point lies on

the lines

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


181

– Side operation

• Check if endpoints of one line lie

on opposite sides of the other line

• The signed area of a triangle build

by a point and a directed line

½(x1y2-x2y1+x2y3 –x3y2+x3y1-x1y3)

determines the side of the point

with respect to the line



182

– Example (100 segments)



183

• Bentley-Ottmann algorithm

– Vertical sweepline

– Priority queue Q for events:

start, end and intersection

points of segments ordered

by the x-coordinates

– Sweepline status T: containing

the set of input line segments

that cross the sweepline

ordered by the y-coordinates



a1

a4

a3

a2

e1 e3

e2 e4

Q (events):

T (active segments):

a1, a2, a3, e1, e3, e2, a4, e4

S2, S1

a3, e1, e3, e2, a4, e4 a2, a3, e1, e3, e2, a4, e4



a1

a4

a3

a2

e1 e3

e2 e4

Q (events):


a1

a4

a3

a2

e1

e3

e2 e4

Q (events):


S1



x1

a1

a4

a3

a2

e1 e3

e2 e4

Q (events):


x1, e3, e2, a4, e4

S2, S3

a1

a4

a3

a2

e1

e3

e2 e4

Q (events):


S2, S1, S3

e1, e3, e2, a4, e4



x1

a1

a4

a3

a2

e1 e3

e2 e4

Q (events):


e2, a4, e4

S2

x1

a1

a4

a3

a2

e1

e3

e2 e4

Q (events):


e3, e2, a4, e4

S3, S2

– Complexity: O((n+k)log n)

n: number of segments,

k: number of segment intersections



n=10

k=0 k=10 k=45

A

D C B

2

2 3

• Directed graphs with weighted edges

• Variants

– Single Source Shortest Path (SSSP)

• Shortest paths from a source

vertex to all other vertices

– Single Destination Shortest Path

• Shortest paths from all vertices in

the graph to a single destination

vertex


2.7 Shortest Path

A

D C B

7 8

2

2

2 3

4

A

D C B

7 8

2

2

2 3

4

A

D C B

7

3

4

– Single Pair Shortest Path

• Shortest path between two vertices

– All Pairs Shortest Path (APSP)

• Shortest paths between every pair of vertices

• Properties

– A sub-path of a shortest paths is also a shortest path

– The shortest paths from one source vertex always form a tree


2.7 Shortest Path

A D

C B E

F A D

C B E

F

• Dijkstra‘s algorithm

– Assign to every node a distance value

– As long as there are unvisited nodes

• Chose the node with the smallest distance (→Greedy)

• Check if a path to one of its neighbors is shorter than the

previously recorded distance

• In this case, overwrite the distance

– Runtime: O(n²)

– Solves SSSP

– Edge weights have to be positive


2.7 Shortest path

A

F

E

D C B

8 2 4

A

F E

D C B 2

8 2 1

2 3 9 5

4








– Runtime: O(n²)

– Solves SSSP



2.7 Shortest path

A

F

E

D C B

2

3

11

5

A

F E

D C B 2

8 2 1

2 3 9 5

4

4








– Runtime: O(n²)

– Solves SSSP



2.7 Shortest path

A

F

E

D C B

2

3

8

5

A

F E

D C B 2

8 2 1

2 3 9 5

4

4

• Bellmann-Ford algorithm – Maximum path length n → n iterations – In each iteration

• Check for every edge if the sum of the previously recorded distance to the start node and the weight of the edge is smaller than the previously recorded distance to the end node

– In the i-th step all shortest paths with length i are determined

– New distances may not be considered until the next step

– Runtime: O(nm)

– Solves SSSP

– Works with negative edge weights


2.7 Shortest path

A

F

E

D C B

A

F E

D C B 2

8 2 1

2 -1 9 5

4

∞ ∞

0

8 2 4





– Runtime: O(nm)

– Solves SSSP



2.7 Shortest path

A

F

E

D C B

A

F E

D C B 2

8 2 1

2 -1 9 5

4

0

2

4 3

1 9





– Runtime: O(nm)

– Solves SSSP



2.7 Shortest path

A

F

E

D C B

A

F E

D C B 2

8 2 1

2 -1 9 5

4

0

2

8

3

1

3

• Floyd-Warshall algorithm

– Dynamic programming

– Nodes are numbered

– Number of Iterations: n

• Paths in the ith step may only contain nodes with numbers < i

• Check if path including node i is shorter than the previously recorded path

– Runtime: O(n³)

– Solves APSP


2.7 Shortest path

A

F

E

D C B

2

8 2

1

2 3

9 5

4

Step 1: direct connections

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 ∞ 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0


2.7 Shortest path

Step 2: connections via A

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 ∞ 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

…


2.7 Shortest path

Step 3: connections via B

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 ∞ ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 10 ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 10 ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 ∞ 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 8 2 4 10 ∞ B - 0 2 12 2 ∞ C - - 0 1 3 9 D - - - 0 14 5 E - - - - 0 ∞ F - - - - - 0

A B C D E F A 0 4 2 3 5 9 B - 0 2 3 2 8 C - - 0 1 3 6 D - - - 0 4 5 E - - - - 0 9 F - - - - - 0

Steps 4-7 omitted, result:

…

• Input: one or more fields

• Output: a resultant field

• Map algebra: system of possible operations on fields in a field-based model

• Five classes of operations (Tomlin 1990)

– Local

– Focal

– Zonal

– Global

– Incremental


2.7 Operations on Fields

• Local operation

– Value of the new field at any

location is dependent only on

the value of the input field(s) at

that location

– Examples: threshold, addition,

division of attribute values




• Focal operation

– The attribute value derived at any

location depends also on the

attribute values of the input

field(s) in the neighborhood of

that location

– Examples: gradient, filter




• Zonal operation

– Aggregates values of a field

over each of a set of zones Z

• Find the zone in which x is

contained

• Compute the value for this zone

• Assign this value to x

– Examples: sum, area, average,

mean, min, max, circumference




• Global operation

– The value at any location may depend on the values at

every location

– Based on the Euclidean or another weighted distance

• Incremental operation

– Following (the boundary of) a

given spatial object

– Examples: determination of

stream directions and paths



• Spatial data modeling – Functions and variants of GIS

– Discreta and continua as objects and fields • Geometry, topology, theme

• Geometry – Vector data model

• Discretization, Greene-Yao algorithm

– Raster data model

– Metric, compression

– Conversion • Rasterization, point-in-polygon

• Vectorization, outline and centerline extraction

• Topological models – Spaghetti, edge list, winged-edge, graphs


2.8 Summary

• Fields – Voronoi diagram


– Interpolation: IDW, Kriging, Spline

• AFIS-ALKIS-ATKIS-Model – DLM, DGM, OK

• Operations – Geometric operations for vector and raster data

– Segment intersection

– Shortest path problems: Dijkstra, Bellmann-Ford, Floyd-Warshall

– Map algebra


2.8 Summary

A D

C B E

F


2.8 Summary

GIS

fields objects

themes topology geometry

data models relations

raster vector

collect

manage analyse

display

conversion


2.8 Summary

GIS

fields objects

functional surface

isolines

point sets

collect

manage analyse

display

interpolation

operations on

vector data

graphs

raster data

Documents

2 Spatial Data Modelling - TU Braunschweig · NR NAME < 6 6 bis 10 10 bis 16 16 bis 20 20 bis 40 ... –In the plane the elements of a spatial ... •The intersection of any two neighbourhoods