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
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
• Several common names for GIS in diverse areas
– Land information system – Urban information system
– Natural resource information system
– Soil information system
– Network information system – Branch information system
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 67
2.1 Introduction
http://www.stadt-kassel.de/
http://www.sueddeutsche.de
• 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
• 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/
2.1 Discreta/Continua
www.wetteronline.de
– 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
2.1 Discreta/Continua
Spatial Framework
Attribute Domain
spatial field
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
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
• 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:
2.1 Discreta/Continua
< 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 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)
2.1 Discreta/Continua
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
2.1 Discreta/Continua
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
2.1 Discreta/Continua
visualization tool: [Lu13]
• Layer
– Combination of the spatial framework and the field that assigns values for each location in the framework
2.1 Discreta/Continua
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
2.1 Discreta/Continua
• Object-based models decompose an information space into objects or entities
– An entity must be
• Identifiable
• Relevant
• Describable
– Entities are spatially referenced
2.1 Discreta/Continua
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)
– Implemented by geometrical data types, based on
• Vector data model
• Raster data model
2.1 Introduction
h tt p :/ /u p loa d .w ik ime d ia.or g /
• Topology
– Spatial relations between spatial objects – “Geometry of the relative position”
– Independent of extent and shape
2.1 Introduction
http://www.openstreetmap.de/
• 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 neighbourhood
– All important spatial relationships (e.g. connectedness, boundary) may be
expressed in point-set topological terms
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
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
• 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)
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 91
2.1 Introduction
geometric data
theme 1 theme 2
theme n
... h tt p :/ /w w w .d ou glasc ou n tyn v. gov /
– 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
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
• 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
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 112
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
– 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) 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
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 123
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
de.wikipedia.org/wiki/Bild:Digitales_Gel%
C3%A4ndemodell.png
• Isoline
– Lines connecting points with the same numerical values or the same properties
– Only suitable 2d-representation for (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
2.5 Fields
www.wetteronline.de
• 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
• Given a set of n points P = {p 1 , p 2 , ..., p n } in the plane
• These points are called sites
• If the plane is divided by assigning every point to its nearest site p i , for every site a Voronoi cell
V(p i ) = {x: | p i – x | ≤ | p j – 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
• 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
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 135
2.5 Voronoi Diagram
www.informatik.uni-trier.de/
.../Voronoi-Diagramme.ppt
• Voronoi cells
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
– 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
http://www.geometrylab.de/applet-16-de/
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 B ij of p i and p j defines the half- space H(p i , p j ), that contains all points being closer to p i than to p j
– The Voronoi cell V of the site p i contains all points x for which holds: x closer to p i than to p 1
and x closer to p i than to p 2 and ...
and x closer to p i than to p n
→ V(p i ) = ∩ i≠j H(p i , p j )
– Intersection of n half-spaces takes O(n log n) time → runtime O(n 2 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
www.diku.dk/hjemmesider/
studerende/duff/Fortune/
– 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
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 – 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 = {d 1 , . . . , d m } of all m=n(n−3)/2 diagonals of P
2. sort D ascending on the length d 1 , . . . , d m
3. triangulation T ← P 4. for i ← 1 to m
5. if d i intersects no segment in T and is in P
6. T ← T d i
Output: triangulation T of P
– 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 of a Delaunay-Triangulation from an arbitrary triangulation
• Consider two triangles (p 1 ,p 2 ,p 3 and p,p 1 ,p 3 ) with a common edge
• Check if the circumcircle of one triangle(p 1 ,p 2 ,p 3 ) contains the other point p
• In that case delete the common edge p 1 p 3 and insert pp 2
2.5 Fields
p 3 p 3
p 1 p 2
p
p 1 p 2
p
– Example: Greedy- and Delaunay-Triangulation
2.5 Fields
– Example: interpolation with Delaunay-Triangulation
2.5 Fields
visualization tool: [Ra10]
– Interpolation of the values of a single triangle
• Arithmetic average v a = ⅓(v 1 +v 2 + v 3 )
2.5 Fields
• Barycentric Interpolation value at point p is computed on the basis of the area of
three sub-triangles v p = (A 1 v 1 + A 2 v 2 + A 3 v 3 ) / A
2.5 Fields
• Inverse Distance Weighting (IDW)
– The interpolant at P is determined by the values z n of the n (nearest) neighbours P 1 … P n and the
normalized reciprocals of their distances d n 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 z d
d P
1 1
)
f( with u 0
and i d
i 0
– 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
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=1 u=0.5
u=9 u=3
– Influence of the
exponent u (50 reading
points, n=20)
2.5 Fields
u=9 u=3
u=0.5
u=1
– Example: wind maps
2.5 Fields
visualization tool: [Sc11]
• (Ordinary) Kriging
– Statistical method
– Principle: utilization of spatial correlation
for the estimation of values between the
reading points
2.5 Fields
– "BLUE": Best Linear Unbiased Estimator
• Unbiased
• Linear: value at the point x 0 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
C: 4
E: 3 D: 2
A: 5
B: 7 dif fer en ce of the v alues
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 z 0 of x 0 is estimated as weighted average of the values of the circumjacent reading points
• Wanted: weights λ i ,so that the estimator is unbiased and the variance is minimized
→ Constraint optimization problem
→ Solution by Lagrange multiplier ν
2.5 Fields
n
j=
j
j z(x )
λ
= z
1 0
1
, , 2 , 1 ,
1
0 1
n
j j
i n
j
j i
j x x x x i n
– The solution of the system of equations supplies the weights
– Insertion of these weights into the estimator supplies the value of x 0
– 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
– Duties and responsibilities
• Joint implementation of projects initiated across all federal states
• Cooperation geared at the
development and application of technological methods
• 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/C894387
1_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
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 181
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
– 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
• 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
x y
x y i i i n
i
i 1 1
1
2 1
1
http://www.ndr.de/
• Vector: minimum
distance between a point and a line
• Raster: distance matrices for distance determination
2.7 Geometric Algorithms
city block metric chessboard metric Euclidean distance
of centers
– Buffering
• Vector:
• Raster: building of a distance matrix, threshold
2.7 Geometric Algorithms
– Centre of gravity (centroid)
• Raster: average of row- and column indices
• Vector:
2.7 Geometric Algorithms
1
1
1 1
1 )( )
6 ( 1 n
i
i i i
i i
i
s x x x y x y
x F
1
1
1 1
1 )( )
6 ( 1 n
i
i i i
i i
i
s y y x y x y
y F
– Examples: centre of gravity
2.7 Geometric Algorithms
• Overlay operations
– Result consists of one or several new spatial objects – Example: combination of parcels and soil types
2.7 Geometric Algorithms
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
2.7 Geometric Algorithms
• Line segment intersection
– Linear equation
• Calculate intersection point
• Check if the point lies on the lines
2.7 Geometric Algorithms
– 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(x 1 y 2 -x 2 y 1 +x 2 y 3 –x 3 y 2 +x 3 y 1 -x 1 y 3 )
determines the side of the point with respect to the line
2.7 Geometric Algorithms
– Example (100 segments)
2.7 Geometric Algorithms
• 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
Spatial Databases and GIS – Karl Neumann, Sarah Tauscher– Ifis – TU Braunschweig 199
