2.1 Geometry
2.2 Conversion between Vector and Raster Models
2.3 Topology
2 Spatial Data Modelling
2.3 Topology 2.4 Fields
2.5 AAA-Project
2.6 Operations
• A geographic information system (GIS) is a computer hardware and software system designed to
– Collect – Manage
2 Spatial Data Modelling
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
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 Spatial Data Modelling
•
Environmental protection
•
Line documentation
– Evolving applications
•
Facility management
•
Traffic management system
•
Radio network planning
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 Spatial Data Modelling
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?
• Several common names for GIS in diverse areas
– Land information system
• Large scale data
• Digital cadastre
• Land-registry and land surveying offices
2 Spatial Data Modelling
• 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)
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 Spatial Data Modelling
• Biotop mapping
• Full description in chapter12
– Soil information system
• Information from soil science und geology
• Soil horizons, humus content, pH-value, parent rock material
• Water balance, erosion prone
http://maps.unomaha.edu/
– Network information system
• Pipes and cables of public utilities
• Graphical representation
• Technical data
• Building measure
2 Spatial Data Modelling
• Building measure
– Branch information system
• Special applications of geographic information systems
• Related to a specific field
• Architecture, geography, geology, hydrology, avalanche and
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 Spatial Data Modelling
• 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 Spatial Data Modelling
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,
2 Spatial Data Modelling
– 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 Spatial Data Modelling
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
Domain
Object Domain
spatial reference
• Example: statistical population data
2 Spatial Data Modelling
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
– A map to visualize this data normally shows boroughs and one distribution diagram
per borough
• Object-based
• Objects: boroughs
2 Spatial Data Modelling
• Objects: boroughs
• Attribute:
– Share of the population per age – Visualisation of the difference
between the average of all
inhabitants and the inhabitants
– Object-based:
• Clump a relation as single or groups of tuples
2 Spatial Data Modelling
id < 6 6 bis 10 60 bis 65 > 65
a + + + +
b + + + -
b + + + -
c + + - -
d - - + +
e + - - -
f - + - -
g - - + -
h - - - -
– Field-based
• Divide the relation into variations of single or multiple attributes (columns)
2 Spatial Data Modelling
NR 111
< 6 -
NR
111 112 113 114
6 bis 10
- + - 112 +
113 114
+ - +
M M
114 120 131
+ - -
M M
NR
111 112
60 bis 65
+ +
NR
111 112
> 65 + -
• Properties of fields
– Continuous – Differentiable – Isotropic
2 Spatial Data Modelling
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 Spatial Data Modelling
– 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 Spatial Data Modelling
assigns values for
each location in the
framework
• 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 Spatial Data Modelling
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 Spatial Data Modelling
• 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)
2 Spatial Data Modelling
spatial reference system (georeferencing chapter 3)
– Implemented by geometrical data types, based on
• Vector data model
• Topology
– Spatial relations between spatial objects – “Geometry of the relative position”
– Independent of extent
2 Spatial Data Modelling
Independent of extent
and shape
• Topological transformations
– Invertible, bijective and continous (homeomorphism)
– Translation, rotation, stretching, reflection, distortion
2 Spatial Data Modelling
reflection, distortion
– Topological properties (invariants):
neighbourhood, connectedness, containedness
– The result of applying a topological
• 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 Spatial Data Modelling
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 Spatial Data Modelling
• The intersection of any two neighbourhoods of any point p
in S contains a neighbourhood of p (T2)
– 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 Spatial Data Modelling
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
• Formal description of binary topological relations:
9-intersection model (Egenhofer)
• Intersections between interior, boundary and exterior of objects
2 Spatial Data Modelling
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 Spatial Data Modelling
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
1 0 0 1 0 0
I B E I
B
1 1 1 0 0 1
• Combinatorial topology
– Base for the concept of triangulation – Theory of simplicial complexes:
tesselation of objects in
2 Spatial Data Modelling
0-simplex
1-simplex
tesselation of objects in
structural identic primitives
2-simplex
• Theme
– Levels of measurement
2 Spatial Data Modelling
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 = ≠ < > metric data with non- length, age
– Layer concept
• Different characteristics of spatial objects are separated in different layers
• Separation based on objects or single attributes
• No hierarchy
2 Spatial Data Modelling
• No hierarchy
• Layers can be analysed and presented separately
• Aggregation and overlay of layers possible
• Deduced from the principle
of separating map layers for
geometrictheme 1 theme 2
– Class concept
• A class comprises objects belonging to the same theme
• Hierarchical classification with subset relation between classes
2 Spatial Data Modelling
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:
2.1 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
chessboard distance
(8 neighbours)
– Points can only be represented by approximation – Lines:
• Connected sequence of pixels
– Areal objects:
• Connected area of pixels
2.1 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
2.1 Geometry
• Connected area of pixels
– Basic morphological operations:
• Dilatation
– Points can only be represented by approximation – Lines:
• Connected sequence of pixels
– Areal objects:
• Connected area of pixels
2.1 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:
2.1 Geometry
but also:
higher memory requirements and computing time – Guideline: raster width half as wide as the smallest
element/distance which should be represented
– Lossless compression techniques
• Chain code/Crack code:
– Stores the direction in which pixels with the same value are
2.1 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
• Block code:
– Decomposition into squares which are as big as possible – Only the position and size of the squares has to be stored
2.1 Geometry
• Vector data model
– Requisite: two or three dimensional cartesian coordinate system with euklidian metric
– Line based model (edge representation)
2.1 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
2.1 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
2.1 Geometry
– Geometry classes
line segment
ring polygon multipolygon
line multiline
2+
2
1+
1+
1+
1+
– Particularly suited to represent discrete objects – Relatively little memory requirements
– Potentially infinite amount of precision
• Discretization
2.1 Geometry
Discretization
– Move intersection point to the nearest grid point
– Split lines so as to join at the
moved intersection point
• Greene-Yao algorithm
– Goal: control drift of lines – Grid points which belong to
the line are never moved
2.1 Geometry
– Partition in 2 to 4 segments – Advantage:
• Well-defined
• Bounded error
– Disadvantage:
• Loss of information
• Point:
– Pixel whose center is closest to the original point
• Line:
2.2 Rasterization
• Line:
– Pixels intersecting the original line – Bresenham algorithm
• Polygon:
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
2.2 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
2.2 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
2.2 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
2.2 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
2.2 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
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
2.2 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
– Deficiencies
• Bumpy lines
• Corner arcs
• Displacement of nodes
• Node bridge
2.2 Vectorization
corner arc node bridge
stubble
• Node bridge
isle• Isles
• Stubble
isle
• Raster topology
– Implicit contained, easy to calculate
– Problems occurring when using chessboard metric
• Lines can intersect without having an intersection point
2.3 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
2.3 Topology
• Access and computations are normally more efficient if the topology is given explicitly
• Basic elements of topological data models:
– Vertex (V)
– Edge (E)
• Relations
– Same kind of elements: Adjacency
– Different kind of elements: Incidence
2.3 Topology
node edge incidence node adjacency
face adjacency edge adjacency
node face incidence face edge incidence
wrt nodes wrt faces
• Spaghetti data structure
– Set of lists of points
– Redundant duplication of data – Inefficient in space utilization
2.3 Topology
F1: V1 V2 V3 V4 V6 V9 F2: V9 V6 V7 V8
L1: V10 V9 V6 V5 P1: V11
Inefficient in space utilization
– Difficult to guarantee consistency – Improvement:
list of nodes
id x y
V7 5 2
V8 5 1
V9 1 9
V10 0 0
id x y
V1 0 1
V2 1 3
V3 1 5
V4 4 5
• Edge list
– Topological relations between points and lines are stored explicitly
2.3 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
• 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
2.3 Topology
explicitly
– The sequence of arcs that bound an area is easily determined
ID left face right left right left right
• Integrity constraints for „maps“ (US Bureau of Census)
– Every edge has two incident vertices – Every edge has two incident faces
2.3 Topology
– Every face is alternately surrounded by edges and vertices
– Every vertex is alternately
surrounded by edges and faces
• These integrity constraints are not always appropriate to model real world phenomena
– Suited for
• Land use
• Administration units
2.3 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
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:
2.3 Topology
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|
2space
• Graph algorithms performing a sequential scan over all
2.3 Topology
• 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)
• Adjacency list
2.3 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
• Field-based models
– Raw data: Measurements, often irregularly distributed – Primary models
• Original data
2.4 Fields
• Usually vector data
– Derivative models
• Interpolated values
• Regular grid
• Usually raster data
– Display formats
• Scatterplot
• Wireframe
• Isoline
• 2,5d representation: functional surface in space
2.4 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
2.4 Fields
(continuous) continua
– Are neither borders nor edges – Are closed
– Do not intersect or touch each other
– Areas are often filled
• Interpolation
2.4 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: stochastic:
• Nearest neighbour
– Every point gets the value of the nearest measurement
– Properties
Exact
2.4 Fields
• Exact
• Local
• Deterministic
• Abrupt
• Suitable for nominal
• Which post office is closest to a residence? How do the catchment areas of the post offices look like?
2.4 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 , for every site a Voronoi cell
2.4 Voronoi diagram
its nearest site p
i, for every site a Voronoi cell V(p
i): 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
• Which post office is closest to a residence? How do the catchment areas of the post offices look like?
2.4 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)
2.4 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
• Voronoi cells
2.4 Voronoi diagram
2.4 Voronoi diagram
2.4 Voronoi diagram
2.4 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
2.4 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
2.4 Voronoi diagram
A Voronoi diagram
with one point? A Voronoi diagram without points?
A uniform grid ?
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
ijof p
iand p
jdefines the half- space H(p
i, p
j), that contains all points being closer to p
ithan to p
j– The Voronoi cell V of the site p
icontains all points x for
2.4 Voronoi diagram
– The Voronoi cell V of the site p
icontains all points x for which holds: x closer to p
ithan to p
1and x closer to p
ithan to p
2and ...
and x closer to p
ithan to p
n→ V(p
i) = ∩
i≠jH(p
i, p
j)
– Intersection of n half-spaces takes O(n log n) time →
• Example: Construction of one Voronoi cell
2.4 Voronoi diagram
• Example: Construction of one Voronoi cell
2.4 Voronoi diagram
• Example: Construction of one Voronoi cell
2.4 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
2.4 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
– 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
2.4 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
2.4 Voronoi diagram
•
After leaving the circle the Voronoi point is determined
– Status:
•
Beach line composed of pieces of parabolas
– Events:
– Site events
• Occur when the sweep line reaches a site
• Generate a new parabola as part of the beach line
2.4 Voronoi diagram
– Circle events
• Are generated by a site event
• Generate a Voronoi point
2.4 Voronoi diagram
– Example: post offices in Braunschweig
2.4 Voronoi diagram
• Some Voronoi points are outside the shown section
2.4 Voronoi diagram
• Surface constructed of triangular faces
– Triangulation of reading points – Shading of the triangles
– Properties
2.4 Fields
Properties
• Exact
• Local
• Deterministic (depends on the triangulation)
• Gradual
• Triangulation
– Valid:
• No degenerated triangles (collinear)
• No overlap
• Intersections between borders
2.4 Fields
• Intersections between borders only at common edges or points
• Covers the whole space
– Regular:
• Domain is connected
– Greedy Triangulation Algorithm
2.4 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 2. sort D ascending on the length
d
1, . . . , d
m3. triangulation T ← P 4. for i ← 1 to m
5. if d
iintersects no segment in T
and is in 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
2.4 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
1,p
2,p
3and p,p
1,p
3) with a common edge
• Check if the circumcircle of one triangle(p
1,p
2,p
3) contains
2.4 Fields
• 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
1p
3and insert pp
2p3 p3
p
p
p
• Inverse Distance Weighting (IDW)
– The interpolant at P is determined by the values z
nof the n (nearest) neighbours P
1… P
nand the
normalized reciprocals of their distances d
nto P
2.4 Fields
– To original data points the measured value is assigned
∑
∑
=
−
=
−
⋅
=
n
i
u i n
i
i u
i
z d
d P
1 1
)
f(
with u > 0and ∀i di ≠ 0
– Properties:
• Exact
• Global/local
• Deterministic
• Abrupt
2.4 Fields
• Abrupt
• Dependent on distance
• Fast calculation
• Direction is not considered
• Problem: „Bull Eyes“
– Influence of the exponent u
2.4 Fields
u = 1
u = 0.5 u = 1
• (Ordinary) Kriging (Daniel Krige (1952))
– Statistical method
– Principle: utilization of spatial correlation for the estimation of values between the reading points
2.4 Fields
– "BLUE": Best Linear Unbiased Estimator
• Unbiased
• Linear: value at the point x
0is estimated as linear combination (weighted mean) of n reading points
• Exact estimator: the estimated values at the reading points
• 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
2.4 Fields
• 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)
– Procedure
1. Develop experimental variogram
2. Choose suitable variogram model
3. Setting up the Kriging equation
2.4 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
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
2.4 Fields
y
Reading points Variogram
C: 4
E: 3 D: 2
A: 5
renceof thevalues
AD BD
BE BC
0.8 1.0 γ(h)
– Choice of a variogram model
γ(h)
2.4 Fields
0.0 0.2 0.4
0.6 spherical
model
exponential model
Gaussian model
linear model
– Setting up the Kriging system
• The value z
0of x
0is estimated as weighted average of the values of the circumjacent reading points
2.4 Fields
∑
nj=
j
jz(x )
= λ z
1 0
• Wanted: weights λ
i,so that the estimator is unbiased and the variance is minimized
→ Constraint optimization problem
→ Solution by Lagrange multiplier ν
( )
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
2.4 Fields
– Properties
• Exact
• Local/global
• Stochastic
• Gradual
• Splines
– Goal: generate a surface with minimal curvature – Utilization of a sequence of different polynomials
(usually ≤ order 3) between the data points
2.4 Fields
– Properties
• Local
• Inexact
• G2 continuous
(curvature is continuous)
• 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
• Working Committee of the Surveying Authorities of
2.5 AAA-Project
• 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
– 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 for technical questions
2.5 AAA-Project
•
Consulting services for technical questions concerning organizational matters, educational affairs, human resource issues, examination matters and the right of use
•
Cooperation with subject-related organizations as well as with geodetic education and research institutions
•
Representation of the field of official surveying in Germany in the
http://www.adv-online.de/
• AFIS-ALKIS-ATKIS-Project (AAA-Project)
2.5 AAA-Project
– ISO/OGC-conform GDI base component – UML-Model
– Conversion: 2006 - 2010 (Lower Saxony from 2009) – Digital landscape model (DLM)
– Basic-DLM (1:25000),
2.5 AAA-Project
– Basic-DLM (1:25000),
• Object oriented vector data
• Complete
• Position accuracy (±3m for road
and stream network)
• Digital Elevation Model (DEM)
– Representative set of terrain points
– Consideration of skeleton lines, terrain edges and prominent points
2.5 AAA-Project
– Grating pitch 12,5m (DEM5) – Gauß-Krüger projection
– Different levels of accuracy
(±0.5m, ±1.5m in DEM5),
are given for every point
• Feature type catalogue contains 226 feature types
– Structured hierarchically
2.5 AAA-Project
– Feature types depend on scale, IS and federal state – Coverage criteria (geometric, semantic)
– Containing definitions and descriptions of the feature types, feature attributes and feature associations
– Modeling regulations define the way the features are
2.5 AAA-Project
– Modeling regulations define the way the features are
to be described and created
• Change in attribute values • Nodes of topological graphs
2.5 AAA-Project
• Change of object type • Multiple spatially seperated
areas
– Object types:
• Simple spatial object (REO: raumbezogenes Elementarobjekt)
• Simple non-spatial object (NREO: nicht raumbezogenes Elementarobjekt )
• Complex object (ZUSO: zusammengesetztes Objekt )
2.5 AAA-Project
• Complex object (ZUSO: zusammengesetztes Objekt )
• Point-set (PMO: Punktmengenobjekt)
– Models:
• Basic-DLM, DLM50, DLM250, DLM1000
• DGM2, DGM5,
– Attributes
2.5 AAA-Project
2.5 AAA-Project
2.5 AAA-Project
2.5 AAA-Project
2.5 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
2.5 AAA-Project
relation
• Metric and Euclidean algorithms
– Area
• Vector: integration
• Raster: number of cells * cell area
– Length, Circumference
2.6 Geometric Algorithms
x y x yi i i n
i
i 1 1
1
2 1
1
+ +
−
=
∑ −
– Length, Circumference
• Vector: euklidian distance of points
• Raster: number of cells
– Distance
• Vector: minimum distance between a point and a line
2.6 Geometric Algorithms
http://worboys.duckham.org/
• Raster: distance matrices for distance determination
– Buffering
• Vector:
2.6 Geometric Algorithms
• Raster: building of a distance matrix, threshold
– Centre of gravity (centroid)
• Raster: average of row- and column indices
• Vector:
2.6 Geometric Algorithms
∑
=
+ +
+ −
+
=
n
i
i i i
i i
i
s x x x y x y
x F
1
1 1
1)( )
6 ( 1
∑
+ −=
n
x y x
y y
y
y 1 ( )( )
∑
=
+ +
+ −
+
=
i
i i i
i i
i
s y y y x y x
y F
1
1 1
1)( )
6 ( 1
