1. Representation of spatial data
GIS architecture, raster and vector, conversion, administrative subdivisions: polygon-ring, topology, extended DCEL, continuous data: contours, DEMs, TINs

2. Thematic map layers
Separate storage of data according to theme: map layers
GIS typically use tens to hundreds of map layers
For example: municipality borders, land use, cadastral boundaries, water pipes, churches, etc.

4. Example map layers
Census data, 1995
(U.S.A.)

5. Geometry, topology and attributes
Geometry: coordinates
Topology: adjacency relations of objects
Attributes: properties, values
Example: Country map of South America
Geometry: coordinates of the borders Topology: which countries border which Attributes: names of countries, population, etc.

6. GIS data architectures
Structure/philosophy of how geometrical/ topological and attribute information should be stored and is accessible 1. Twofold or dual architecture 2. Layered architecture 3. Integrated architecture

7. Pure database approach
Geometry and attributes in same relational data model
+ More concurrent users possible
- Objects must be reduced to atomic parts, and partitioned over various tables
- Retrieving original objects is expensive (join)
- Query language doesn’t know spatial concepts (area, intersects, …)

8. Twofold architecture Attributes in a DBMS
Geometry in separate files (by theme)
Connection by unique identifier
Two subsystems: the DBMS and one for the geometry
E.g. ArcGIS

9. Twofold: pros and cons + DB generally known (at organizations)
+ Geometry fast and easily accessible
- More users difficult to incorporate concurrently
- Maintaining consistency between 2 systems is tricky
- Efficient transactions (optimizations) tricky because of two systems

10. Layered architecture
RDBMS with additional, geographically intelligent layer
Layer contains extension with geographical data types, e.g. Point, pointcluster, polygon, line
Layer offers extension to query language, and translates for the actual RDBMS

11. Layered: pros and cons + More concurrent users possible
+ Spatial object types and concepts are present
- Middle layer not extendible
- Topological relations must be determined when they are needed
- Practice: no object type for subdivision

12. Integrated architecture
Spatial object types and functions in the database itself
RDBMS or OO
E.g. Postgres

13. Integrated: pros and cons
+ No translation in middle layer necessary
+ Extendible with additional types and functions
- Extension is rather complex
- Practice: less GIS-functionality present by default

14. Representation of geometry
Two main approaches: raster and vector
Can also be mixed in a GIS, any map layer
Conversion raster-vector and vice versa possible
Representation depends on type of data, way of acquisition, desired operations, etc.

15. Raster structure
Division of space into equal-size cells (squares, pixels)
Theme gives cells a value (nominal, ordinal, interval, ratio, vector, …)
Cells should not contain any further spatial information (more detail)

16. Data in raster form Point object in raster form Line object in
Plane object in
raster form

17. Raster maps

18. Raster: pros and cons Simple structure Simple operations
Obtained after scanning, remote sensing
Less suitable for point and line objects: representation does not follow intuition
Network analysis difficult
Not adaptive: no difference in detail possible in different regions
Either expensive in memory, or little precision
Not obtained after digitizing

19. Raster: memory reduction
Run-length encoding: no 2-dim array but coding start pixel with value and length of run
Block encoding: 2-dim version
Disadvantage: makes structure and operations much more complex
(34,67) forest 9
(34,67) forest 4,6

20. Vector structure Objects stored as points, lines and areas
Points have coordinates; lines connect points; areas are delimited by lines
Attributes are stored with the objects (point, line or areal)

21. Vector: pros and cons
Elegant structure; fits with both point, line and areal objects
Small storage consumption
Precise
Adaptive: additional control points possible
Network and cluster analysis possible
Obtained after digitizing
Relatively complex
Map overlay and buffer computation complex

22. Vector representation of a region
Not necessarily simply-connected:
NL has islands
NL has holes (Baarle-Nassau / Baarle-Hertog); there are even regions in these holes

23. Representation of subdivisions

24. Subdivisions: spaghetti model
Every chain is represented by a list with coordinate pairs
Split nodes are doubly stored
Areas are not present explicitly C1 C2 C5 C4 C3 C6
C1: (..,..), (..,..), (..,..), ...
C2: (..,..), (..,..), (..,..), ...
C3: (..,..), (..,..), (..,..), ...

25. Subdivisions: polygon ring structure
Every area is represented by a list with coordinate pairs
Control points are doubly stored
Neighbor areas are difficult to determine
Consistency is difficult to maintain P1 P2 P3
P1: (..,..), (..,..), (..,..), ...
P2: (..,..), (..,..), (..,..), ...
P3: (..,..), (..,..), (..,..), ...

26. Subdivisions: topological structure
Nodes are objects with coordinates
Edges are connections of nodes
Sequences of edges along polygon boundaries are connected
Polygons are objects of which the boundary is stored
Doubly-connected edge list

27. Subdivisions: topological chain structure
Splitting nodes are objects with coordinates
Chains are connections of splitting nodes and contain zero or more nodes with coordinates
Sequences of chains along polygon boundaries are connected
Polygons are objects of which the boundary is stored
Doubly-connected chain list

28. Vector structures Memory Duplication Polygon Topology
retrieve retrieve
Spaghetti Polygon ring DC edge list DC chain list

29. Raster-vector conversion
E.g. for data integration
Vector-to-raster: Like in computer graphics: scan-conversion of lines, etc.
Raster-to-vector: Consider pixel sides between pixels with different values as boundary and put in vector representation  Thinning, line simplification

30. Thinning
Raster-vector
conversion
Thinning

31. Line simplification Douglas-Peucker algorithm from 1973
Input: chain p1, …, pn and error   p1 pn

32. DP-algorithm  Draw line segment between first and last point
If all points in between are within error: ready
Otherwise, determine farthest point and recursively continue on the part until farthest point and the part after farthest point

33. DP-algorithm DP-standard(i, j, )
Determine farthest point pk between pi and pj
If distance(pk, pi pj) >  then DP-standard(i, k, )
DP-standard(k, j, )
Return the concatenation of
the simplifications

34.   

35.   

36. Properties of the DP-algorithm
DP-algorithm does not minimize the number of points in the simplification   
DP-algorithm
Optimal

37. Properties of the DP-algorithm
Determining farthest point takes O(n) time
Whole algorithm takes T(n) = T(m) + T(n-m+1) + O(n), T(2) = O(1) time, splitting in m and n-m+1 points
“Fair” split gives O(n log n) time
Worst case gives quadratic time

38. Properties of the DP-algorithm
DP-algorithm may give self-intersections in the output 
Solution: test output for self-intersections and continue adding control points if necessary

39. Improved DP-algorithm
DP-improved(i, j, )
Simp = DP-standard(i, j, )
V = set of intersecting segments of Simp Repeat
For all segments s  V Refine(s) in Simp Do 1 refinement à la DP by adding the farthest point V = set of intersecting segments of Simp Until V is empty

40. Continuous data representation
Digital Elevation Model (DEM)
Data on interval or ratio measurement scale
Data values of points near by will usually be not very different
Representation is necessarily an approximation: finite representation of information with infinite detail
Raster (1x) or vector (2x)

41. Elevation models Raster Vector Vector (Elevation) grid21 20 21 20 15 20 19 25 10 10
(Elevation) grid
Contour line model
Triangulation (TIN; triangulated irregular network)

42. Grid elevation model

43. TIN elevation model

44. Elevation models
Contour model well-suited for visualisation, not for representation or storage
Interpretations grid: - elevation whole cel: not a continuous model - elevation middle cel: interpolation needed; how?
Advantage grid: simple storage, operations simple too
Advantage TIN: more efficient in storage, adaptive

45. Interpolation for grid20 18 20 18 18 22 18 22 20 18
Linear interpolation; saddle point problem 18 22 20 18 20 18 18 22 18 22
Linear interpolation; additional point
Non-linear
interpolation
( ) / 4 =19.5

46. Topological TIN structure
With explicit vertex and triangle representation t2 w t3 t1 t1 t2 t t u v u w t3 v
x, y-coordinates and elevation

47. Topological TIN structure
With explicit vertex and triangle representation t2 w t3 t1 t1 t2 t t u v u w t3 v
Because t1 has pointers to two the same vertices as t, we can determine their shared edge, even though it is not represented explicitly

48. Topological TIN structure
With explicit vertex and triangle representation w w t1 t2 t2 t1 t u v t t3 v u t3

49. Topological TIN structure
Alternatively, edges have an explicit representation too w t1 t2 w t1 t e1 e2 e1 e2 u e3 v t3 t u e3

50. Summary representation
Objects have geometry and attributes, at least the attributes are in a database
Geometry can be stored in raster or vector form; each has advantages and disadvantages
Important geometric types of representations are those for subdivisions and for elevation models
For subdivisions, the doubly-connected chain list is the most suitable structure
For elevation models, grids or TINs are most useful