1. L9 – Generalization algorithms
NGEN06 & TEK230:
Algorithms in Geographical Information Systems
by: Sadegh Jamali (source: Lecture notes in GIS, Lars Harrie)

2. Background Cartographic generalization:
L9- Generalization algorithms
Background
Cartographic generalization:
an application where previous algorithms support it.
has always been a central issue in map production.
the process of simplifying a representation to suit the scale and purpose of a map.
such as generalizing a map at a scale of e.g. 1: to produce a map at a smaller scale of e.g. 1:
By the use of digital maps, in the 1970s, studies of automating the generalization process was initialized.
since then some of the developed algorithms have been implemented in commercial mapping and GIS software.

3. Aim Content 1. Introduction to cartographic generalization
L9- Generalization algorithms
Aim
to learn about cartographic generalization
to get examples of the use of previous described algorithms (foremost in LN4 to LN8)
Content
1. Introduction to cartographic generalization
2. Generalization operatros

4. Introduction Definition of cartographic generalisation
L9- Generalization algorithms
Introduction
Definition of cartographic generalisation
“Selection and simplified representation of detail appropriate to the scale and/or the purpose of a map” (ICA, 1973, p ).
Types of cartographic generalization methods
Raster versus vector generalisation
Raster methods: applied to raster data
Vector methods: applied to vector data
Real-time versus map production generalization
Real time: - only concerned with displaying data (no new data-set)
- a multiple-scale geographic information system
- low computational complexity prerequisite
Map production:
- the computational efficiency demands are not so important
- higher requirements for cartographic quality

5. Examples of cartographic generalisation
L9- Generalization algorithms
Examples of cartographic generalisation
Scale = 1:50 000
Scale = 1:10 000
Figure: manually generalization of a vector-format topographic map.
© Lantmäteriverket, Sweden. Printed by permission

6. L9- Generalization algorithms
Figure: Automatic raster-based generalization of CORINE land-cover data (modified from Jaakola, 1997).

7. Generalization is a complex problem!
L9- Generalization algorithms
Generalization is a complex problem!
There is no simple algorithm to generate generalized maps!
To simplify the automation of generalization, the process is divided into some
sub-processes (operators)

8. Generalization operators
L9- Generalization algorithms
Generalization operators
Simplification
Smoothing
Aggregation
Amalgamation
Selection
Classification
Displacement
Collapse

9. L9- Generalization algorithms
Simplification
It reduces the number of points in a line, while retaining the most representative points.
Figure: Simplification transformation.

10. Douglas-Peucker algorithm (Douglas and Peucker, 1973)
L9- Generalization algorithms
Douglas-Peucker algorithm (Douglas and Peucker, 1973)
the most commonly used simplification algorithm
computationally efficient
it requires O(n log n) time where n= the number of points on the line.
the algorithm is based on computing the distances between points and lines (LN5)
Figure: The Douglas-Peucker algorithm

11. Wang and Müller algorithm (1998):
L9- Generalization algorithms
Wang and Müller algorithm (1998):
line generalization based on an analysis of bends using rules, such as:
- small bends should be removed
- two similar consecutive bends should be combined into one bend
- three bends can be represented as two, etc

12. L9- Generalization algorithms
Figure: A comparison of a line simplified with Douglas-Peucker algorithm and Wang- Müller algorithm. (Wang and Müller, 1998, p. 14)

13. Smoothing Smoothing aims at reducing the angularity of lines
L9- Generalization algorithms
Smoothing
Smoothing aims at reducing the angularity of lines
Figure: Smoothing transformation
Commonly used smoothing algorithms:
spline-functions and B-splines, Gaussian smoothing and frequency filters.
B-splines are 𝒄 𝟐 continuous (continuous first and second derivatives) and, accordingly, have a smooth appearance (LN5).

14. Gaussian smoothing The original line is represented by a vector:
L9- Generalization algorithms
Gaussian smoothing
The original line is represented by a vector:

15. L9- Generalization algorithms
Aggregation
This operator reduces the number of objects by aggregating objects of any dimension to area objects.
Figure: Aggregation transformation

16. Joubran and Gabay (2000) algorithm:
L9- Generalization algorithms
Joubran and Gabay (2000) algorithm:
- creating Delaunay triangulation to the original objects
- removing the edges longer than a given threshold
Figure: Aggregation of a building cluster. (Joubran and Gabay, 2000, p. 422)

17. L9- Generalization algorithms
Amalgamation
It combines area objects to accomplish a simpler appearance of the objects.
Figure: Amalgamation transformation.
Note: it is mainly performed for land use and land cover data (raster data).

18. Morphological operators
L9- Generalization algorithms
Morphological operators
Expand
Shrink

19. An example the simplified and amalgamated map the original raster map
L9- Generalization algorithms
An example
the simplified and
amalgamated map
the original raster map
Figure: Example of simplification and amalgamation of a raster map (Schylberg, 1993, p. 103)

20. L9- Generalization algorithms
Selection
Selection aims at solving problems associated with too high density of objects by leaving some of them out.
Figure: Selection transformation

21. Steps of a selection algorithm:
L9- Generalization algorithms
Steps of a selection algorithm:
identifying regions of high density of objects
This is performed by a clustering technique of building objects (e.g., a minimum spanning tree (LN3; O’Rourke, for details)).
2) removing some of the objects within these regions
When clusters were defined as connected building objects, a density analysis is performed and then objects are removed.
Radical law: the ratio of the number of objects in two maps should equal the square root of the ratio of the map scales

22. Classification reclassifying objects to a new object type
L9- Generalization algorithms
Classification
reclassifying objects to a new object type
Figure 9.12: Classification transformation

23. L9- Generalization algorithms
Problem: maps use classical set theory (i.e. unique object type for each object) but classic set operations do not always give satisfactory results for e.g. geographic entities.
Membership
function
Note:
non-classically set based operators are useful for thematics data such as soil and vegetation.
Figure: Membership functions for classical set theory (a), fuzzy set theory (b) and rough set theory (c).

24. L9- Generalization algorithms
Displacement
Visualization of geographic data on small-scale and mid-scale maps often leads to spatial conflicts, mainly due to the exaggeration of the symbol size. In the displacement transformation spatial conflicts are solved by moving and/or distorting objects
Figure: Displacement transformation

25. Types of optimization techniques used for displacement:
L9- Generalization algorithms
Types of optimization techniques used for displacement:
1) combinatorial
2) optimization with continuous variables.
n objects, each with k possible positions, so it’s a combinatorial optimization problem with a search space of size 𝒌 𝒏 .
Figure (the combinatorial method):
a) Original map. b) Map at reduced scale, c) Spatial conflicts have been resolved by displacement. d) Trial positions; tp1 is the original position of the object in question; Ddisp is the maximum displacement distance.
(Ware and Jones, 1998)

26. Optimization with continuous variables
L9- Generalization algorithms
Optimization with continuous variables
The main idea is to define a set of constraints (rules) that the generalized map should obey, such as:
there should be no proximity conflicts between the objects and that the objects’ characteristics should not change.
Then by using mathematical methods the map is stretched (as in a rubber sheet transformation, cf. LN1) to find a solution that is optimal (according to the constraints).

27. L9- Generalization algorithms
Harrie (2001)
Bader (2001)
Figure: Automatic displacement of road objects using continuous variables.

28. L9- Generalization algorithms
Collapse
The collapse operator decomposes an area object into a point (or line) object, or two line objects into one line object.
Figure: Collapse transformation
Note: a common approach to collapsing an area object into a point object is to set the geometry of the point object equal to the center of the minimum bounding rectangle of the area object.