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

L9 – Generalization algorithms

NGEN06 & TEK230: Algorithms in Geographical Information Systems

by: Sadegh Jamali (source: Lecture notes in GIS, Lars Harrie)1

L9- Generalization algorithms

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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. 1:10 000

to produce a map at a smaller scale of e.g. 1:50 000.

• 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.


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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


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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. 51.10).

Types of cartographic generalization methods

Raster versus vector generalisationRaster methods: applied to raster dataVector 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


© Lantmäteriverket, Sweden. Printed by permission 507-98-4091

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Examples of cartographic generalisation

Figure: manually generalization of a vector-format topographic map.

Scale = 1:10 000 Scale = 1:50 000

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

2525m 100100m

225225m 500500m


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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)

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Generalization operators

• Simplification • Smoothing • Aggregation • Amalgamation• Selection• Classification • Displacement • Collapse


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Simplification

It reduces the number of points in a line, while retaining the most representative points.

Figure: Simplification transformation.


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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


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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


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


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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 continuous (continuous first and second

derivatives) and, accordingly, have a smooth appearance (LN5).


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Gaussian smoothing The original line is represented by a vector:


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Aggregation This operator reduces the number of objects by aggregating objects of any dimension to area objects.

Figure: Aggregation transformation


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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)


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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).


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Morphological operators

Shrink

Expand

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Figure: Example of simplification and amalgamation of a raster map (Schylberg, 1993, p. 103)

the original raster map

An example

the simplified and amalgamated map


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Selection Selection aims at solving problems associated with too high density of objects by leaving some of them out.

Figure: Selection transformation


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Steps of a selection algorithm: 1) 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,

1993 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

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Classification reclassifying objects to a new object type

Figure 9.12: Classification transformation


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Figure: Membership functions for classical set theory (a), fuzzy set theory (b) and rough set theory (c).

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.

Note: non-classically set based operators are useful for thematics data such as soil and vegetation.


Mem

bers

hip

func

tion

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Displacement

Figure: Displacement transformation

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


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Types of optimization techniques used for displacement: 1) combinatorial 2) optimization with continuous variables.

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)


n objects, each with k possible positions, so it’s a combinatorial optimization problem with a search space of size .

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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).


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Harrie (2001)

Bader (2001)

Figure: Automatic displacement of road objects using continuous variables.


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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.


Documents

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