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

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. 1:10 000 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.

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

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. 51.10). 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

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 507-98-4091

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

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)

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

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

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

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

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

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).

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

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

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)

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).

Morphological operators L9- Generalization algorithms Morphological operators Expand Shrink

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)

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

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, 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

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

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).

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

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)

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).

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

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.