A conservation law for map space in portrayal and generalisation

Transcription

1 A conservation law for map space in portrayal and generalisation Alistair Edwardes University of Zurich, Department of Geography, Winterthurerstrasse 109, 8057 Zurich, Switzerland, Themes: Theoretical Cartography, Cartographic Generalization and Multiple Representation, Internet Location-Based Services, Mobile Mapping and Navigation Systems. Abstract The paper describes new approaches in computational cartography to assist online map portrayal e.g., in web map and location-based services. Such services demand increasing levels of flexibility to support portrayal operations such as the on-the-fly combination (conflation) of data from different sources in response to ad hoc database queries and dynamic generalisation of information in response to different device profiles, user preferences and interaction. The research described in the paper takes the position of a conservation law for mapping to build models of representation that can assist in portrayal tasks, specifically cartographic generalisation and data conflation (to support online information retrieval services). It uses a grid mesh that is fitted to the map objects, such that the grid shares the same topology as the features. That is, a feature s geometry lies on a grid vertex (if it is a point object), follows a grid edge or crosses a grid face diagonally (if it is a linear object). This grid is then used as a parameter space to compute coordinate transformations that model the deformations that occur when features are symbolised. By adding structural constraints to the grid it is then shown how these transformations can be used to perform conflation and generalisation operations. Two cartographic activities where this issue is encountered are conflation and cartographic generalisation. Here, conflation refers to the task of portraying and combining data from ad hoc queries on geo-referenced databases with a topographic base map. This task is performed frequently in online mapping systems, for example web- and location based services. Cartographic generalisation is a portrayal task that seeks to balance requirements for good map legibility with accuracy and optimal information content. It does so by abstracting both the spatial relationships and semantics of the map information and reorganising features accordingly. Introduction Maps are an analogue representational media (Palmer, 1978) that communicate information in two ways; symbolically and spatially (Berendt et al, 1998). Symbolic information is represented explicitly by presenting attribute information of features pictographically. Spatial information is represented implicitly by using the spatial characteristics that constrain the map medium as analogous to those constraining geographic space (Sloman, 1985). Hence spatial relationships in geographic space can be said to be represented more or less faithfully in the map space. These two dichotomous forms of information description do not generally sit happily together. Symbolising features on a map, beyond their real world footprint, necessarily impacts on the ability of the map to represent spatial relationships. Two inter-related mapping activities can be distinguished affecting this relationship: portrayal and scale selection.

2 ?? Portrayal relates to the selection and application of a set of signs that will be used to communicate information symbolically. Different forms of portrayal will have different effects on the ability to describe spatial relations.?? The scale selection affects the way in which the dimensions of the symbol scale relative to how properties of space scale. Figure 1 illustrates this for a circular point symbol, shown in the inset for different scales. 300 Distance (Map Units) distance 0.5 mm sq 1 mm sq 2 mm sq Scale Denominator Figure 1. Scaling, representation and spatial relationships Figure 1. shows how the distance in map units between two point circle symbols of areas 0, 0.5, 1 and 2 mm 2 changes with scale. It can be seen that the scaling rates are dependent on the property of the symbol, here varied by size. Hence the larger the symbol the faster the degradation of spatial relationships as scale is decreased. Maps are always portrayed at some scale so these effects are always evident. However they are particularly marked in mobile information services where the resolution of the device requires larger symbols than would be required for other display mediums. The operation of cartographic generalisation can be thought of as mediating in the relationship between these two forms of information representation. On the one hand, generalisation can omit and reconfigure information in order to preserve the communication of spatial relationships. On the other, generalisation can change the form of portrayal to describe phenomena at a different level of organisation Symbolisation and spatial relations This impact of symbolisation on space can be considered in one of two ways according to whether we view space as relative or absolute. Viewing space as absolute, the symbol is thought

3 of as occupying a region of space, or consuming white space. Viewing space relatively the impact of symbology is to distort space around the point. Figure 1 illustrates these perspectives for a single symbolised point feature and a set of point features viewed in terms of absolute and relative space. a) b) c) d) Figure 1. Distortions of space and spatial relations caused by symbology using views of space as absolute a) and c) and relative b) and d). Viewing space as absolute is the usual perspective taken in cartometric analysis and map generalisation. Measurements are made using the underlying unsymbolised data points. This allows the use of standard geometric tools for analysis. However, when the points are considered including their symbology, this absolute view does not always represent the space satisfactorily. The region of space in the direct neighbourhood of symbolised point can no longer be considered since it is now covered. At large cartographic scales this effect is generally unimportant, however at smaller and smaller map scales the ability for the map space to act as an analogue to the real world is increasingly compromised. The relative view of space attempts to rectify the analogy by is replacing a static uniform view of space with a more elastic, deformable one. The effect of symbology is then to distort spatial relations. The relative view accepts that local distortions of space mean that not all the relationships that can be determined in a completely metric space can be supported. Instead a less constrained space should be considered that preserves fewer properties. Habel (2003) terms this representational commitment, where the map author commits to a specific order within an axiomatic system of geometries. Habel s description of representational commitment relates mainly to global aspects of map design, such as choosing whether to represent data in a geometrically accurate or a highly abstract schematized form. Barkowsky and Freska (1997) also suggest an ordered system for representational commitment, which they term representational correspondence, that they suggest might be used to guide more local generalisation processing. An example of such a hierarchy is shown in Figure 2.

4 8-sector orientation 4-sector orientation Dichotomic orientation Precise Angles No orientation information Fuzzy angles Decreasing levels of precision in correspondence to geographic reality Figure 2. An example of a hierarchical structure for representation correspondence from Barkowsky and Freska (1997) Law of conservation applied to the map space An alternative way of considering this perspective is as a conservation law applied to the properties of the map space. This is simply to say that the amount of space in a map remains constant independent of symbolisation. Map space around a feature is expanded by symbolisation and, because of the conservation law, compressed elsewhere to compensate for this. Hence the properties of the space are distorted across the surface of the map. This view raises a number of possibilities for map generalisation.?? Modelling the distortions of the underlying space can help to: Identify areas of conflict in the map caused by symbols overlapping. Describe information density across the map space Provide transformation based generalisation operators, for example for displacement?? Committing to different levels of a system of representation, based on distortions can help to: partition the map space based on regions of the space holding the same level of commitment and therefore localise the generalisation processing; reduce the search space of generalisation operators by only considering solutions that satisfy the properties of the commitment; and guide the generalisation process by suggesting the types of generalisation operation and the order in which they are applied. Define generalisation operators that generalise spatial relations explicitly, e.g. typification Taking this perspective suggests two classes of generalisation operation; continuous operations and discontinuous operations. Continuous operations are essentially smooth functions driven by changes in map resolution due to scale. These include simplification of shape, smoothing, morphing between representations and displacement. Discontinuous operations involve sharp changes. They are driven more by changes in the semantics of the information due to levels of detail and levels of organisation of geographic phenomena. Such operations include; collapse,

5 aggregation and elimination. The application area for models described here is to continuous operations A map space conserving model In order to model a deformable space two components are required. To model the structure of space a system of discretisation is needed, for example a quadrilateral mesh or triangulation. A topology should be defined on this to allow the model to be traversed and so that changes to the model can be propagated through the space. It must be possible to associate attributes to this structure, for example material properties and stresses. Secondly, a method for modelling the deformation, due some attribute that is forcing it, is required, for example using a finite element method or particle system. For this research a quadrilateral mesh was used to model the structure of map space. The symbolisation of map features were embedded into the mesh through an antialiased rasterisation (e.g. Gupta and Sproull, 1981) type process. This associated a scaling value to each quad, which described how much the symbolisation of a map feature was causing the map space to locally expand. Finally the process of deformation was using the method of non-linear magnification (Keahey, 1997), described next. This propagated the deformation through the map space. These steps are shown in Figure 3. a) b) c) d) Figure 3. Steps of map space deformation. a) regular quadrilateral mesh used to structure space, b) line feature with associated symbology added to mesh. c) Symbol width attributed to quad cells using anti-aliased rasterisation process. d) Non-linear magnification used to deform space Non-Linear Magnification. The technique of non-linear magnification (Keahey 1997) was used to model the local magnification of space due to symbolisation and ensure that the resulting deformation was smooth. This technique works by considering the current area around each grid cell together with the area required to support the symbolisation (the area magnified). If the current area is too small the mesh vertex attempts to increase the area around it by pushing its neighbours away. If the area is too big the vertex pulls its neighbours towards it to reduce the area. If there is no symbolisation in a grid cell it still must attempt to preserve its initial unit area. This is to try to ensure that the distribution of density between the symbolised and unsymbolised map space is maintained. The process is applied as an iterative process with vertices pushing and pulling until they converge in a smooth state. Figure 4 describes the process.

6 Target area (magnification * unit area) Target area Error Error Current area Current area Push Neighbours Away Pull Neighbours Towards Figure 4. Two principal operations of non-linear magnification The application of this technique to cartographic features is shown in Figure 5. Figure 5. Two principal operations of non-linear magnification Figure 5 illustrates the underlying mesh representing space. The space required by symbolisation has been exaggerated to emphasise the effect.

7 Deformation as an operator One use of the techniques was to model displacement. As can be seen, the process deforms mesh cells from squares to arbitrary quadrilaterals. A four point transformation (perspective or bilinear transformation) can be computed using this information and a displacement projection defined for a point. Figure 7. The displacement of a set of points based on the symbol size By revering the processes and minifying instead of magnifying, a smoothing operation can also be demonstrated. The operator is shown in Fig. 8, though it is far from ideal. Figure 8. Line smoothing using a minification process Control problems Two problems were found in applying the procedure. First, the regular grid used to structure space introduced a sampling problem when attributing the symbolisation. It was difficult to define a grid size optimally for the whole map which sampled only a single point or segment of a line feature in a cell without generation many redundant cells. Secondly, the process displaced all features indiscriminately which would result in shape changes such as exaggeration in bends. What was desired instead was to lock the position of some features and displace others with respect to these. In this way an additional level of hierarchical control can be afforded. The solution to both these problems was the same. The mesh edges were aligned with the linear map features, such that a feature always ran along the edge of a cell or diagonally across a cell face. The vertices incident to these edges could then be locked, and hence the feature would not

8 move due to the pushing or pulling of a neighbour. It could itself induce a deformation on its neighbouring verticies. Feature aligned mesh Implementing this solution followed the method described in Biermann et al (2001) and Boier- Martin et al (2004), who developed it for computer graphics to embed sharp features in a subdivision surface. The principal for creating the representation is to construct a one-to-one mapping between the grid and the features such that any piecewise linear feature curve either follows along mesh edges or crosses mesh faces diagonally. Biermann et al (2001) term this the approximation property. To achieve this property a two step iterative processes of mid-point subdivision of the grid followed by a snapping of grid points to feature curve is used. The stages are shown in figure 9. Starting grid Grid refinement Snapping Figure 9. Grid alignment The application of the method to cartographic data is show Figure 10. Figure 10. Grid alignment

9 The integration of the techniques with non-linear magnification is the subject of on going research. Conclusion. The paper presents on-going research into the use of continuous, deformable models of space for map generalisation and cartometric analysis. The philosophy of applying a conservation law to map space offers an interesting new alternative for graphical generalisation operations, providing a top-down, transformation based approach. The approaches operate in real-time for use in online (web and mobile) GIS. Acknowledgements This work is part of the European Union Framework 5 project WebPark: Geographically relevant information for mobile users in protected areas" (IST ). We gratefully acknowledge the financial support of the Swiss Office of Education and Science (OFES) within the scope of this project (BBW Nr ). References Palmer, S. E. (1978). Fundamental aspects of cognitive representation. In E. Rosch & B. B. Lloyd (Eds.), Cognition and categorization, pp ( Hillsdale, NJ: Lawrence Erlbaum) Sloman, A (1985) Afterthoughts on Analogical Representations. In Readings in Knowledge Representation; R. J. Brachman and H. J. Levesque (Eds), pp , (Los Altos,CA: Morgan Kaufman) Berendt, B., Barkowsky, T., Freksa, C., and Kelter, S. (1998). Spatial representation with aspect maps. In C. Freksa, C. Habel, & K. F. Wender (Eds.), Spatial cognition - An interdisciplinary approach to representing and processing spatial knowledge. (Berlin: Springer) Biermann H., Martin I. M., Zorin D., Bernardini F., (2001) Sharp Features on Multiresolution Subdivision Surfaces. Pacific Graphics 2001 Conference Proceedings. Tokyo, Japan. October 16-18, Boier-Martin, I. Ronfard, R., and Bernardini F. (2004).Detail-Preserving Variational Surface Design with Multiresolution Constraints, Proceedings of Shape Modeling International 2004, pp , Genova, Italy. Habel, C. (2003) Representational Commitment in Maps In Duckham, M., Goodchild, M.F., Worboys, M.F. (Eds.): Foundations of Geographic Information Science, pp , (Taylor & Francis, London and New York) Barkowsky, T., and Freksa, C. (1997). Cognitive requirements on making and interpreting maps. In S. Hirtle & A. Frank (Eds.), Spatial information theory: A theoretical basis for GIS pp , (Berlin: Springer) Edwardes, A., Burghardt, D and Weibel, R., (2003). WebPark location based services for species search in recreation area. Proceedings of the 21st International Cartographic Conference (ICC), Cartographic Renaissance, August 10-16, 2003, Durban, South Africa de Berg, M., Bose, P., Cheong, O., and Morin. P. (2004) On simplifying dot maps. Computational Geometry: Theory and Applications, 27(1), pp Keahey, A. (1997) Nonlinear Magnification. PhD thesis, Department of Computer Science, Indiana University, December 1997.

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