Using Vector and Raster-Based Techniques in Categorical Map Generalization

Transcription

1 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Using Vector an Raster-Base Techniques in Categorical Map Generalization Beat Peter an Robert Weibel Department of Geography University of Zurich Winterthurerstrasse Zurich (Switzerlan) Phone: Fax: {beatp weibel}@geo.unizh.ch Abstract Categorical ata are a frequent ata type in GIS an thematic cartography. Therefore, comprehensive methoologies for the generalization of categorical ata in both the vector an the raster moel are urgently neee. After the presentation of a general framework an recommene workflow, generic cartographic constraints governing the generalization of categorical ata are specifie. In the next step, these constraints will be translate into tools for assessing the nee an the quality of generalization solutions as well as tools for achieving the necessary generalization transformations. These tools are usually associate with a particular ata moel. Typical an frequent generalization problems are use to stuy how the constraints can be best translate an parameterize in a particular ata moel, an how particular solutions perform in comparison to others. The last component of this paper relates to the conversion of ata between the vector an the raster moel an vice versa an explores how such operations can be usefully integrate into a generalization strategy. 1. Introuction Categorical maps are a frequent ata type in GIS applications an in thematic cartography. Examples inclue maps (or atabases) of soil, geology, vegetation, or classifie remote sensing images. Networks of political or aministrative bounaries can be consiere a special case of this maps type. Categorical maps are commonly moele as either vector ata (i.e., as polygonal maps or polygonal subivisions) or as raster ata. Raster categorical ata mainly originate from gri samples, remote sensing imagery, or interpolate an classifie point samples. Vector ata are usually igitize from the corresponing categorical maps. Although there are tools available in current commercial GIS an cartography systems that allow processing raster an vector categorical ata for purposes of analysis an isplay, specific methos for automate generalization of such ata are less well evelope. They represent mere aaptations of methos evelope elsewhere to the problem of categorical map generalization. For vector categorical maps line generalization algorithms are use instea of polygon-oriente methos. Techniques to generalize raster ata sets are essentially equivalent to simple pixel-base image processing operations, not respecting the object nature of raster polygons (regions, connecte components). More sophisticate methos of preliminary nature for both raster an vector categorical maps have been propose in the research literature, such as Schylberg (1993), Su et al. (1997) an Jaakkola (1998) for raster ata, or Muller an Wang (1992), e Berg et al. (1998) for vector moels respectively. However, they nee further improvement an integration into a coherent framework an workflow if the generalization of categorical maps is to be solve more comprehensively. The research reporte here buils on previous work (Weibel 1996, Baer an Weibel 1997, Peter 1997) an has two main objectives: to improve current methos for categorical map generalization an to evaluate if an how vector an raster-base techniques can be usefully integrate into a comprehensive generalization methoology. In orer to meet these objectives, three elements are looke at. These elements will be iscusse in sections 3 to 5, following the presentation of a general framework in section 2. The first element concerns the specification of so-calle generalization constraints, that is, conitions of geometric, topological, semantic, an Gestalt nature, which govern the process of categorical map generalization. The secon element of an integrate methoology has to o with the translation of the constraints into tools for assessing the nee an the effect or quality of generalization (assessment tools, measures) an tools for achieving the necessary generalization transformations (transformation tools, generalization algorithms). In-

2 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August variably, these tools will be associate with a particular ata moel. Using typical an frequent generalization tasks, it is stuie how the constraints can be best translate an parameterize in a particular ata moel, an how particular solutions perform in comparison to others. Finally, the thir component of this research relates to the conversion between raster an vector ata in both irections. Conversion may be necessary because categorical ata in raster or vector format nee to be combine with or aligne to other feature classes, which exist in another ata moel. It is also potentially of help to substitute for missing generalization functionality in one omain by exploiting generalization algorithms in the other, as has been suggeste by Su et al. (1998), for instance. Section 5 iscusses some of the problems that may occur uring the conversion process, an proposes ways to cope with these ifficulties as well as possible applications. The final section of the paper presents conclusions an irections for future research. 2. Framework for Categorical Map Generalization While most frameworks for the generalization of cartographic ata, like for instance those by Brassel an Weibel (1988) or Ruas an Plazanet (1996), provie generic proceural information, the one presente in this paper is esigne more specifically for categorical ata an provies more etail (see figure 1). Large parts of this framework may be consiere generic as well (e.g., constraints efinition an constraints translation). However, all parts are efine an instantiate specifically for categorical generalization, an apply almost exclusively to categorical ata (e.g. thematic generalization). MAP CONTROLS ANALYSIS THEMATIC GENERALIZATION - aggregation of categories - resampling of raster gri - map scale - cartographic principles - map purpose - source ata: - graphic limits - ata moel - output meium - acquisition metho - symbology - classification metho - classification probabilities CONSTRAINTS DEFINITION classes: spatial scope: - graphical - object - topological - category - structural - partition/map - Gestalt MEASURES DEFINITION / CONSTRAINTS TRANSLATION purpose: classes: - conflict etection - size - quality evaluation - istance an proximity - ata structuring/ - shape partitioning - topology - ensity an istribution - pattern an alignment Measures for Quality Evaluation PROCESS MODELLING QUALITY EVALUATION strategic: tactical: - operator selection - algorithm selection - operator sequencing - ata moel transformation - operator prioritizing - object-algorithm mapping - parameter selection - efault settings processing sequence functional relationship PROCESS EXECUTION Figure 1: Framework for categorical ata generalization

3 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August When analyzing map controls relevant for a generalization problem, specific issues for categorical ata arise mainly from the way ata has been obtaine. Categorical ata is frequently acquire from remote sensing sensors, point sampling methos or fielwork. Therefore, aspects like the classification metho use play a major role when eveloping a global generalization strategy. These issues influence the quality of the results that can be usefully achieve as well as the methos that are applicable to a wie egree. For instance, when a maximum likelihoo classifier was use, the resulting classification probabilities can be exploite in generalization operations where a istinction can be mae between ambiguously an unambiguously classifie regions. The next step in the process, thematic generalization, is unique to categorical ata. While in small-scale topographic maps features like minor roas can just be omitte, this is not possible in categorical maps since a continuous network of polygons is require by efinition. Thematic generalization is in general possible for both vector an raster ata an oes not require any geometric transformations. Resampling of ata to a coarser resolution using appropriate interpolation methos is, of course, a metho that can only be applie to raster ata, while aggregation of categories (e.g., eciuous an coniferous forest to forest) is not boun to a specific ata moel. The next two steps in process flow irection, constraints efinition an measures efinition/constraints translation, are generic an apply to constraint-base generalization in general. These two items together roughly represent what is terme structure recognition in the Brassel an Weibel (1988) conceptual framework. Both constraints efinition an constraints translation will be iscusse in more etail an with respect to categorical ata in the following sections. Process moeling is the next step in the framework. Here, in contrast to thematic generalization, geometric transformations of the ata are applie. This is vali for both vector an raster ata since although, in principle, raster ata is generalize by reclassifying cells, the concepts on how to o this rely on (iscrete) geometrical transformations. Process moeling can be ecompose into a strategic an a tactical part. The first part is to evelop a generalization strategy within the limits of the efine an formalize constraints as well as the preefine map controls. For categorical ata, global structural accuracy is in most cases more important than local positional accuracy, especially when ata is use as thematic backgroun (e.g. for linear or network ata). After appropriate operators have been selecte, their sequence an their tolerances regaring preefine values or banwiths has to be ecie upon (prioritizing). After a strategy has been formally eclare, tactical consierations follow. The possibility of ata moel transformation has to be inclue in tactical consierations since, as we will see, some algorithms can are consierably easier to implement, require less computational effort or generate visually more convincing solutions in either the vector or the raster moel. Data moel transformation is possible on a global or local basis an can be temporary or permanent epening on the esire ata structure of the resulting map. Evaluation an valiation of the results follow the execution of the selecte strategy. Measures for this purpose are provie by the constraints translation step, as the corresponing arrow in figure 1 inicates. The main goal of the evaluation step is to verify that all ientifie conflicts have been resolve, that no new conflicts have been introuce uring generalization (e.g., self-intersection of outlines ue to line filtering algorithms) an to ecie if values that implement structural an Gestalt measures lie within the specifie banwiths. Valiation of results inclues, of course, also visual inspection since, as we will see, complex concepts like visual balance cannot be formalize comprehensively. Therefore, a result can be rejecte, although all specifie parameters may lie within acceptable limits. For this reason, the last part of the framework forms a feeback loop. Several passes with changes of parameters, tactics or even the entire strategy might be necessary until a formally an visually acceptable result is achieve. Shoul this not be possible with the available means, the loop to the starting point has to be chosen, requiring a complete re-evaluation of the map controls. 3. Constraints to Categorical Data The goal of this section is to ientify constraints that apply to the generalization of categorical ata. No restrictions regaring the unerlying ata moel (vector or raster) are mae at this point. After iscussing basic issues of constraints with respect to the generalization of categorical ata, some basic properties for this ata type are introuce an a classification scheme is presente. A etaile list of generic constraints to the generalization of categorical ata follows, which allows eveloping a strategy for the representation of categorical ata at various scales an for a variety of purposes. 3.1 Key Aspects of Constraints A constraint in the context of generalization can be efine as a esign specification to which the solutions to a generalization problem shoul ahere (Weibel an Dutton 1998). A constraint is meant to limit the number of possible solutions without bining it to a particular action. This concept reflects the iea that more than one acceptable solution may exist to a given generalization problem. It is therefore well suite for eveloping flexible systems. Constraints originate from specific map controls applicable to a generalization problem. They are usually specifie as something to maintain or to avoi. The expression to respect is use for something to be ahere to as far as possible while to preserve is

4 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August use for topology relate constraints in the same sense as to maintain. Some constraints can be terme absolute (e.g., minimum size) while others esignate issues to be optimize (e.g., respect size istribution). Constraints like minimum size are terme intrinsic since they consier only one state of an object in a atabase while extrinsic constraints require two states (before an after generalization) for quality evaluation. Most constraints o not work inepenently, they are contextually relate an affect one another. A sub-system for prioritizing an managing constraints is therefore require inicating the priority an maximum tolerable severity for each constraint. Priority esignates the importance of a constraint in relation to others, while severity inicates the egree to which a constraint can be violate uner certain conitions. Finally, it is important to mention that not every constraint is neee in every situation. Constraints an appropriate mechanisms for their management provie the means for the evelopment of global strategies for holistic solutions to generalization problems, incluing both spatial an semantic aspects. 3.2 Constraints vs. Properties an Rules For categorical ata, a number of properties are efine. Although they coul also be formulate as constraints they are liste here separately since they reflect either intrinsic, low level aspects of this kin of ata or just efinitions mae for ata ealt with in this paper. Since only few properties are efine, the applicability of the presente concepts to most types of categorical ata shoul not be compromise. Properties of categorical ata enforce in this work are: Data covers the entire plane either as polygonal subivision or as a gri of raster cells. No unclassifie polygons or raster cells are permitte to occur as a result of generalization. In the case of vector ata, a ata moel using share primitives (i.e., share arcs an noes) within every feature class must be use. If technically possible, share primitives can also be use between feature classes to constrain common bounaries to each other. No object has common bounaries with other objects of the same category. If such a case occurs, the separating bounary is roppe. Rules are, compare to constraints, more fixe an not ynamically moifiable since they usually clearly inicate what particular action to take uner a certain conition. They follow a notation of the type IF <conition> THEN <action>. Since for the treatment of generalization problems many variations in spatial an attribute characteristics have to be consiere, a very large an harly manageable number of rules woul result. Working with constraints is therefore better suite for eveloping flexible generalization systems (Bear 1991). 3.3 Constraints an Data Moel Datasets with categorical ata consist either of polygons for vector ata or an array of cells for raster ata. Using the technique of connecte component labeling, so-calle regions can be forme for raster ata from connecte cells of the same category. Depening on the resolution an spatial structure of the ata, 4 or 8 cell connectivity can be use to form a region. In many cases 4 cell connectivity is more appropriate since very large an complex regions can result if 8 cell connectivity is chosen. GIS systems in use provie functionality for connecte component labeling but no topologic information is usually compute an associate with the forme regions. Instea of the term object, the expression patch is use from now on if both polygons an raster regions are meant. It is important to mention that constraints, in principle, express cartographic esign specifications which shoul be completely inepenent from specific ata moels. Cartographic constraints are first of all efine in terms of continuous geometry. Vector moels offer the most irect translation of continuous to iscrete geometry, while raster moels entail a significant iscretization by virtue of iscrete an systematic spatial sampling. Hence, some constraints particularly size an istance constraints cannot be easily an usefully accommoate in raster moels. The sampling interval (i.e., spatial resolution) has a premier influence on the potential of translating constraints into raster moels. Depening on whether the resolution is below or above the minimum visual separability istance, some constraints are simply not applicable an cannot be consiere. 3.4 Classification of Constraints For the classification of constraints the scheme of Weibel an Dutton (1998) is use. Constraints are classifie accoring to their function, which seems appropriate for generalization problems in a igital environment. Further subivision relates to the spatial application scope of constraints, which can either be a single patch, all patches of a category or a group of patches, a partition of the map or the whole map respectively. It has to be pointe out that, although constraints for single patches can be ientifie, at least two patches are always involve in the actual generalization process since patches in categorical maps share bounaries as per efinition. Therefore, constraints for single patches are mainly relate to the selection process rather than to the actual transformation. Four types of constraints are istinguishe: Graphic constraints mainly eal with aspects of perceptibility such as size, with an separability. Categorical ata consist entirely of area features where each category is assigne a ifferent color fill. Depening on the color, ifferent parameters for minimum perceptibility may apply. Furthermore, symbolization effects have to be consiere since the

5 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August line weight of the outlines of polygons or regions is affecte by scale change. Figure 2 presents an overview of istance relate constraints within an between patches for both vector an raster ata. Figure 2: Distance relate constraints within or between patches Topological constraints eal with basic topological relationships like connectivity, ajacency an containment, which shoul be maintaine when generalizing ata. Self-intersection an overlapping patches are also issues relate to topology. Since a ata moel using share primitives is use by efinition, overlapping patches o not exist an cannot be introuce with generalization. However, self-intersecting bounaries or intersections between ifferent bounaries can occur as a result of erroneous line generalization algorithms. Structural constraints efine criteria that escribe spatial an semantic properties of the ata. Spatial structural constraints eal mainly with the preservation of typical shapes (on the patch level) or with the preservation of patterns an alignments if multiple patches are involve. Semantic structural constraints eal with the preservation of the logical context of patches. For these constraints auxiliary ata such as roa an river networks or terrain moels are necessary as well as heuristics an omain knowlege about the nature of the ata being generalize. Gestalt constraints relate to aesthetic aspects. These inclue the preservation of the patch characteristics as well as the retention of the overall visual balance when multiple patches or the whole ataset is consiere. Gestalt constraints are complex an ifficult to formalize for use in igital systems but nevertheless important since they represent aspects of cartographic knowlege, which is not necessarily formalize. Gestalt constraints are enforce by the global strategy rather than by tactical ecisions. Constraints Relate to Patches 1. Minimum size (graphical): Patches, which are too small, can be either elete or enlarge 2a. Minimum istance (graphical, vector): The istance between consecutive vertices of a polygon outline shoul not be less than the minimum visual separability istance (see figure 2a) 2b. Minimum istance (graphical, raster): The istance between any parallel eges of the outlines (horizontal or vertical) of a region shoul not be less than the minimum visual separability istance (see figure 2b) 2c. Self-coalescence (graphical, vector): The istance between any vertices of a polygon outline shoul not be less than the minimum visual separability istance (see figure 2c) 3. Separability (graphical): The istance between two patches shoul not be less than the minimum visual separability istance (see figure 2 an e) 4. Separation (topological): Avoi separation of patches when eleting parts of it 5. Islans (topological): Patches, which can be ientifie as islans may be elete or enlarge but shoul not be amalgamate with other patches of the same category 6. Self-intersection (topological): Avoi introuction of self-intersection of patch outlines 7. Amalgamation (structural): Disjoint patches of the same category may be amalgamate 8. Collabsability (structural): The area of eliminate patches shoul be istribute among the neighboring patches 9. Shape/Angularity (structural): Respect the global shape an angularity of patches Constraints Relate to Categories 10. Size ratio (structural): Respect the size ratio for each category relative to the total area 11. Shape/Angularity (structural): Respect typical shapes an angularity of patches of each category 12. Size istribution (structural): Respect the given size istribution of patches for each category 13. Alignement/Pattern (Gestalt): Preserve typical alignments an patterns of patches of a category

6 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Constraints Relate to Partitions or Groups of Patches 14. Neighborhoo relations (topological): Preserve given neighborhoo relations 15. Spatial context (structural): Avoi introuction of illogical neighborhoo relations (e.g., house in a lake) 16. Aggregability (structural): Allow aggregation of categories if require an suitable super-categories exist 17. Auxiliary ata (structural): Observe constraints impose by auxiliary ata (e.g., roas, rivers, point features) 18. Alignment/Pattern (Gestalt): Preserve typical alignments an patterns of patches within the map or within a group of patches 19. Visual balance (Gestalt): Avoi gross changes in shape an istribution of patches, unless require by extreme scale change 20. Equal treatment (Gestalt): Ensure equal treatment within a partition of the map an avoi highly unequal treatment across all partitions The analysis of the above list of constraints shows that, compare to topographic maps, only relatively few topological issues have to be observe for the generalization of categorical ata. Accoring to Weibel (1996), it can be hoppe that, if all other constraint classes are satisfie, aesthetic principles (Gestalt constraints) are met to a large egree as well. The above list is ominate by graphical an structural constraints with the graphical constraints exclusively on the patch level. This inicates that a generalization strategy shoul tackle the problem from two sies simultaneously. On the patch level, methos take care of conflict ientification between or within patches (intrinsic graphic constraints) while simultaneously ecisions on which alternative to chose for conflict resolution are mae on a higher spatial level (category, whole map), consiering an monitoring the structural changes for an entire category or the whole ataset. 4. Translating Constraints to Measures an Generalization Algorithms The first part of this section iscusses key aspects an general requirements for geometric an semantic measures. Due to the overwhelming number of measures that can be foun in the literature their review within this paper seems neither possible nor useful. Instea, the secon part of this section emonstrates the process from instantiation of constraints over the selection of appropriate measures to the enforcement of constraints with the help of two frequent an typical generalization problems. This proceure shoul allow to better illustrating problems an alternatives as well as avantages an restrictions that arise from the ata moel. 4.1 Key Aspects of Measures A measure is efine as a proceure for computing measurements (numerical values). Measures are the basis for formal escriptions of relevant characteristics of geographical entities at the patch, category an map level. They allow assessing the nee for an the success of generalization. A measure can be a simple formula (e.g., area calculation) or a complex algorithm, which may even require the computation of auxiliary ata structures like a Delaunay triangulation. Measures can be either absolute (intrinsic), meaning that they can be interprete an applie accoring to the analysis of one state of the atabase or be relative (extrinsic), which means that measurements of two states of the atabase have to be compare an evaluate to ecie if a solution is acceptable or has to be rejecte. This inclues also subsequent testing for sie effects (e.g., self-intersection) that can be introuce by certain generalization algorithms (e.g., line simplification). Most measures exist for vector an raster ata but employ ifferent methos for their computation. The key concept for using measures in generalization systems is atabase enrichment. The measures compute are ae to the atabase as attributes. This inclues numerical values as well as topological information (if not compute automatically) or just flags that ientify a patch e.g. as islan or uneletable. Computation of statistical measures (e.g., histograms) is also consiere very useful for the analysis of the istribution an variability of patches an the evaluation of changes. 4.2 Classification of Measures Measures can be classifie accoring to the main characteristic they represent. This schema is influence to a large egree by the constraints efine in section 3. However, some measures may express more than one property, for example core area (FRAGSTATS 1994) which is use to characterize size in the first place but contains also information about the shape of a patch. The following classes of measures are istinguishe: Size measures Distance an proximity measures Shape measures Topological measures Density an istribution measures Pattern an alignment measures

7 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Requirements for Measures A useful measure shoul satisfy the following criteria: Ieally, it shoul escribe the intene property as precise as possible an shoul not be influence by other effects (orthogonality), be insensitive to outliers (robustness), be invariant to geometric transformations (geometric invariance), prouce ifferent results for ifferent configurations of characteristics an similar results for similar configurations (ifferentiation), be easy to calculate (ease of calculation), be easy to use (with only a limite number of parameters) (ease of use), an be easy to interpret (ease of interpretation). Ieally for a certain value (a measurement) only one possible configuration of the measure property shoul exist. While simple measures (e.g., area, perimeter) can fully meet the above criteria, this may not be the case for complex measures, especially those formalizing structural or Gestalt constraints (e.g., escription of pattern an alignment of patches). Abstract measures such as the fractal imension (representing shape aspects) are very ifficult to interpret an shoul therefore be use with care. In many cases, a set of measures is neee to cover the main characteristics of a spatial or thematic entity sufficiently. Many statistical measures such as patch size stanar eviation assume normal istribution of ata. Variability measures (e.g., patch size coefficient of variation) shoul only be interprete together with the total number of patches to avoi misinterpretation. For raster ata measures are compute for regions that were forme using the connecte component labeling metho. Measurements can vary ramatically epening on the connectivity rule employe for region builing (4 or 8 cell). Of course, measures for quality evaluation of generalize raster ata can only be usefully applie if the spatial resolution of the raster gri is not change uring the generalization process. 4.4 Translating Constraints to Measures The translation process of constraints to formal measures is execute by the system esigner at research an esign time. Optimally, at run time, system users nee only to specify the priority an the maximum tolerable severity of the various constraints for a given mapping task. Translating constraints to measures is a complex an crucial process within a constraint base generalization system. The unerlying concepts of constraints an measures are quite ifferent. As state in section 3, the goal of a constraint is to limit the number of acceptable solutions to a problem without bining it to a particular action. A measure on the other han is a formal mathematical concept, which makes use of clearly efine formulae or algorithms. Only few constraints, e.g. minimal size, can be translate to a measure on a 1:1 basis. Most concepts, such as shape or visual balance are rather fuzzy an ill-efine terms. Hence, it is almost impossible to formally escribe all properties that characterize such a spatial concept comprehensively. Translating constraints to measures is therefore also a selection process. The goal of the process is to make the main properties of a spatial entity available to formal mathematical escriptions. The egree to which this goal can be achieve has a major influence on the results of the generalization process. Generalization algorithms cannot eal with properties of patches or spatial entities that have not been formalize nor can changes be evaluate for possible rejection of solutions. In general, graphical constraints can be formalize more easily an precisely than structural or Gestalt constraints. 4.5 Examples Using typical an frequent generalization problems, the following two examples emonstrate how constraints are instantiate an how they can be translate to measures. While the constraints themselves are unspecific to any particular ata moel, algorithms for ata enrichment an conflict etection (e.g., measures) an ata generalization must be specialize for vector or raster ata. Problems an avantages of each ata moel resulting from this fact are iscusse. Only basic properties of categorical ata have been incorporate in the escription of generalization algorithms. Countless others can be specifie to moify the generalization process, respecting specific aspects of the ata use. Furthermore it shoul be pointe out that the problems an solutions presente in both examples are normally part of an integrate generalization strategy of interrelate operators an algorithms an shoul not be looke at in isolation. Example 1: Detecting an Resolving Conflicts Impose by the Minimum Size Constraint Minimum size constraints are straightforwar an easy to translate to a measure. Conflict ientification is simple an methos for conflict resolution are not very challenging. However, we will show that for the selection of the appropriate generalization operator aitional information nees to be consiere. The measure for size is area. For polygons, area calculations are stanar GIS functions. For raster ata, the area of a region is represente by the number of consecutive cells of the same category. The lower limit of the value for minimum area is the minimum perceptibility size but shoul be selecte higher ue to other relevant map controls (e.g. map purpose or output meia). In principle, two operators exist for conflict resolution. A patch violating the minimum size constraint can be either elete or enlarge until the size is above the specifie minimum. The possibility to amalgamate patches of the same category will be iscusse in example 2 an is not consiere here. Every patch, even if it is

8 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August very small, represents not only itself but also its category in the map. To be able to ecie which operator to use for a particular patch, further constraints, especially structural an Gestalt constraints have to be instantiate an translate to measures allowing to look at a patch in its spatial context. Such measures are: Total number of patches of a category Total area of a category relative to total map area Ratio of the area of a category relative to areas of other categories Size istribution of the patches of a category Spatial istribution or concentration of the patches of a category Topological information about neighborhoo relations of patches Semantic information about neighborhoo relations of patches Observing these statistical measures, or their moification by generalization respectively, allows assessing potential structural changes that influence the visual appearance of a map. The strategic goal when resolving size relate conflicts is to preserve the given istribution of the patches of a category as far as possible. If, for example, most patches of a frequent category are very small, the overall structure woul not be maintaine if the eletion operator was selecte for all patches. Structural an Gestalt constraints (e.g., maintain visual balance) woul be violate. With the help of the above mentione measures, ifferent scenarios can be compute to assess if changes in total area an istribution can be tolerate before the actual generalization is carrie out. For a first scenario, patches with an area just below the efine minimum size coul be enlarge while patches where conflict violation is severe coul be elete. As a secon criterion, istances to nearest patches of the same category coul be observe as well. The further away a patch is from others of the same category, the more structurally important it is for its category. Further criteria (e.g., the importance of the categories involve) are possible epening on specific properties of the ata. Several algorithms that implement these operators can be foun in the literature. For patches with only a single neighboring patch (islans), elimination is easy. The polygons coorinates are simply elete from the atabase (vector) or all cells of a region are assigne the value of the surrouning region. Cases where a polygon has more than one neighbor require more effort. Baer an Weibel (1997) have evaluate methos for this operator. They propose a solution base on the computation of a skeleton for the polygon to be eliminate (see figure 3). The area of the polygon is istribute equally among its neighbors an introuction of topological error is avoie.!! " Figure 3: Elimination of a polygon using a skeleton algorithm (Baer an Weibel 1997) A simpler metho works with the polygons noes, which are isplace in the irection of the center of gravity. With this metho, however, introuction of topological error is possible if complex polygons are eliminate (see figure 4). # $ % & ' ( ) * + *, -. /,. # * % $, # $ 0 * ) - $.,. / - 1 * ) *, - * 0. / 2 0 ( 3 $ - 4 Figure 4: Topological error after application of the noe isplacement metho (Baer 1997) Methos for raster ata require less algorithmic effort ue to the ata moel. Normally, regions are eroe from the outsie using for example a majority filter. More sophisticate rules reflecting structural an semantic knowlege can be efine to control the erosion process (Peter 1997). Polygons can be enlarge raially by scaling the vector between the center of gravity an the polygon noes. This is basically the same metho as the one escribe for elimination; therefore the same restrictions apply. Uneven expansion is possible (e.g., weighte with respect to strong an weak neighbor regions). Raster regions are enlarge by reclassification of cells of ajacent regions. As for vector ata, rules for controlling enlargement can be implemente easily.

9 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Detecting an resolving minimum size conflicts while respecting structural an Gestalt constraints can be conucte equally well for both vector an raster ata. No moel offers significantly better methos that woul justify ata transformation from one moel to the other. In general, methos for vector ata are computationally more complex but working with continuous ata allows better control of results which might not always be possible with raster ata. The avantage of raster ata is that implementation is straightforwar an rules respecting specific properties can be easily integrate. Example 2: Amalgamation of Disjoint Patches of the Same Category This example iscusses measures an generalization algorithms available for amalgamation of patches of the same category. Amalgamation may be require for resolving minimum istance conflicts an, more generally, to reuce the number of patches an spatial variability in a map to meet specific map controls (e.g., map purpose). In aition to the measures presente here, the statistical measures mentione in example 1 have to be observe as well to prevent violation of structural an Gestalt constraints. Searching for caniate patches for amalgamation requires istance measures to be compute. One possibility is the computation of buffers for each iniviual polygon to both of its sies. Buffer with woul be set to half the istance up to which patches shoul be amalgamate (i.e., half the minimum visual separability istance). Intersecting buffers will ientify the esire situations. For polygonal ata, a cell can be calculate as a measure for the egree of overlap (Baer 1997, Baer an Weibel 1997). As illustrate in figure 5 this cell can serve as a basis for the actual amalgamation process as well as for other operators such as isplacement (Baer an Weibel 1997) : ; 8 < : = : >? : = A B C : ; 8 < < A ; D > : E 8 = 8 Figure 5: Buffer operations for amalgamation an isplacement operators (Baer an Weibel 1997) A secon possibility is the computation of a conforming Delaunay triangulation. In this case, a global triangulation representing istance is calculate before ecisions are mae where amalgamation woul be possible an useful (Baer 1997). The main avantage of this metho is that, once compute, several alternatives can easily be teste. Triangulations may also be of goo use for the actual generalization process. In general, triangles connecting the polygons are reclassifie to the category of the polygons to be amalgamate. This may prouce visually not very convincing solutions., More sophisticate solutions using curves to connect polygons are possible but require complex an computationally expensive algorithms. Methos base on Delaunay triangulations have for instance been implemente by Baer (1997) an Jones et al. (1995) who use constraine Delaunay triangulations not only for the amalgamation operator but also for polygon exaggeration an collapse (see figure 6). Figure 6: Amalgamation of polygons base on a Delaunay triangulation after Jones et al. (1995) The use of cost-istances instea of Eucliean istances is a major avantage of raster ata. This concept allows easy integration of semantic information an knowlege in the amalgamation operator. Important regions or their category respectively can be given very high costs to prevent parts of them from being eliminate ue to amalgamation. On the other han, assigning low costs to the respective cells can facilitate amalgamation over objects of unimportant categories. With this metho it is, for instance, possible to prevent amalgamation of two forest regions over a narrow lake.

10 Z ^ ^ Z Z ^ \ \ \ _ { { { } } } Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Algorithms for the amalgamation operator have been implemente by Schylberg (1993), Jaakkola (1998) an Peter (1997) for lanuse/lancover raster ata. Schylberg (1993) uses a simple grow-an-shrink algorithm. Objects overlapping or touching in grown state remain connecte after re-shrinking by the same amount of cells. Although very simple to implement, this metho might not be aequate in situations where spatial variability is high (see figure 7). F G H I N O J K L M P L Q Q R J S G T U V J K L M N O P L Q Q R W X W Q F W X W K L Y G L F T H U J Figure 7: Grow-an-shrink algorithm for the amalgamation operator Peter (1997) has aapte an moifie a metho propose by Brown et al. (1996) using cost-istances. Cells are weighte accoring to the cost-istance to the least cost path between caniate regions. As an aitional criterion, Eucliean istance to the nearest caniate region is consiere as well. This results in a classification of the cells between an aroun caniate regions with the lowest values right between them provie the respective categories were given low costs. With this metho naturally looking results can be achieve. Figure 8 illustrates this metho schematically. Amalgamation of regions of category A is promote over cells of category C (low costs) while cells of category B act as a barrier (high costs). Z Z Z [ \ \ [ [ [ [ \ ] [ [ \ \ \ ] Z Z ] Z Z ] \ ] ] h i j k l j m n o p m q r s i j o p t k p m p u v s w i o q o p m q r s i x y z _ ` a _ b _ c a e f c ` f c a c c c a c b b g c a c c f _ g a ƒ ƒ ˆ ƒ Š Œ Ž ƒ Ž ƒ Š ƒ Š Œ Ž ˆ Ž Ž { { { } } { { { } ~ { { } } } ~ { { ~ { { ~ } ~ ~ Ž Œ ƒ Š Ž ˆ Ž Š ƒ Œ ˆ ƒ ˆ Š Figure 8: Amalgamation of regions base on cost-istance methos In general, raster base methos for the amalgamation operator offer more flexibility an are easier to implement than methos for vector ata. Appropriate measures are more easily compute an the use of cost istances is a major avantage. Furthermore, computation of triangulations is not trivial since numerous special cases have to be respecte (Baer 1997). On the other han, an this is the major problem with raster ata, quality of the visual appearance of solutions epen to a large egree on the spatial resolution of ata. 5. Integration of Vector an Raster-Base Methos The purpose of this section is to show that although technically speaking the conversion of categorical ata is a straightforwar process, it is by no means trivial to hanle when integrate with the generalization process. After iscussing general issues relate to generalization proceures where ata moel transformations are involve, examples are presente to emonstrate when an how conversion from raster to vector an vector to raster can be usefully integrate in a generalization strategy. The remainer of the section iscusses specific possibilities of local ata transformation. 5.1 Data Moel Transformation: Applications an Limitations In the previous sections it has been shown that integrate generalization systems can be evelope for categorical ata in a vector as well as in a raster environment. Means for translating constraints an generalization algorithms are commonly available for both ata moels. Since some methos can be implemente more easily or more precisely in one moel, a generalization strategy incorporating algorithms from both ata moels, using their respective avantages has to be consiere. Several authors, for instance Bo et al. (1998), have propose methos where vector ata is converte to the raster structure, then generalize an finally transforme back to the vector moel (vector raster vector). In general, this strategy makes more sense than the reverse (raster vector raster) since most raster operators, especially those in-

11 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August volving neighborhoo an contextual operations, are simpler an easier to implement than their counterparts in the vector omain. This may compensate for the relative loss of precision an semantic expressiveness that occurs when ata is transforme. When ata is reconverte after generalization, the source ata (vector) might help with the interpretation of possible ambiguities that arise in the raster to vector phase. No convincing argument, however, can be thought of that woul justify a generalization strategy where raster ata is converte to the vector moel for generalization with subsequent re-transformation back to raster structure. When a generalization strategy involving bi-irectional conversion is implemente, various effects an problems have to be analyze. Unavoiably, transformation of ata between continuous an iscrete reference systems an vice versa results in a loss of information an/or precision. Piwowar et al. (1990) have implemente several conversion algorithms an have evaluate them base on qualitative, quantitative an efficiency criteria. None of them coul satisfy all requirements at the same time, meaning for example that an algorithm which minimizes changes in region area (for a given cell resolution) may isplace an istort the same region heavily. Since bi-irectional conversion usually oubles the effects mentione an these effects cannot be easily controlle, we o not recommen it for general use espite the fact that some generalization algorithms might be easier to implement in a raster environment. In aition, parameters for vector to raster conversion (e.g., sampling interval) are usually efine globally for the whole ataset. This may result in partial or complete loss of important local geometric information that cannot be taken care of uring the generalization process. Using local conversion, which will be iscusse later, might provie a solution for this kin of problem. We propose to istinguish between the source representation an the target representation of a map or a ataset. Commonly, one shoul ten to maintain the representation of the source ata. That is, conversion is to be avoie unless it serves a specific purpose, certain operators are only available in a particular representation, or the intene target representation is ifferent from the source representation. A specific purpose is, for instance, if ata from ifferent sources nee to be integrate. Given a raster lanuse ataset an several vector atasets, the lanuse ata woul then be generalize in raster moe, transforme to vector an finally integrate (i.e. matche) with the vector ata. This ata or the constraints impose by them shoul alreay be consiere an respecte uring ata generalization as far as possible to avoi integration problems as well as topological an semantic error (e.g., isolate rivers coul exist after a small lake has been elete). Smoothing of the outlines of patches is an operator that is execute more precisely an flexibly with continuous than with iscrete ata an may therefore require ata structure transformation from raster to vector. Converting ata to the raster moel applies a iscrete sampling istance to a vector ataset. Theoretically, if the sampling interval (i.e., the spatial resolution) was chosen to be equal to the machine precision (e.g., float), then a regular raster coul represent the geometry as precise as vector ata. However, that seems impractical. Even if storage costs coul be neglecte, the excessively high resolution woul o away with the avantages of raster in neighborhoo operations because the instantaneous fiel of view (e.g., a 5x5 kernel) woul only cover minute portions of the ataset. The sampling theorem is more practical to efine an optimizes the resolution of a raster ataset. It can make sure that geometric accuracy is not lost unintentionally. In that case the resolution of a ataset has to be selecte twice as high as the imensions of the smallest patch that shoul be resolve. However, oversampling as mentione above can be use eliberately in orer to obtain a smoothing effect. 5.2 Raster to Vector Conversion Smoothing of the outlines of complex patches is a typical example, which may require conversion from raster to the vector structure. The visual quality that can be achieve by smoothing algorithms in the raster omain, for instance moe filtering, or eroe smoothing (Monmonier 1983) is limite by the resolution of the ataset. Other raster base methos work with resampling of the raster gri to a higher resolution (oversampling). Depening on the oversampling factor chosen (e.g., 4) each raster cell woul then consist of a number of sub-cells (e.g., 16 for factor 4). The smoothing effect is achieve by removing or aing sub-cells as illustrate in figure 9. This metho can result in a consierable increase of the amount of ata an steppe lines may still be visible unless the size of the sub-cells is below the minimum visual separability istance. š œ ž Ÿ ž œ ž œ œ Ÿ š ª «Figure 9: Raster moe generalization base on oversampling Continuous geometry provies better means for smoothing the outlines of patches. Conversion to the vector moel prouces polygons that exactly match the outlines of the regions they represent, meaning that only right angles occur. The main task of the smoothing operation is to remove steppe lines. As Peter (1997) has shown, the commonly use line simplification algorithms (e.g., Douglas Peucker) will not yiel the esire results an may even estroy the effects of the previous generalization operations. A simple yet effective algorithm has been evelope by Herzog et al. (1983).

12 Thir ICA Workshop on Progress in Automate Map Generalization, Ottawa, August Designe for the simplification of bounaries extracte from raster ata, the algorithm ientifies regularly steppe portions of polygon outlines an replaces them by straight lines (see figure 10). This metho can easily be moifie to meet specific requirements. After its application, various line simplification an smoothing algorithms can be employe for further refinement. ± ² ³ µ ³ µ µ ¹ º µ º ² ¹» ² ¼ ½ ¾ ³ µ ±»» ³ À ² º ³»» º ³ µ» ² ² ¼ Á  µ ¾» ² ± ³ µ Ã Ä ± µ ³ Å Æ Ç È É Ê Ë Ì Í µ ¾ µ À º ² º Î µ» ¹ ±  ² µ ³ ¾ µ µ ¾ ³ ² µ À ³ µ Î ²» µ Ï µ ¾ ¹ µ» º ³ Ð ³ º Í ³ ² À ² ³ µ µ ¾ » л ² µ µ ¾ ±  ² µ ¼ Ì ² ³ µ ¾ º ³ Ï ¹ µ» º  ³» ² Î ² º ¾ ³ ² ± ¼ Figure 10: Smoothing of transforme raster regions in vector moe after Herzog et al. (1983) 5.3 Vector to Raster Conversion Cases where vector ata is converte to the raster structure for generalization without subsequent re-transformation (biirectional conversion) are rather rare. A possible application might be that the target representation is raster an that vector ata (e.g., line or point features) nee to be integrate with an existing raster ataset. In such cases integration shoul take place before any generalization process is execute. Since positional precision an semantic information are partly lost uring the conversion process, it woul not be of great use to generalize vector ata prior to vector-raster transformation. Furthermore, the structure of the existing regions will alter when ata is integrate, as will the preconitions for the generalization process. Basically, all cells that intersect with a line are assigne the value of the newly integrate category. A thinning algorithm can be applie to reuce the with of rasterize linear features to one cell for atasets with a course resolution. This operation improves the visual quality of rasterize lines but may cause consierable isplacement. If regions are built, 8 cell connectivity shoul be use to allow iagonal connectivity of cells. The size of a point feature in the raster moel is always one cell. Symbolization will be lost uring conversion as will, at least to a certain egree, positional accuracy. If the resolution of the ataset is below the minimum visual separability size at target scale, rasterize point features have to be either enlarge or cannot be represente at all. Rasterize point features, or their containment information respectively, can be use to avoi topological error from being introuce by generalization. Rasterize lines can be given high costs to control amalgamation when cost-istance algorithms are use as mentione in section 4. Figure 11 illustrates the concept of converting an integrating vector ata with raster ata an subsequent raster base generalization. Ñ Ò Ó Ô Õ Ö Ó Ñ Ø Ò Ú Ñ Ò Ô Ö Ò Û Ø Ù Õ Ô Ò Ô Ö Ú Ñ á Ó Ñ Ø Ò â ã Ú Õ ã Ó Ñ Ø Ò ä Ü Ø Ñ Ò Ó Ú Ñ Ò Ô Ù Ô Ó Ý Ö Ô Þ ß Ó à Ñ Ò Ò Ñ Ò Õ Ö Ô Þ Ü Ô å Ó Ñ Ò Õ Ñ Ò Ó Ô Õ Ö Ó Ô Û Û Ó Figure 11: Integration of vector ata through vector to raster conversion 5.4 Local Conversion Some of the above mentione problems with generalization strategies involving bi-irectional ata conversion can be avoie if the transformation is kept local. The principle is illustrate in figure 12. Such transformations normally only occur from vector to raster, where generalization operations are applie, followe by subsequent re-transformation to the vector moel (vector raster vector). The main avantage of local conversions is that the sampling resolution for the raster part can be coorinate with the specific properties of the patches involve an the planne generalization algorithms. Algorithms in question are mostly those which are more easily implemente for raster than vector ata, like methos that use cost-istances or involve neighborhoo operations. For the implementation of a local transformation, the minimum bouning rectangle of the esire polygons or area of interest is calculate. A margin is ae to avoi ege effects. After applying the generalization algorithms, ata is re-converte to the vector moel. A major rawback of this metho is that the computational effort might be consierable for large atasets an/or smaller portions where