Algorithmica 2001 Springer-Verlag New York Inc.

Transcription

1 Algorithmica (2001) 30: DOI: /s Algorithmica 2001 Springer-Verlag New York Inc. Contextual Building Typification in Automated Map Generalization 1 N. Regnauld 2 Abstract. Cartographic generalization aims to represent geographical information on a map whose specifications are different from those of the original database. Generalization often implies scale reduction, which generates legibility problems. To be readable at smaller scale, geographical objects often need to be enlarged, which generates problems of overlapping features or map congestion. To manage this problem with respect to buildings, we present a method of selection based on the typification principle that creates a result with fewer objects, but preserves the initial pattern of distribution. For this we use a graph of proximity on the building set, which is analysed and segmented with respect to various criteria, taken from gestalt theory. This analysis provides geographical information that is attached to each group of buildings such as the mean size of buildings, shape of the group, and density. This information is independent of scale. The information from the analysis stage is used to define methods to represent them at the target scale. The aim is to preserve the pattern as far as possible, preserve similarities and differences between the groups with regard to density, size and orientation of buildings. We present some results that have been obtained using the platform Stratège, developed in the COGIT laboratory at the Institut Géographique National, Paris. Key Words. Building generalization, Gestalt, Minimum Spanning Tree. 1. Introduction. Cartographic generalization is an abstraction process that seeks to reduce the quantity of information in accordance with particular objectives, typically scale and theme. When the objective is a reduction in scale, generalization is used to convert the data into a legible form. When the objective is thematic, generalization is used to emphasize specific themes while reducing information relative to other ones. Automation of the generalization process has been an issue for the last 20 years. Early studies have focused on developing independent algorithms, especially on line simplification [1], [2]. More recent studies focus more on management of tools to achieve specific complex generalization, for example see [3] for urban generalization at large scale, and see [4] for generalization of lake patches. In an attempt to make better use of available tools, some studies have been done to classify the algorithms and measures according to the type of design problem [5] [7]. The latest developments deal with contextual generalization, aimed at controlling the triggering of algorithms depending on the spatial context in which they are applied. This includes studies on modelling issues [8], [9], spatial analysis tools development [10] (graph theory to generalize network), [11], [12] (extention of the Delaunay triangulation to support displacements), and studies on 1 This research was partially funded by the EU under the ESPRIT programme of research through the AGENT Project. 2 Department of Geography, University of Edinburgh, Drummond Street, Edinburgh EH8 9XP, Scotland. nrr@geo.ed.ac.uk. Received January 26, 1999; revised September 30, Communicated by M. van Kreveld. Online publication March 9, 2001.

2 Contextual Building Typification in Automated Map Generalization 313 strategies, involving planning, assessment and reaction [13]. At every stage of the generalization process, the key issue is to provide the system with knowledge (condition of use of an algorithm, how to tune parameters, which sequence gives the best results in which situation). This has justified researches on knowledge acquisition, such as [14] [16]. In this research we focus on the development of an algorithm for generalizing buildings from 1:15,000 to medium scales (between 1:25,000 and 1:50,000). Our focus is not on the generalization of building shapes, which has already been studied [3], [17] [19], but on the generalization of groups of buildings in order to transform them into a readable form at a smaller scale. This paper discusses the automation of a process traditionally done using manual techniques and attempts to achieve the goals associated with the manual approach, i.e. reducing the number of buildings in a built-up area while preserving the pattern and local characteristics. The issues that need to be resolved can best be discussed by looking at some examples. Figure 1(a) shows a sample map at 1:25,000 and the corresponding map at 1:50,000 (Figure 1(b)), both from maps produced by the national mapping agency of France, the IGN. Figure 1(b) is the result of a manual generalization of Figure 1(a). In the upper left of each are their reductions. We can see that the reduction of the 1:25,000 is less readable than the 1:50,000 map. The objects are too small and too close to be readable. By comparing the two, we can see more easily what the cartographer did when they generalized the map. Looking at individual objects, we notice that they have been enlarged. At a higher level, we see that some objects have been removed, but use of space has been preserved. Buildings are still aligned along the roads, and the differences in density are preserved, as is the general pattern. The discussion of an automated solution to this problem is presented under six headings. We begin by describing all the constraints that should be satisfied. These are used to guide the map generalization process. A second part describes the general method we use to satisfy these constraints. The third part describes the analysis method that makes explicit the relationships between buildings prior to generalization. The analysis segments the dataset into groups by using minimum spanning trees, size and orientation homogeneity, and other perception criteria. These groups are used to control our method of global typification, which reduce the number of buildings in each group, preserving the visual separation between the groups and their intrinsic characteristics. This typification method is described in the fourth part. The fifth section describes the criteria necessary to check the validity of the result and as information for further generalization. The last part presents some results obtained via our prototype implemented in Stratège, one of the experimental generalization platforms developed at the IGN in Paris. 2. An Ontology of Buildings. When human cartographers generalize a map, they know by looking at the original map, the relationships between each object, what information is conveyed by each object, both as a single entity (the house) and collectively (a residential area). An automated generalization process starts from a limited set of information. It has at its disposal a database describing each object individually and independently from one another. This means that a building is just a list of co-ordinates describing its boundary together with a semantic code. It is quite easy to compute its size,

3 314 N. Regnauld Fig. 1. Cartographic generalization of groups of buildings in an urban context.

4 Contextual Building Typification in Automated Map Generalization 315 Fig. 2. Perception constraints. but its size relative to other buildings is just as important. For this relative information we need to compare each building among a set of buildings. In a similar vein, the system has no information about the distribution of the objects: which ones are neighbours, or part of the same town, or which ones are isolated. It is important to identify such additional information in order to generalize buildings in an urban area successfully. This information is needed to control the generalization process in response to a set of (sometimes) competing objectives. Some objectives are well defined (legibility constraints) whilst others are not because they include aesthetic criteria or are difficult to formalize (such as spatial pattern or homogeneity). When generalizing buildings it is very important to maintain (1) legibility, (2) the visual identity of each building, and (3) the pattern among a local group of buildings. The fourth goal is to maintain the overall density of buildings across a region of the map. Each of these four issues is now discussed in detail Legibility Constraints. The legibility constraints ensure that the map will be readable by a human observer. Perception capabilities of the human eye depend on the distance of reading, the contrast between objects and their brightness or luminosity. Legibility constraints can be divided in three classes: Perception. Perception constraints are those that specify a minimum size for objects or the detail of objects. Figure 2 shows two constraints. The first one specifies the minimum size of buildings (a square of 0.5 mm), and the second one specifies the minimum length for an edge of a building boundary (0.3 mm). These values come from [20]. Separation. The separation threshold is the minimum distance between two features (0.15 mm). Maximum density. The maximum density is the number of objects per unit area, and is the point at which the map becomes locally unreadable. When generalizing, these constraints become somewhat in competition with one another. For example enlarging objects to satisfy minimum size constraint tends to reduce the distance between them and increase the density of objects. Some solution have already been presented to solve conflict using displacement [21], [22], but these method may not provide any solution when space is limited. A solution may require considerable change including elimination of some buildings. To control these changes, we need constraints to ensure that the overall information conveyed by the map will be preserved, even if some object are missing or displaced. This is the reason for defining the following three criteria.

5 316 N. Regnauld Fig. 3. Proximity Visual Identity. The following qualities are intended to preserve the visual characteristics that help the reader to identify an object as a building. We identified three qualities: The shape. In general, a building is represented in a map by an area whose boundary has orthogonal angles. Airault [23] defines a method to improve this characteristic of buildings after distortions in shape have occurred. Size. The size of a building helps to convey its type. Usually the smallest ones are residential buildings, while the largest are industrial or administrative. There tends to be a correlation between shape and size. The smaller a building is, the less detailed its boundary can afford to be for reasons of legibility. Colour. The colour of an object can help to identify a building and sometimes its type. The focus of this paper is in the presence/absence and non-overlapping of buildings and therefore colour is not considered as part of this focus Spatial Organization. In addition to individual characteristics, the spatial relationships between the objects in a map contribute collectively to the information conveyed by the entire map the gestalt of the map. This is the reason why we need to add constraints governing the spatial organization of objects. To express these constraints, we use ideas drawn from gestalt theory (the study of the factors influencing grouping perception). Three of them proximity, similarity and continuity, drawn from [24] are relevant to the distribution of buildings. Proximity. Proximity is one of the most critical criteria for the visual grouping of objects. In Figure 3 three groups of points can easily be identified. For our purpose this is of great importance because it helps us to identify objects in conflict due to their proximity. Similarity. This criterion relates to our capacity to group objects that look similar. In Figure 4 we can divide the set of objects into two groups: the squared ones and the round ones. Buildings can also be grouped in terms of shape, size and orientation. Fig. 4. Similarity.

6 Contextual Building Typification in Automated Map Generalization 317 Fig. 5. Continuity. Continuity. The last criterion is that of continuity. We can identify groups of objects according to their regular linear disposition, as in Figure 5. This characteristic is important for our purposes because buildings are usually located along roads, often with a strong regularity in their position, especially within small residential areas Homogeneity. In considering all the above issues, it is equally important that any modifications are done in a homogeneous manner (in order to preserve overall balance). This should be done at various levels of detail. For example at the city level, the decrease in the number of buildings and the road density from the centre to the peripheral of the city must be preserved. More locally, in a particular area of the city, major differences between building sizes and between densities of local groups of buildings should also be preserved. More generally, differences and similarities should be preserved as much as possible so that the map is well balanced and has good aesthetic quality Summary. We described in this section the need to generalize within a set of constraints to insure a good quality map overall. Having identified four main kinds of constraints: legibility, visual identity, spatial organization and homogeneity, we must now address the following four issues: How to manage these competing constraints to ensure a generalized solution in every case? How to compute all the characteristics described above? How to use these constraints during the generalization process? How to assess the result: how far are the goals reached? The next four parts propose solutions for these four issues. 3. Methodology. We are concerned with finding a method to derive from a source map, a representation of a set of buildings at the target scale. The proposed method ensures that a good compromise is reached between all the constraints. Related research has produced some interesting results. For example Ware and Jones [12] address the problem of generalizing buildings in urban areas, but at larger scales, where selection is rarely needed. Muller and Zeshen [4] address a similar problem but with respect to lakes. The process of elimination uses a strategy of rich get richer before making some selection with regards to a size threshold. This strategy does not fit for the buildings problem because of the bad effects that amalgamation has on buildings.

7 318 N. Regnauld The methodology that we used is based on the divide and conquer principle. We segment the initial set of buildings into homogeneous groups. Then the characteristics of each group are used to build a new representation of the group according to the new scale, whilst preserving their essential characteristics. The method is comprised of three main steps: The first step consists of analysing the source set of buildings in order to partition the set into groups. The analysis is based on information relative to the pattern of buildings and their local characteristic. This results in a partition of the source set, into groups of buildings whose characteristics make them visually distinguishable. Then the global typification step processes each group in turn and creates a graphical representation suitable for the target scale. The term global typification is based on the typification operation, which has often been identified as part of the main generalization algorithms [6], [7], but is extended so that it processes each group whilst taking into account the interrelationships between groups. The general principle is that typification involves enlarging buildings, eliminating some and gives the remainder a pattern that reflects the distribution of the source data. The last step consists of assessing the result with regard to the initial constraints. This is an important step since our method is constrained by various conditions, and alternative solutions may be more appropriate. So we need to assess the result in order to be able to consider alternate strategies. 4. Analysis and Partitioning. Analysis is undertaken to partition the space into meaningful groups of buildings. Meaningful groups of buildings are those that would be typically identified by visual inspection. We therefore use criteria from grouping perception theory (gestalt theory, see Section 2.3). The road and river networks are used to partition the buildings in the first instance (Figure 6). Roads are considered to be a pragmatic and meaningful way of partitioning buildings into groups [25] (there being a close interdependence between roads and buildings). Partition of the source set of buildings is done in three steps: 1. Initially, we have a set of buildings that belong to the same partition. The first task is to make explicit the proximity relationships between these buildings. This is done Fig. 6. Using roads to partition the map space.

8 Contextual Building Typification in Automated Map Generalization 319 Fig. 7. The partitioning process. by computing a proximity graph. These proximity relationships are of considerable interest because they show where proximity conflicts occur. 2. The constitution of groups works by successive segmentation of the initial graph. The point of separation depends on the level of homogeneity among the groups. The aim is to find groups with the greatest homogeneity because typification performs better on groups with a regular pattern. For this reason groups may be further subdivided if the group is not sufficiently homogeneous. 3. A final step of reorganization is needed because the previous step leads to groups that may be locally too fragmented where no regular pattern exists. This results in bad typification. The second factor influencing the quality of a typification is the size of the group. The larger a group the easier it is to conceal the absence of an eliminated object. This is why a local reorganization of the groups is sometimes necessary. It seeks to merge the small groups that are close and have similar characteristics. The final result is groups of buildings ready for typification. Figure 7 illustrates the entire process. Square boxes represent the three action steps of the process, and elliptic ones are data that are either input or output from the process Building a Proximity Graph. Proximity is the primary criterion for grouping buildings. Ahuja and Tuceryan [26] analyse the capabilities of different methods to extract proximity relationships in order to detect perceptible structures among dots. Ahuja and Tuceryan observe that the minimum spanning tree (MST) is not an ideal method of defining proximity between clouds of points because the tree is sensitive to small changes in the position of points. However, in this research, the distribution of buildings made it

9 320 N. Regnauld Fig. 8. Building an MST from a set. an ideal method for the following reasons: It links each building with its nearest neighbour, thus making explicit a very important relationship (proximity). It stores chains of buildings, preserving the order between the buildings and implicitly conveying the linear shape of the group. This enables us to generalize whilst respecting the continuation criterion (see Section 2.3). It has no cycles, making it easy to segment or aggregate the graph. This is important when reconstituting homogeneous groups. In addition, the absence of cycles makes it easier to enlighten continuity characteristics (gestalt criterion relating for buildings their frequent alignment along roads). The principle of computing an MST is the same as the classical algorithms described in [27] [29]. The only adaptation is to weight the edges of the graph with the minimum distance between the two boundaries of the linked buildings. We used an iterative process for that: at the first step, each building is linked with its nearest neighbour, and then each group is linked to its nearest neighbouring group and so on. Figure 8 illustrates this process for a set of points Constituting the Groups. The creation of groups, starting from the initial MST, is an iterative process which has been proposed by Zahn [30] for grouping points and that we have adapted to the buildings. Starting from the initial MST, the group is analysed to determine how regular the pattern of buildings is. If it is insufficiently regular, the graph is segmented into two subgraphs. Then the process starts again for each of the subgraphs, giving a tree hierarchical decomposition of the initial MST. The process stops in a branch when a regular pattern is found or when the group becomes too small. The process recursively divides the MST into homogeneous groups, as shown in Figure Analysing the groups. The goal is now to determine which criteria define the homogeneity of a group. We limited our study to two types of criterion: the size and orientation of a building. Homogeneity and differences of size are important in order to distinguish residential areas from town centres or industrial areas. Orientation is studied because buildings are often oriented with regard to the bordering road, and their orientation can make them homogenous in the case of straight roads. The last type of similarity

10 Contextual Building Typification in Automated Map Generalization 321 Fig. 9. Constitution of groups. criterion (shape) has not been used because our process focuses on generalizing areas of dense individual housing where shape is rarely a defining characteristic. Furthermore, the representation of such buildings at 1:50,000 scale cannot support shape detail. The following method is used to determine if a group is homogeneous with regard to a given criterion: Let c be a criterion: c {size, orientation}. Let V c : {c 1, c 2,...,c n } be a collection of values for a group of n buildings and a particular criterion. Find the largest subset V c V c that satisfies c i c 2 σ(v c ), where c is the mean value of V c and σ(v c ) is the corresponding standard deviation. This is done by successively removing the value that is the most different (and higher) from the mean, and recomputing the mean and standard deviation. This results in a decrease in the standard deviation. In the end, if this subset is greater than 80% of the initial collection, then the entire collection is deemed homogeneous for the chosen criterion, even though it includes exceptions. The value of 80% was chosen because we deemed that the smallest group in which an exception can be represented is four elements (three similar and one exception). To get such a configuration at the target scale after a reduction of the number of buildings, the maximum initial proportion is one exceptional building among four similar buildings (80%). At the completion of this stage, we know if a group is homogeneous with regard to one, two or none of our criteria. If the group is not homogeneous for the two criteria, we segment it, seeking homogeneity in the subgroups.

11 322 N. Regnauld Segmenting a group. The segmentation of a group of buildings is done by the elimination of one edge of the MST, the critical issue being the choice of edge. The objective is to segment a graph where there is a break in the distances separating neighbouring buildings. This means that the choice is not done with regard to a threshold distance, but on a threshold variation of distances. To do that, each edge is compared with the neighbouring edges at both ends of an edge. Neighbours on the side A of an edge AB are defined as edges reached by moving through the graph from vertex A up to a distance of two edges from A with the exception of the edge AB. For each edge, two coefficients of homogeneity are deduced by comparing each end and its neighbouring edges: Let e be an edge of length l e. E l (E r ) is the set of the neighbouring edges to the edge s left (right) vertex. m l (m r ) is the mean of the lengths of the edges of E l (E r ), and σ l (σ r ) is the corresponding standard deviation. h (e,el ) (h (e,er )) is the coefficient of homogeneity of a with regard to its neighbours at the left (right) end of e: h (e,el ) = l e 1.2 m l + 2 σ l (same principle for h (e,er )) The coefficient 1.2 allows minimum tolerance around the mean value in case the standard deviation is null. The value 2 increases the tolerance around a mean value when the respective standard deviation shows an inhomogeneous set of values. These two values were set empirically and are discussed in [31]. The final coefficient assigned to an edge is h e = max(h (e,el ), h (e,er )). The eliminated edge by the segmentation process is the one whose coefficient h is the highest. Figure 10 shows an example where the maximum h is being calculated for AB. The solid edges are the 2-depth neighbours of AB for vertex B. This set of edges has a low standard deviation and a low mean length with respect to AB. For the edge AB, the coefficient h has a higher value than for A. This means that AB is at variance with the regular pattern near B, and should be deleted. Four cases govern the point at which this iterative process stops: 1. Size of the group: if the group has less than four elements, it will not be segmented anymore. Further decomposition would be meaningless for typification purpose. 2. Homogeneity: the group is homogeneous for both criteria. The group is distinct and homogeneous. Fig. 10. Selection of the edge to be eliminated.

12 Contextual Building Typification in Automated Map Generalization Distance regularity: the group is very regular (maximum coefficient value is very low). There is no place to segment the group because the spacing between the buildings is very regular, making this characteristic more important than the similarity one. 4. Density: the longest edge is shorter than the separation threshold (see Section 2.1), thereby defining a dense group. Segmentation is therefore stopped Reorganization of the Groups. Some rearrangement is sometimes required once this recursive process has stopped. Two situations can occur: 1. Groups are too fragmented. In the case of inhomogeneous groups, the segmentation does not stop before having segmented the group into fragments with less than four elements. This very fragmented result is difficult to typify, which is why an aggregation step in charge of unifying two adjacent groups is added. The principle of the aggregation step is to find where the segmentation process segmented a group but no additional similarity has been found within the subgroup. It means that the segmentation was irrelevant. These subgroups are merged, resulting in a backtrack of the segmenting process. In effect, adjacent groups (directly linked in the initial MST) that have the same characteristics of homogeneity are merged. 2. Groups are not regular (but contain regular subgroups). Sometimes segmentation can stop prematurely because groups are very similar in size and orientation, yet irregular in distribution. In such cases a final step of segmentation is added in order to isolate subgroups that have regular spacing (thus respecting the continuity criterion). This segmentation process uses the same method as used during segmentation of the initial groups, except that the only criterion used to decide if an edge must be eliminated is the comparison of its coefficient with a threshold value. The end result is a collection of groups of buildings. These groups are characterized with regard to their homogeneity (orientation and size) and the regularity of their pattern. The next step is to typify each group. 5. Global Typification. Global typification is a generalization tool, based on the typification operation often identified as one of the main basic generalization operations. The tool is global because instead of just processing one group of buildings, we process an entire region. Each group is typified by preserving its relative position with respect to the neighbouring groups, and by preserving its own characteristics. Global typification is useful for processing dense regions of buildings where we wish to preserve the individual character of buildings. It is typically suitable for generalizing urban areas from 1:25,000 to 1:50,000 IGN style maps. We rebuild each group with new buildings with sizes suited to the target scale, and that preserve the global pattern. This requires two main steps: (1) first positioning of some buildings to ensure the preservation of the relative position between the groups, and (2) second positioning, where each group is locally typified. Before discussing these two steps, we need to describe two functions which we use during all the process to control the harmony between the building sizes and between the building inter-distances.

13 324 N. Regnauld Fig. 11. Harmonizing sizes The Harmonization Functions. The size of buildings must be scaled via an harmonization function to ensure that the increase in the minimum size (due to scale reduction) is mirrored by changes in the size of larger buildings. Figure 11 shows what this interpolation function of buildings areas does: the smallest building is set to the threshold size for the target scale (a), the largest one keeps its size (c), and all intermediate buildings are increased proportionally to their size (b). Building (b) should not be excessively enlarged otherwise its difference relative to (a) or (c) would be lost First Positioning. The first step is to place buildings that insure the visual separation between groups. The required distances are calculated by the distance harmonization function, which is an interpolation function for the distances between buildings, defined following the same schema than the size harmonization function (see Section 5.1). To begin with, all pairs of buildings at the ends of an eliminated edge are placed at their original position. For homogeneous groups, the character of an individual building (size, orientation and shape) is defined by the dominant characteristic of the group, otherwise the building adopts the character of the building closest to the new location. Most of the time, the size of the new building, computed by the size harmonization function, is increased, which leads to a decrease in the distance between buildings. To ensure that group perception is preserved, the buildings are displaced until their distance is in accordance with the one given by the distance harmonization function. Buildings are displaced along an imagined line defined as the connections between the centroid of the buildings linked by the initial MST (Figure 12(a)). Buildings are placed at the ends of the eliminated edge (Figure 12(b)), then buildings are displaced along their group to ensure the visual perception of the groups (Figure 12(c)). Fig. 12. First positioning.

14 Contextual Building Typification in Automated Map Generalization Second Positioning. Once buildings at the margins have been placed, each group is in-filled by first positioning important buildings and then filling the gaps while maintaining group characteristics Positioning important buildings. Three kinds of building are placed at this stage: Buildings at the end of the initial MST. They are at the end of a group, but not at the end of an eliminated edge. Exceptional buildings in the group (with respect to characteristics found during the analysis). Buildings at the junctions of the graph inside a group. These are buildings that are connected to more than two edges inside their group. Placing these buildings ensure that the pattern is linear between two consecutively placed buildings. At this stage, several kinds of problem can arise. For example, an exceptional building can be too close or can overlap a building placed at a junction. All the different configurations of conflict are described together with their method of resolution (amalgamation, selection, displacement,...) in [32] Filling the gaps. To fill the gaps between important buildings, we start from one end of the gap and following the line defined as connections between the centroids of the original buildings, we add buildings with a step corresponding to the mean step of the original group, but revised using the distance harmonization function (see Section 5.1). The size and orientation of each building added depends either on the mean value of the original buildings if the group is homogeneous, or in accordance with the characteristics of the nearest initial building. Marginal buildings not involved in the proximity relationships between the groups can also be used to fill gaps by moving them closer to the centre of the group (Figure 13). Some other method should be developed to fill the last gap. Depending on the size of this gap we can reduce it by slightly increasing the inter-distance between buildings or by making the buildings more elongated (or a combination of the two), or increase it (modifying the same parameter in the opposite direction) by adding a building. This has not been done but should be added to improve the results. Figure 14 summaries the global typification process, together with images showing the solution at each stage. The first image shows the result of the analysis step. The groups of buildings are denoted by black edges, while segmentation edges are shown in white. The second image shows the first building placement. The next image shows the addition of the important buildings, and the last one presents the final result, after filling the gaps. Fig. 13. Filling the gap.

15 326 N. Regnauld Fig. 14. Global typification process.

16 Contextual Building Typification in Automated Map Generalization Evaluation. Evaluation is a critical element of automated generalization and is required to validate the result and to quantify the changes that have occurred. There are two responses to an invalid operation: the algorithm can backtrack and seek an alternate solution (indicated by the two arrows in Figure 14), or, if no solution is found, radically different alternative must be considered. For example, if the scale reduction is too great and the area too dense, then global typification may not work (the minimum building symbol size is larger than the space between two streets). In this case the global process should probably select another operation, such as amalgamating buildings. Quantifying the result is important for two issues: (1) It provides metadata for alternate/future generalization operations. For example if a group typification has led to a slightly more dense result than expected, then a subsequent process should seek to reduce the density of that group. (2) It provides information about what deformation has occurred, with respect to the original dataset. This is important metadata for the user. The question now becomes: which are the indicators that best summarize these changes? Considerations of our initial objectives help us to select appropriate measures. Three measures that could be considered are: Characteristics of groups: The relative size and orientation of buildings are implicitly preserved by the algorithm. A change in character can arise from the removal of exceptional buildings. A ratio between the exceptional buildings and the original ones can be computed. It is hard to envisage how this information could be used by other generalization algorithms, but it is important to inform the user about such a change. Distribution of buildings: It is of critical importance that we know if the resulting groups are visually discernible from each other, i.e. has the original grouping perception been preserved? To do that, we apply the partitioning process (described in Section 4) on the initial data and on the generalized ones. Then we compare the results, analysing three type of differences: Whether buildings have changed group. By analysing groupings before and after generalization we can identify those buildings that have become part of a different group. We have such an example in Figure 15. The building above number 4 on the right figure (after generalization) has changed group. This is due to the incompleteness of our typification algorithm which has not filled all the gaps (see Section 5.3.2). Thus the resulting groups are not as regular as they should be, which has an effect on group perception. Whether groups perception has changed. As our partitioning process is iterative and seeks at each step the most relevant place to segment, the order in which the cutting is done is important. In Figure 15 the numbers represent the order between the eliminated edges. The sequence has been preserved both before and after generalization, which means that group perception has been preserved, except for edge 3 on the left, which has not been eliminated after generalization. This is due to the fact that the original more dense group of three buildings has been reduced to two, which is too small to be considered as homogeneous. Whether neighbourhood relationships have changed. Sometimes, after generalization, a group is not linked to the same groups as it was in the original dataset. This

17 328 N. Regnauld Fig. 15. Analysis on an area (a) before and (b) after global typification. means that the neighbourhood link is not obviously linear, there are several candidates for being the nearest neighbour of this group. Such a case exists in Figure 15. The edge labelled 1 links one group to two different neighbours before and after generalization. If the length of these edges is large, this has no effect on grouping perception (as in this case). However, if these distances are short, then the local distribution tends not to be linear, which makes the MST less suitable for expressing the neighbourhood relationships. If such cases occur frequently in an area, an alternative method for representing proximity relationships should be considered, such as cluster analysis. Where nonlinear groups occur relatively infrequently, then some modification to the MST approach may be sufficient to deal with this problem. Harmony between the groups: The harmony between the groups can be assessed by comparing obtained values with the ideal values specified by the harmonizing function. This information can be used in the subsequent application of other movement /enlargement related techniques such as displacement or exaggeration. All these assessments are needed for integration of the typification into a global generalization system. Depending on them, the global system can decide if the result is conform to what was expected, or if not the evaluation should provide some information helpful to determine which alternative method should be used. We have only discussed the criteria that should be used to qualify our results, without explicitly defining descriptors, as they would depend on the context of use of the method: the needs of the user and characteristics of other algorithms available. 7. Results and Comments. The grouping and the typification presented in the previous chapters have been implemented on an experimental platform called Stratège, developed at the COGIT laboratory of the national mapping agency of France (IGN). Stratège is devoted to research on contextual generalization. Stratège has an object-

18 Contextual Building Typification in Automated Map Generalization 329 Fig. 16. First grouping. oriented database and an expert system that can manage rule bases as well as state trees. This is of critical importance when we try to model strategies that require backtracking. In the following figures we present some results produced automatically by our methods. In Figure 16 the result of the first step of the distribution analysis (first grouping) is shown. The process has been applied independently to the seven topological faces (labelled from 1 to 7 in Figure 16 and delimited by the road network). Black edges link buildings belonging to the same group while white ones have been eliminated by the segmentation step. Black buildings are part of a group with an orientation and a size characteristic and buildings with a large black boundary are exceptions with regard to one of those two characteristics. Grey buildings are part of a group with a size characteristic. In this dataset the size of the buildings is homogeneous, this is why all groups have at least a characteristic of size for the buildings. Figure 17 shows the result of the aggregation step. Some stars have been added to show places where the aggregation should not have taken place because of the gap that they fill. This is why the last segmentation (Figure 18) is computed. Figure 19 shows the result of the typification process. Black buildings are those positioned to the prior positioning step (to ensure separation between the groups). The grey ones are characteristic buildings (at an end of a group, or at a node of the MST), which are positioned and used at the end of the linear intervals. Finally, white buildings have been added to fill the intervals. Figures 20 show how global typification gives a more readable result at a smaller scale. In addition, most of the distribution characteristics have been preserved, as well as maintaining the different density between the groups. It is important to stress that no user intervention is required or expected during the creation of the generalized solution shown in Figure 19. The only information needed at the beginning is the target scale, from which all the thresholds are automatically derived. The time needed to produce this result is about 5 minutes, including all the intermediate printings, on a SPARC 10 machine running Stratège.

19 330 N. Regnauld Fig. 17. Aggregation. Conclusion. This paper has presented a method for typifying a set of buildings. The approach has illustrated the importance of characterization and analysis prior to generalization. It has also stressed the importance of evaluation during and after typification. These two stages are critical to the development of algorithms capable of intelligent and meaningful generalization without intervention from the user. We began by describing the objectives of the method. We presented the analysis step that makes explicit the distribution of the buildings. This method takes into account criteria of gestalt theory Fig. 18. Final segmentation.

20 Contextual Building Typification in Automated Map Generalization 331 Fig. 19. Typification. [24] in order to group buildings as much as possible in the same way as is done visually. We used graphs (Minimum Spanning Tree) to make explicit building distribution, which makes another application of graph theory in the field of cartographic generalization, after the studies of road network generalization from [33] and [34]. Then the global typification uses these groups and uses their description to transform the building set into a form readable at a smaller scale, whilst preserving the distribution of the buildings. In the final result there are fewer buildings than in the original, they are larger, better spaced, yet their distribution pattern is very similar to the original pattern. Future research will focus on algorithm integration with other generalization algorithms. Of particular interest is the generalization of buildings with respect to the road. For example the enlarged symbolization of the road will require subsequent displacement of buildings, as well as local amalgamation or elimination. Several researches in this area are currently being followed at the COGIT laboratory, including displacement with propagation [21], interrelationships between buildings and street network [35], and strategies for automating the generalization process in urban areas [13], [36]. As most of the studies in urban generalization involve partitioning of space using blocks delimited by roads, further studies will be necessary to address the problem of integrating road network generalization, since this process modifies the composition of buildings within a partition [37]. Fig. 20. Before and after generalization.

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