Gestalt principles describe how humans perceive and organize individual elements into groups. The quantification of Gestalt principles serves as a basis for predicting human perception. It has been found that achieving reliable predictions by quantifying a single principle is challenging because two or more principles usually have joint effects. Therefore, quantitative models for joint effects are required. Although some quantitative models have been developed, the model for quantifying the joint effect of continuity and similarity (two commonly used principles) remains lacking. To fill this gap, this study proposes a permutation entropy-based model to quantify this joint effect on grouping segments of cross-curves and evaluates the grouping performance of the proposed model through a set of simulated curves and a set of real-life curves (e.g., roads). The evaluation considers two types of benchmarks: (1) grouped results by the existing commonly used model and (2) comparison with pre-determined correct groupings. The results from the proposed model are consistent with the pre-determined correct groupings, while those of the existing commonly used model fail when complexity differences between segments increase. These findings suggest that the entropy measure used in the proposed model can effectively quantify the joint effect of continuity and similarity of curves.

Landmark buildings are salient features for spatial cognition on maps. Distinctive outlines are the major visual characteristics that separate landmark buildings from their surrounding environments. The automatic symbolization of landmark outlines facilitates recognition and map production. As users often recognize landmarks by the outlines of their façades from a street view, this study proposes an automatic method for automatically generating representations of the outlines of landmark buildings in four steps: (1) extract outlines from street-view photographs using GrabCut method, (2) vectorize the extracted building outlines, (3) simplify outline shapes, and (4) symbolize the simplified building outlines in three dimensions (3D). We used the proposed method to generate test data with symbolized outlines for eight buildings in a real-world environment for a wayfinding experiment in which the subjects used the building representations to identify landmark buildings and evaluated their perception of the generated maps. The subjects successfully recognized these buildings based on the symbolized outlines on a map, expressed satisfaction with the manually generated 3D symbols, and reported the same or similar ease of building recognition using 2D or 3D symbolized outlines.

Cities are spatially evolving complex systems. The order and pattern beneath the apparent chaos and diverse physical forms of cities are still unclear. How the form and structure of a city evolve to improve its functions needs further exploration. To fill thisgap, we examine the geometric fractal (GF), the topological fractal (TF), and the hierarchical fractal (HF) evolution of cities by taking Hong Kong street networks from year 1971 to 2018 as an example. We find that these networks keep to be fractals both in form and structure. The values of GF, TF, and HF dimensions increase with fluctuations, revealing a more mature and complex street network. The radius-length GF dimensions demonstrate the bi-fractal property, with values ranged 1.653–1.832 and 0.677–0.892, respectively, reflecting a core-periphery pattern. The values of TF dimensions increase steadily with a wider gap to GF dimensions, indicating progressively structural optimization of street networks. These street networks keep showing fractal properties in form and structure through spatial extension, local densification, vertical stratification, hierarchies enrichment, and shortcuts construction. Moreover, street networks are GFs and TFs at the city, county, and MSA scales. The discoveries advance our understanding of urban development.

An image-map is a compromise between an image and a map. The quality of such maps is affected by several factors, such as (a) the matching between the features on images and the graphic symbols from maps, (b) the complexity of background images, and (c) the representation of graphic and text symbols on the images. This project deals with the first issue. The current solution is that the accuracy of images should satisfy the accuracy standard of maps. However, images with different resolutions can satisfy the standard for a specific map scale. This may lead to a situation in which the levels of detail (LoD) in images may not match the complexity of map features although the planimetric accuracy is matched. To solve this problem, we developed a complexity-based matching between the image resolution and map scale. More precisely, the matching is based on the complexity of line features. Experimental evaluations were conducted in 15 representative areas in Hong Kong using maps at seven scales and eight image resolutions. Results show that the proposed complexity-based method is capable of obtaining good matching between image resolution and map scale in terms of both accuracy and users’ preference.

Schematic maps are popular for the representation of transport networks. Many automated methods have been developed to generate such maps. In these methods, optimization techniques work with various sets of constraints. Most of these constraints govern geometric properties of individual features. A few constraints address relationships among features, but none explicitly deal with the main structure of an entire network. We believe that preservation of main structure is the most important and preservation of relative relations is helpful. This is because human perception follows a global-to-local process. These constraints have recently been formed into four general principles, with two for global structure and two for relativity of features. This study develops an automated method by integrating these principles into the mixed-integer programming (MIP) framework. Experimental evaluations have been conducted with two sets of real-world transit networks. In comparison to the existing method, the proposed method has smaller fractal dimensions, better computational performance and higher scores in terms of clarity, recognition of major lines, visual simplicity and satisfaction. Therefore, it is concluded that the proposed method can generate schematic maps with improved clarity and aesthetics. The idea in this study is also helpful for the design of other visual representations.

Selective omission in a road network is a necessary operation for road network generalization. Most existing selective omission approaches involve one or two geometric parameters at a specific scale to determine which roads should be retained or eliminated. This study proposes an approach for determining the empirical threshold for such a parameter. The idea of the proposed approach is to first subdivide a large road network, and then to use appropriate threshold(s) obtained from one or several subdivisions to infer an appropriate threshold for the large one. A series of experiments was carried out to validate the proposed approach. Specifically, the road network data for New Zealand and Hong Kong at different scales (ranging from 1:50,000 to 1:250,000) were used as the experimental data, and subdivided according to different modes (i.e. administrative boundary data, a regular grid of different sizes, different update years, and different road network patterns). Not only geometric parameters, but also structural and hybrid parameters of existing selective omission approaches were involved in the testing. The experimental results show that although the most appropriate thresholds obtained from different subdivisions are not always the same, in most cases, the appropriate threshold ranges often overlap, especially for geometric parameters, and they also overlap with those obtained from the large road network data. This finding is consistent with the use of different subdivision modes, which verifies the effectiveness of the proposed approach. Several issues involving the use of the proposed approach are also addressed.

Nowadays, the design of the London Tube map (as a kind of schematic map) has been popularly adopted for transport network maps worldwide because of its great clarity of representation. In such types of map, the shape of the network is simplified and the topology between lines is preserved while the congested areas are enlarged to a desirable scale. Efforts have also been made to automate the production of such maps. However, to our best knowledge, no existing methods have explicitly taken into consideration the automated enlargement of congested areas. As such an enlargement is vital to the improvement of clarity, this paper proposes a new automated method to generate schematic network maps, consisting of (a) automated detection of congested areas, (b) automated enlargement of congested areas to a desirable scale and (c) automated generation of the schematic representation of the deformed network maps using a stroke-based approach. The new method has been tested with two real-life network data sets, i.e. the London Tube and Hong Kong metro data sets, and evaluated by fractal analysis and experimental studies. The results of the evaluation indicate that the new method is able to automatically generate the schematic maps with improved clarity and aesthetics.

Intersection relations are important topological considerations in database update processes. The differentiation and identification of non-empty intersection relations between new updates and existing objects is one of the first steps in the automatic incremental update process for a land parcel database. The basic non-empty intersection relations are meet, overlap, cover, equal and inside, but these basic relationships cannot reflect the complex and detailed non-empty relations between a new update and the existing objects. It is therefore necessary to refine the basic non-empty topological relations to support and trigger the relevant update operations. Such relations have been refined by several researchers using topological invariants (e.g., dimension, type and sequence) to represent the intersection components. However, the intersection components often include only points and lines, and the refined types of 2-dimensional intersection components that occur between land parcels have not been defined. This study examines the refinement of non-empty relations among 2-dimensional land parcels and proposes a computation model. In this model, an entire spatial object is directly used as the operand, and two set operations (i.e., intersection (∩) and difference (\)) are applied to form the basic topological computation model. The Euler number is introduced to refine the relations with a single 2-dimensional intersection (i.e., cover, inside and overlap) and to distinguish the refined types of 2-dimensional intersection components for the relations with multiple intersections. In this study, the cover and overlap relations with single intersections between regions are refined into seven cases, and nine basic types of 2-dimensional intersection components are distinguished. A composite computation model is formed with both Euler number values and dimensional differences. In this model, the topological relations with single intersections are differentiated by the value of the dimension and the Euler number of the resulting set of the whole-object intersection and differences, whereas the relations with multiple intersections are discriminated by the value of the resulting set at a coarse level and are further differentiated by the type and sequence of the whole-object intersection component in a hierarchical manner. Based on the refined topological relations, an improved method for automatic and incremental updating of the land parcel database is presented. The effectiveness of the models and algorithms was verified by the incremental update of a land cover database. The results of this study represent a new avenue for automatic spatial data handling in incremental update processes.

This article proposes an efficient solution algorithm to aid travelers' route choice decisions in road network with travel time uncertainty, in the context of advanced traveler information systems (ATIS). In this article, the travel time of a link is assumed to be spatially correlated only to the neighboring links within a local ‘impact area.’ Based on this assumption, the spatially dependent reliable shortest path problem (SD-RSPP) is formulated as a multicriteria shortest path-finding problem. The dominant conditions for the SD-RSPP are established in this article. A new multicriteria A* algorithm is proposed to solve the SD-RSPP in an equivalent two-level hierarchical network. A case study using real-world data shows that link travel times are, indeed, only strongly correlated within the local impact areas; and the proposed limited spatial dependence assumption can well approximate path travel time variance when the size of the impact area is sufficiently large. Computational results demonstrate that the size of the impact area would have a significant impact on both accuracy and computational performance of the proposed solution algorithm.

Spatial data can be represented at different scales, and this leads to the issue of multi-scale spatial representation. Multi-scale spatial representation has been widely applied to online mapping products (e.g., Google Maps and Yahoo Maps). However, in most current products, multi-scale representation can only be achieved through a series of maps at fixed scales, resulting in a discontinuity (i.e., with jumps) in the transformation between scales and a mismatch between the available scales and users' desired scales. Therefore, it is very desirable to achieve smoothly continuous multi-scale spatial representations. This article describes an integrated approach to build a hierarchical structure of a road network for continuous multi-scale representation purposes, especially continuous selective omission of roads in a network. In this hierarchical structure, the linear and areal hierarchies are constructed, respectively, using two existing approaches for the linear and areal patterns in a road network. Continuous multi-scale representation of a road network can be achieved by searching in these hierarchies. This approach is validated by applying it to two study areas, and the results are evaluated by both quantitative analysis with two measures (i.e., similarity and average connectivity) and visual inspection. Experimental results show that this integrated approach performs better than existing approaches, especially in terms of preservation of connectivity and patterns of a road network. With this approach, efficient and continuous multi-scale selective omission of road networks becomes feasible.

The study of road networks has been a topic of interest for some time. A road network in a database is often represented by intersections and segments. However, in many cases (e.g., traffic flow analysis and map generalization), one needs to consider individual roads as a whole, instead of individual segments. Thus, it is sometimes very desirable to concatenate road segments into long lines – ‘strokes’ as they are called in the literature. For stroke building, a number of strategies are available and the effectiveness of using these strategies needs to be evaluated. This article presents a comparative analysis of 17 such strategies, including 3 of the geometric approach, 1 of the thematic approach, and 13 of the hybrid approach for road network generalization purposes. Three sets of real-life data with different patterns are used to test these strategies. Corresponding road maps at smaller scales are used as benchmarks and a new measure called the accuracy rate is proposed to indicate the correctness of the concatenated strokes. The results show that if only the geometric approach is considered, the every-best-fit strategy performs best; if thematic attributes are also added, road class can be more effective than road name. Also significance tests (the chi-square test and the Marascuilo procedure) are carried out to give all pairwise comparisons of these strategies. The results indicate that 45 of the 136 pairs of strategies have statistically significant differences; the purely geometry-based every-best-fit performs significantly better than the purely geometry-based self-fit; and the inclusion of thematic attributes, especially road class, sometimes improves the accuracy rate but the improvement is not significant.

Studies on the structural properties of road network and its close relationship with the traffic flow distribution have received intensive interdisciplinary attention. However, most of these attempts were theoretical. It is also a challenge to understand the relationship between the structure and morphology of a road network and peoples' movement. We developed a new methodology to deal with this challenge in this study. The first attempt was to apply the ego network analysis (which is rooted in social science) to the formation of hierarchical road networks. Then, the ego network was improved to become weighted ego network by assigning a weight to each of the links in a network. A measure called weighted average centrality rank is developed to define the order of links in a complex network. The ego network and the weighted ego network are both evaluated with a notional network and two sets of real-life road networks. Traffic flow data were used as a benchmark for the evaluation of the two approaches. The results show that they both perform well. But the hierarchies formed by weighted ego network analysis are more consistent with the real-life traffic flow, and the improvement is clearly observable.

This article deals with the graphic simplification of a network by schematization. A new method employing a stroke-based and progressive strategy is proposed to generate schematic network maps. This method treats a stroke (which is a long line with segments concatenated together) as a basic unit for the implementation. The procedure is as follows: (a) strokes are formed from line segments, (b) the strokes are re-orientated along grid lines and/or diagonals, and (c) two endpoints and all intersection points on (sub-)strokes are projected onto re-oriented straight lines, and (d) spatial inconsistency is detected and resolved. A methodology for each of these steps is described. This new method has been tested with a set of real-life road network data and evaluated by fractal analysis and empirical study. Experimental results show that this new method is more effective than segment-based methods and is able to produce graphics with great simplicity and clarity. Based on the results obtained, the stroke-based schematization with four primary directions is recommended.

In the process of topographic map updating, spatial conflicts (inconsistency) between rivers and contours may be created. This project studies the particularity and complexity of the relationships between rivers and contours, and develops a method for automatic detection of the spatial conflicts between them. This method consists of a refined descriptive spatial model (called a topological chain), an algorithm for the computation of spatial relations, and a set of rules for the determination of spatial conflicts. In the spatial relation model, the topological relationships, order relationships, and metric relationships are integrated to describe the line–line spatial relationships. The rules are derived from the natural relations between rivers and contours in the real world. The effectiveness of this method has been verified by using the national 1:50 000 topographic map databases. An accuracy of 91% has been achieved.

Distance is a fundamental concept in spatial sciences. Spatial distance is a very important parameter to measure the relative positions between spatial objects and to indicate the degree of similarity between neighbouring objects. Indeed, spatial distance plays an important role in many areas such as neighbourhood analysis, structural similarity measure, image (or object) matching, clustering analysis, and so on. In this paper, existing computational models for the distance between spatial objects are evaluated and their problems pointed out; then, the concept of the Hausdorff distance is introduced as a metric indicator for different types of spatial objects. This distance is extended to a uniform representation by the introduction of the quantile, leading to the extended Hausdorff distance. Indeed, the so‐called extended Hausdorff distance is, in fact, a kind of metric characterized by the minimum distance, the Hausdorff distance, and the median Hausdorff distance. The first two can be used for measuring the dispersion and the last one for measuring the central tendency of the distance distribution between spatial objects. A method termed the ε‐buffer has been proposed for the computation of the median Hausdorff distance. Finally, potential applications are discussed.

Building generalization is a difficult operation due to the complexity of the spatial distribution of buildings and for reasons of spatial recognition. In this study, building generalization is decomposed into two steps, i.e. building grouping and generalization execution. The neighbourhood model in urban morphology provides global constraints for guiding the global partitioning of building sets on the whole map by means of roads and rivers, by which enclaves, blocks, superblocks or neighbourhoods are formed; whereas the local constraints from Gestalt principles provide criteria for the further grouping of enclaves, blocks, superblocks and/or neighbourhoods. In the grouping process, graph theory, Delaunay triangulation and the Voronoi diagram are employed as supporting techniques. After grouping, some useful information, such as the sum of the building's area, the mean separation and the standard deviation of the separation of buildings, is attached to each group. By means of the attached information, an appropriate operation is selected to generalize the corresponding groups. Indeed, the methodology described brings together a number of well-developed theories/techniques, including graph theory, Delaunay triangulation, the Voronoi diagram, urban morphology and Gestalt theory, in such a way that multiscale products can be derived.

The map is a medium for recording geographical information. The information contents of a map are of interest to spatial information scientists. In this paper, existing quantitative measures for map information are evaluated. It is pointed out that these are only measures for statistical information and some sort of topological information. However, these measures have not taken into consideration the spaces occupied by map symbols and the spatial distribution of these symbols. As a result, a set of new quantitative measures is proposed, for metric information, topological information and thematic information. An experimental evaluation is also conducted. Results show that the metric information is more meaningful than statistical information, and the new index for topological information is more meaningful than the existing one. It is also found that the new measure for thematic information is useful in practice.

Models of spatial relations are a key component of geographical information science (GIS). Efforts have been made to formally define spatial relations. The foundation model for such a formal presentation is the 4-intersection model proposed by Egenhofer and Franzosa (1991). In this model, the topological relations between two simple spatial entities A and B are transformed into pointset topology problems in terms of the intersections of A's interior and boundary with B's interior and boundary. Later, Egenhofer and Herring (1991) extended this model to 9-intersection by addition of another element, i.e. the exterior of an entity, which is then defined as its complement. However, the use of its complement as the exterior of an entity causes the linear dependency between its interior, boundary and exterior. Thus such an extension from 4- to 9-intersection should be of no help in terms of the number of relations. This can be confirmed by the discovery of Egenhofer et al. (1993). The distinction of additional relations in the case where the co-dimension is not zero is purely due to the adoption of definitions of the interior, boundary and exterior of entities in a lower dimensional to a higher dimension of space, e.g. lines in 1-dimensional space to 2-dimensional space. With such adoption, the topological convention that the boundary of a spatial entity separates its interior from its exterior is violated. It is such a change of conventional topological properties that causes the linear dependency between these three elements of a spatial entity (i.e. the interior, boundary and exterior) to disappear, thus making the distinction of additional relations possible in such a case (i.e. the co-dimension is not zero). It has been discussed that the use of Voronoi-regions of an entity to replace its complement as its exterior in the 9-intersection model would solve the problem (i.e. violation of topological convention) or would make this model become more comprehensive. Therefore, a Voronoi-based 9-intersection model is proposed. In addition to the improvement in the theoretical aspect, the Voronoi-based 9-intersection model (V9I) can also distinguish additional relations which are beyond topological relations, such as high-resolution disjoint relations and relations of complex spatial entities. However, high-resolution disjoint relations defined by this model are not purely topological. In fact, it is a mixture of topology and metric.

Generalization is a fundamental function in GIS. It has been an important research theme for many years in cartography and GIS. A number of generalization operations have been identified, however most of them, especially those rule-based operations, remain at the conceptual level. This paper describes a set of mathematical (algebraic) models for area aggregation based on the operators developed in mathematical morphology. In this paper, the process of area aggregation is decomposed into two components, viz., combination and shape refinement, and algebraic models for both components are developed. These are demonstrated using various examples. The models provide a mathematical basis for area aggregation in digital generalization of map and other spatial data. The results show that these algebraic models have the potential for successful application.

This article describes a new set of algorithms for locally–adaptive line generalization based on the so-called natural principle of objective generalization. The drawbacks of existing methods of line generalization are briefly discussed and the algorithms described. The performance of these new methods is compared with benchmarks based on both manual cartographic procedures and a standard method found in many geographical information systems.