Academic journal article Cartography and Geographic Information Science

B-Spline Functions and Wavelets for Cartographic Line Generalization

Academic journal article Cartography and Geographic Information Science

B-Spline Functions and Wavelets for Cartographic Line Generalization

Article excerpt

ABSTRACT: Most line processing algorithms developed so far in cartographic generalization focus on polygonal curves (or polylines). This representation model is sometimes not sufficient for certain processes due to its lack of continuity or smoothness. Indeed, it may provide poor results for lines having "smooth" initial shapes such as roads. Thus, we suggest using a modeling method based on B-spline curves. A maritime case study described in this paper shows that this representation provides good results at a fixed scale and is suitable for several automatic line cartographic generalization operators (smoothing, displacement, aggregation and compression). Lastly, we discuss the application of B-spline wavelets used in dealing with multi-scaling.

KEYWORDS: B-splines, line generalization, B-spline wavelets


Cartographic generalization is the process of data abstraction used for cartographic visualization (Weibel and Dutton 1999). It is often required when the scale of a map is changed. It involves data modification in such a way that data can be represented in a smaller space (or scale), while preserving meaningful and legible visualizations. In this paper, we focus on independent line generalization (i.e., generalization of a single line), and not on the generalization of several lines and multiple objects.

Polygonal curves (or polylines, where the data points are connected by straight line segments) are mainly used to model data in line generalization. Advantages are simplicity and fast computing time. Nevertheless, they are not appropriate for modeling smooth initial shapes, because we often have to use many points (e.g., during the digitizing process) in order to produce the necessary smoothing effect. In addition, cartographers often need to zoom in on a part of a line and preserve its smoothness (Figure 1). One of the solutions is to introduce new continuous representations in cartographic line generalization (Fritsch 1997; Mitas and Mitasova 1999).


Our approach uses continuous representations such as B-spline curves in addition to polylines. Our goal is not to substitute for the usual polygonal representation but to adapt the modeling method according to the initial shape of the line. We suggest that lines having "smooth" initial shapes can be defined by means of B-spline curves. Geographical features such as harbors should not be modeled by means of B-spline curves, however. The jagged lines of a pier, for example, must remain on the generalized map, and polylines ought to be applied in this case. Furthermore, and in comparison with other typical continuous modeling methods, we show that B-spline curves are suitable for generalization and that their geometrical properties provide useful geometrical information for generalization needs. In order to give substance to my claim, readers are referred to my previous work on the definition of certain line generalization operators with B-spline curves (Saux 1998; Saux and Daniel 1999). In the present paper I introduce new results of linear generalization through different resolutions using B-spline wavelets.

The paper is organized as follows. The following section summarizes the methods usually applied in the automated line generalization process. In the next section I describe the application of B-spline curves to this problem. Then I apply B-splines to line generalization operators at a fixed scale and review my previous published results on smoothing, compression, displacement, and aggregation operators applied in a maritime case study. The following section focuses on new research based on multi-scale analysis, detailing the notion of using B-spline wavelets (i.e., B-spline functions adapted to deal with multi-scale processing). In the last section, I analyze the potential of these generalization operators using a multi-resolution approach employing B-spline wavelets.

Cartographic Line Generalization: State of the Art

Line generalization has received considerable attention in recent years (Buttenfield 1985; Plazanet 1995, 1996; Plazanet et al. …

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