Morphing Polylines Based on Least-Squares Adjustment

Digital maps such as Google Maps or Open Street Map have become one of the most important sources of geographic information. When users interactively browse through such maps on computers or small displays, they often need to zoom in and out to get the information desired. Often, zooming is supported by a multiple representation database (MRDB). This stores a discrete set of levels of detail (LOD) from which a user can query the LOD for a particular scale (Hampe et al 2004). A small set of LODs leads, however, to large and sudden changes during zooming. Since this distracts users, hierarchical schemes have been proposed that implement the generalization process based on small incremental changes, for example, the BLG-tree (van Oosterom 2005) for line simplification or the GAP-tree (van Oosterom 1995) for area aggregation. The incremental generalization process is represented in a data structure that allows a user to retrieve a map of any desired scale. Still, the generalization process consists of discrete steps and includes abrupt changes. To achieve a continuous generalization, Sester and Brenner (2004) suggested to simplify building footprints based on small incremental steps and to animate each step smoothly. Also aiming at a continuous generalization, several authors have developed methods for morphing between two polylines (Cecconi 2003, Nöllenburg et al. 2008). Most of these methods consist of two steps (Cecconi 2003, Nöllenburg et al. 2008, Peng et al. 2012). The first step usually identifies the corresponding vertices of the two polylines. The second step defines a trajectory for each pair of corresponding vertices. Most often, straight-line trajectories are defined on which, when morphing, the vertices move at constant speed. In this paper we address morphing, but we relax the requirement that the vertices of the polyline move on straight lines. Our concern with straight-line trajectories is that characteristic properties of the polylines change drastically during the morphing process. In particular, we suggest that the angles and edge lengths of the polyline should change linearly during the morphing process. As Figures 1(a) and 1(b) show, this is clearly not accomplished with straight-line trajectories. In contrast, the new method that we present in this paper yields a close-to-linear relationship, for example, between time and edge lengths; see Figures 1(c) and 1(d). The paper is organized as follows. We review related work in Section 2. The details of our method are presented in Section 3, which include a list of soft and hard constraints, estimates for the unknown parameters, and the iterative process of our model. We present a case study in Section 4, which shows that our method generally performs well but also reveals new problems. We conclude the paper in Section 5.