Local length ratio as a measure of critical points detection for line simplification

Critical points detection is a research topic related to cartographic generalization as well as computer science in disciplines like computer vision, pattern recognition, or signal processing. In the present paper a short review of existed critical points detection algorithms is given. Most of these algorithms are related to the estimation of curvature of the points defining the line. Instead of calculating the curvature, local length ratio (LLR) is introduced as a measure of critical points detection based on geometric principles. Four lines (three geomorphological lines and one arbitrary) selected from the existed cartographic literature were used in order to assess the new method of detecting critical points. Finally, the proposed measure of critical points detection is used to evaluate the results of two line simplification algorithms applied for six line simplification tasks on the coastline of a small island. Although the present research approached the problem at a preliminary stage, promising results have been produced. It is apparent that further research is required based on a more in depth analysis for concluding about the method power. Key-words: critical/dominant points, local length ratio, line generalization, line simplification algorithms Introduction In a study of clarifying several concepts of visual perception in the context of information theory, Attneave (1954) concluded that the points of an object are divided into two categories according to the information they convey. The first category is partitioned in a portion of points that are dominant, and the remaining predictable from the former, thus highly redundant. It could be assumed, that the dominant points are able to describe and characterize the shape of the object, since they transmit more information than the others. These points are termed “critical points” in the cartographic literature and “dominant points” in the computer science literature (Li 1995). Marino (1979) underlines the impact of critical points in line generalization because these points are recognizable both by cartographers and non-cartographers at the same degree. Critical points according to Attneave (1954) are concentrated at those locations on a line where its direction changes more rapidly. Specifically he states that, “It is clear that subjects show a great deal of agreement in their abstractions of points best representing the shape, and most of these points are taken from regions where the contour is most different from a straight line” (Attneave 1954, p.185). Freeman (1978) includes in his definition of critical points the points along a digital line that are: 1. curvature maxima, 2. curvature minima, 3. end points, and 4. points of intersection. Critical points have attracted interest in research of the cartographic generalization domain (Marino 1979, White 1985, Thapa 1987,1988a, 1988b, 1989, McMaster 1987, Jenks 1989, Visvalingam and Whyatt 1990, Li 1995). In the computer science domain, especially topics such as computer vision, pattern recognition, or signal processing, a large number of critical points detection algorithms have been developed mainly addressing the problems of line approximation, curve segmentation, or feature detection (Pavlidis and Horovitch 1974, Iñesta et al 1998, Cronin 1999, Neumann and Teisseron 2002). The majority of the existing algorithms of critical points detection are mainly focused on the estimation of the curvature of the contour of a graphical object at any location. In addition, the reader can find algorithms for detecting critical points based on wavelets (Antoine et al 1997), or adaptive