Fractal Enhancement of Cartographic Line Detail

In plane geometry curves have a dimension of exactly 1 and no width. In nature, all curvilinear features have width, and most have dimension greater than 1, but less than 2. Many phenomena, such as coastlines, have the same "look," even when viewed at greatly varying scales. The former property is called "fractional dimensionality," and the latter is called "self similarity." Curves digitized from maps may be analyzed to obtain measures of these properties, and knowledge of them can be used to manipulate the shape of cartographic objects. An algorithm is described which enhances the detail of digitized curves by altering their dimensionality in parametrically controlled, self-similar fashion. Illustrations show boundaries processed by the algorithm. Measuring and Modelling Irregularity in Nature Only in the mind and works of man do straight lines exist. Rarely does Nature rule with a straightedge, and even these lines are rough, seldom extending very far. But surrounded by rectilinear artifacts, it is understandable why humans try to measure and model the world with Euclidean precepts. Frustration in making certain measurements and in modelling many natural forms can be attributed to this view of space itself, in which distance between two given points is assumed to be Pythagorean. Suppose one is surveying a section of coastline and wants to calculate its length accurately and map it. A series of closelyspaced sightings must be made at the high-water mark. The cumulative distance along these points can then be accurately computed, and it is invariably greater than the crow's-flight distance spanning the stretch of coast. Fig. IA represents the profile of a fictitious coastline. Its surveyed approximation is plotted in Fig. IB, and the crow's-flight version of it is shown in Fig. 1C. A greater Geoffrey Dutton is a programmer/analyst at the Harvard Laboratory for Computer Graphics and Spatial Analysis. His mailing address is Graduate School of Design, Harvard Univ., 520 Gund Hall, Cambridge, Mass. 02138. number of sightings yields a closer approximation to actual length and shape, even though the rate of increase of length slows. This lesson in approximation has several morals. One is that surveyors run into a real law of diminishing returns when trying for centimeter accuracies in the lengths of complicated boundaries. Another is that the difficulty, hence the probable error, in measuring coastlines and the like varies from place to place. In Fig. 1A, it is obvious that there is much more irregularity in the lower part of the coast than in the upper part. This may be due to the former being composed of rock outcroppings and the latter being a sandy beach. But in trying to express this qualitative difference quantitatively, one finds scientific vocabulary confusing and inadequate. Literature in geography and image processing abounds with indices that characterize the shapes of point sets and linear and areal features (Stoddard, 1965; Boyce and Clark, 1964; Bunge, 1961; Boots, 1972). However, to paraphrase Pavlidis (1978), these indices are destructive of information and provide neither a general linguistic model nor a measure suitable to allow manipulation of cartographic detail. One wishes for a measure of geometric complexity and irregularity that is as general as that of The American Cartographer, Vol. 8, No. 1, 1981, pp. 23-40 The American Cartographer 11 0 Fig. 1. A coastline and its approximations: (A) the original coastline, (B) segmented approximation. (C) original coastline with superimposed trend line. entropy in thermodynamics. Fortunately, foundations for such a vocabulary and for such measures have been developed. Irregularity as Fractional Dimensionality and Self Similarity A suitably general approach to quantifying the complexity of irregular forms, and one that directly confronts the dilemmas of Euclidean measurement, is that of Mandelbrot (1977). The phenomena that he addresses-natural forms arising from forces such as turbulence, curdling, Brownian motion, and erosion-have at all scales two related properties, self similarity and fractional dimensionality. Self similarity means that a portion of an object when isolated and enlarged exhibits the same characteristic complexity as the object as a whole. The shapes revealed may be highly irregular, and none may be exactly alike, but they will have the same kind of irregularity over a wide range of scales. Fractional dimensionality means that the Euclidean dimension that normally characterizes a form (1 for lines, 2 for areas, 3 for volumes) represents only the integer part of the true dimension of the form, which is a fraction.' Mandelbrot treats dimension as a continuum, in which the integer Euclidean dimensions merely represent limiting cases of topological genera, unlikely to occur in nature. Thus the coastline in Fig. 1A might have an approximate overall dimensionality of 1.2, but its two dissimi lar subsections have different structure and dimensionality. The more irregular lower portion may have a dimension of nearly 1.3, while the smoother upper part may be of a lower dimension, less than 1.1. There is only one version of the coast that has a dimensrionality of exactly I (its Euclidean dimensionality), and that is the trendline shown in Fig. 1C. This difference in dimensionality is quantified in Fig. 2. On this graph the abscissa symbolizes the number of sightings (or line segments) used to approximate the entire coast (top curve) and its lower (middle curve) and upper (bottom curve) portions. The values read from the Fig. 2. Dependence of length on fractal limensionality and scale of measurement.