A rigorous definition of axial lines: ridges on isovist fields

We suggest that ‘axial lines’ defined by (Hillier and Hanson, 1984) as lines of uninterrupted movement within urban streetscapes or buildings, appear as ridges in isovist fields (Benedikt, 1979) as Rana first proposed (Rana, 2002). These are formed from the maximum diametric lengths of the individual isovists, sometimes called viewsheds, that make up these fields (Batty and Rana, 2004). We present an image processing technique for the identification of lines from ridges, discuss current strengths and weaknesses of the method, and show how it can be implemented easily and effectively. Introduction: from local to global in urban morphology Axial lines are used in space syntax to simplify connections between spaces that make up an urban or architectural morphology. Usually they are defined manually by partitioning the space into the smallest number of largest convex subdivisions and defining these lines as those that link these spaces together. Subsequent analysis of the resulting set of lines (which is called an ‘axial map’) enables the relative nearness or accessibility of these lines to be computed. These can then form the basis for ranking the relative importance of the underlying spatial subdivisions and associating this with measures of urban intensity, density, or traffic flow. To date, progress has been slow at generating these lines automatically. Lack of agreement on their definition and lack of awareness as to how similar problems have been treated in fields such as pattern recognition, robotics and computer vision have inhibited explorations of the problem and only very recently have there been any attempts to evolve methods for the automated generation of such lines (Batty and Rana, 2004; Ratti, 2001). One obvious advantage of a rigorous algorithmic definition of axial lines is the potential use of the computer to free humans from the tedious tracing of lines on large urban systems. Perhaps less obvious is the insight that mathematical procedures may bring about urban networks, and their context in the burgeoning body of research into the structure and function of complex networks (Albert and Barabási, 2002; Newman, 2003). Indeed, on one hand urban morphologies display a surprising degree of universality (Batty and Longley, 1994; Carvalho and Penn, 2003; Frankhauser, 1994; Makse et al., 1995; Makse et al., 1998) but little is yet known about the transport and social networks embedded within them (but see (Chowell et al., 2004)). On the other hand, axial maps are a substrate for human navigation and rigorous extraction of axial lines may substantiate the development of models for processes that take place on urban networks which range from issues covering the efficiency of navigation, the way epidemics propagate in cities, and the vulnerability of network nodes and links to failure, attack and related crises. Further, axial maps are discrete models of continuous systems and one would like to understand the consequences of the transition to a discrete approach. In what follows, we hypothesise a method for an algorithmic definition of axial lines inspired by local properties of space, which eliminates both the need for us to define convex spaces and to trace “(...) all lines that can be linked to other axial lines without repetition” (Hillier and Hanson, 1984, p 99). A definition of axial lines (global entities) with neighbourhood methods (local entities) implies that transition from small to large-scale urban environments carries no new theoretical assumptions and that the computational effort grows linearly (less optimizations) with the number of mesh points used. Our main goal is to gain insight into urban networks in general and axial lines in particular. Therefore we leave algorithm optimizations for future work. It is, however, beyond the scope of the present note to address generalizations of axial maps or to integrate current theories with GIS (but see (Batty and Rana, 2004; Jiang et al., 2000)). The method: Axial lines as ridges on isovist fields Axial maps can be regarded as members of a larger family of axial representations (often called skeletons) of 2D images. There is a vast literature on this, originating with the work of Blum on the Medial Axis Transform (MAT) (Blum, 1973; Blum and Nagel, 1978), which operates on the object rather than its boundary (see (Tonder et al., 2002) for a link between Visual Science and the MAT applied to a Japanese Zen Garden). Geometrically, the MAT uses a circular primitive. Objects are described by the collection of maximal discs, ones which fit inside the object but in no other disc inside the object. The object is the logical union of all of its maximal discs. The description is in two parts: the locus of centres, called the symmetric axis and the radius at each point, called the radius function, R (Blum and Nagel, 1978). The MAT employs an analogy to a grassfire. Imagine an object whose border is set on fire. The subsequent internal convergence points of the fire represent the symmetric axis, the time of convergence for unity velocity propagation being the radius function (Blum and Nagel, 1978). An isovist is the space defined around a point (or centroid) from which an object can move in any direction before the object encounters some obstacle. In space syntax, this space is often regarded as a viewshed and a measure of how far one can move or see is the maximum line of sight through the point at which the isovist is defined. We shall see that the paradigm shift from the set of maximal discs inside the object (as in the MAT) to the maximal straight line that can be fit inside its isovists holds a key to understanding what axial lines are. As in space syntax, we simplify the problem by eliminating terrain elevation and associate each isovist centroid with a pair of horizontal coordinates ( ) , x y and a third coordinate the length of the longest straight line across the isovist at each point which we define on the lattice as where max , i j ∆ ( ) , x y is uniquely associated with . Our hypothesis states that all axial lines are ridges on the surface of . The reader can absorb the concept by “embodying” herself in the landscape: movement along the perpendicular direction to an axial line implies a decrease along the surface; and is an invariant, both along the axial line and along the ridge. Our hypothesis goes further to predict that the converse is also true, i.e., that up to an issue of scale, all ridges on the landscape are axial lines. Most of what follows is the development of a method to extract these ridges from the surface, in the same spirit that one would process temperature values sampled spatially with an array of thermometers. ) , ( j i max , i j ∆ max , i j ∆ max , i j ∆ max , i j ∆