Automatic Extraction of Hierarchical Urban Networks: A Micro-Spatial Approach

We present an image processing technique for the identification of ’axial lines’ [1] from ridges in isovist fields first proposed by Rana [2, 3]. These ridges are formed from the maximum diametric lengths of the individual isovists, sometimes called viewsheds, that make up the isovist fields [4]. We discuss current strengths and weaknesses of the method, and show how it can be implemented easily and effectively. 1 Axial Maps as Skeletons for 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 [1, 5, 6]. 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 [7, 8, 4]. 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 [9, 10]. Indeed, on one hand urban morphologies display a surprising degree of universality [11—15], but little is yet known about the relation between this observed universality and the transport and social networks embedded within urban space (but see [16, 17]). 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, and the vulnerability of network nodes and links to failure, attack and related crises. 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) [18, 19]. 2 Axial Lines as Ridges on Isovist Fields An isovist is the space defined around a point (or centroid) from which an object can move in any direction before it encounters some obstacle. 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∆ i,j . We extend previous work by Rana [3], where he noted that "the ridge lines give an indication of the disposition of the axial lines", by using a modification of the Medial Axis Transform [18, 19] and the Hough Transform [20]. The hypothesis states that all axial lines are ridges on the surface of∆ i,j . The reader can absorb the concept by ’embodying’ herself in the ∆ i,j landscape: movement along the perpendicular direction to an axial line implies a decrease along the ∆ i,j surface; and ∆ max i,j is an invariant, both along the axial line and along the ridge. The hypothesis goes further to predict that the converse is also true, i.e., that up to an issue of scale, all ridges on the ∆ i,j landscape are axial lines. Here we sample isovist fields by generating isovists for the set of points on a regular lattice [2, 21, 8, 22]. Specifically, we are interested in the isovist field defined by the length of the longest straight line across the isovist at each mesh point, (i, j). This measure is denoted the maximum diametric length, ∆ i,j [4], or the maximum of the sum of the length of the lines of sight in two opposite directions [8, p 204]. To simplify notation, we will prefer the former term. First, we generate a Digital Elevation Model (DEM) [23] of the isovist field, where ∆ i,j is associated with mesh point (i, j) [21, 8]. Our algorithm detects ridges by extracting the strict maxima (i.e. a cell with value stricly greater than any of its nearest neighbours [24]) of the discrete DEM. Next, we use an image processing transformation (the Hough Transform) on a binary image containing the local maxima points which lets us rank the detected lines in the Hough parameter space. Finally, we invert the Hough transform to find the location of axial lines on the original image. The process of using the HT to detect lines in an image involves the computation of the HT for the entire image, accumulating evidence in an array for events by a voting (counting) scheme and searching the accumulator array for peaks which hold information of potential lines present in the input image. The peaks provide only the length of the normal to the line and the angle that the normal makes with the y-axis. They do not provide any information regarding the length, position or end points of the line segment in the image plane [25]. Our line detection algorithm starts by extracting the point that has the largest number of votes on parameter space, which corresponds to the line defined by the largest number of collinear local maxima of ∆ i,j , and proceeds by extracting lines in rank order of the number of their votes on parameter space. One of us [4] has previously proposed rank-order methods as a rigorous formulation of the procedure originally outlined of “first finding the longest straight line that can be drawn, then the second longest line and so on (. . . )” [1, p 99].