Comments on "ground from figure discrimination"

In a recent paper Amir and Lindenbaum [1] described a method for discriminating between foreground and background features. Over the last decade or two there has been considerable interest in computational solutions to this problem, much of which has focussed on alternative search techniques [2, 6, 9, 12] and neural network implementations [8, 5, 3, 4, 11]. The basis of Amir and Lindenbaum’s algorithm is the commonly used characteristic of foreground shapes that they tend to be smooth in comparison with background objects. In their paper they take as input binary edge points which are then connected to their nearest k neighbours, creating the so-called “underlying graph”. To enforce goodness of shape a binary smoothness function is applied to connected edgel pairs in the graph, and is computed as a combination of proximity, cocircularity, similar orientation, and low curvature. Only arcs satisfying all the constraints are retained to form the “measured graph”. Nodes with a low proportion of edgel neighbours that pass the smoothness test are considered to be part of the background. A form of relaxation labelling is applied to the two graphs in which background nodes are iteratively deleted and new arcs are inserted into the graphs so that nodes are connected to at least k neighbours. The final “measured graph” contains the foreground edge set. Our criticism of this scheme is that it is unnecessarily complex. For the sort of data that Amir and Lindenbaum used to demonstrate their algorithm (i.e. standard edge detector output) it is possible to use simpler, more efficient, and more robust edge thresholding methods. In particular, this is possible if the connectivity and gradient magnitude information generally readily available from the edge map is used, rather than discarded (or at any rate not required) as in Amir and Lindenbaum’s scheme. It is also possible to use many other edge characteristics in the thresholding process such as clutter, lifetime, regularity, etc. [7]. For example, Venkatesh and Rosin [10] described an approach in which, prior to thresholding, all the connected edgel curves are processed as single elements. A plot of curve length versus the mean gradient magnitude over the curve shows a distinctive triangular cluster formed by the edges arising from background noise. Based on simple robust statistics (medians and median absolute deviations) the triangle can be reliably located and eliminated, leaving just the foreground edges. It is interesting to note that in both the Venkatesh/Rosin and Amir/Lindenbaum algorithms the same asymmetry is present – the background is determined first, and the foreground is simply whatever remains after the background is eliminated. An important consideration in the practicality of an algorithm is the number of tuning parameters it contains. In Venkatesh and Rosin’s scheme there is only one parameter which determines the cut-off line for eliminating the triangle, but in practise this is fixed to six. In contrast, each of Amir and Lindenbaum’s four smoothness tests requires a threshold, while the node pruning phase needs yet another threshold to determine the transition value for the proportion of connected neighbours that distinguishes foreground and background nodes. In terms of complexity Amir and Lindenbaum’s algorithm is also at a disadvantage, principally because it requires the k nearest neighbours to be found. For n edgels, ifm iterations are run then its complexity isO(m(n log n+kn)). In comparison, once the lengths of the linked edge curves have been computed (which can be done on the fly during edge linking), Venkatesh and Rosin’s algorithm is O(e) where e is the number of edge curves (assuming a linear median algorithm is used). Since the number of edge curves is relatively small compared to the number of edges, the total computation is negligible. This is confirmed by the run times. On a Sparc 10 Venkatesh and Rosin’s algorithm took 48 seconds to process the sixty images described in section 3 (despite its inefficient I/O; this also includes the time for edge linking) while Amir and Lindenbaum’s algorithm took 1606 seconds.