6.2 The criticality graph and zone of influence algorithm.

We now show how to compute the points of $\bf {Z}^n$ in the zone of each criticality and the criticality graph.

One can pass the data and record all critical points, sets, and values in linear (disk access) time, An issue here is the identification of connected sets of identically valued vertices and their immediate neighbors by a breadth first labeling.

Once one has identified criticalities and associated values, one sorts the values and subdivides the range of data values by the critical values. One can then plot sample families of iso-surface bounded objects by selecting a sample value $\tau$ from each critical value-free interval and running SpiderWeb for each such value.

One can compute the zone of influence of each criticality by Latombe style wavefront propagation [19] or grass-fire labeling algorithms. The labeling algorithm is similar to the methods employed in the distributed representation approach to robot motion planning ( See, for examples, [20], [19], [21], [22], [23] ). The algorithm is also used to compute the criticality graph (see figure 19 through figure 24). Figure 19 shows the criticalities, figure 24 shows the criticality graph, and the remaining figures show the zones.

The algorithm can remind one of various watershed algorithms, except that the borders are precisely defined as are the peaks and pits, and it extends to higher dimensions(See [24], [25], [26], [27], [28], [29], [30]). As we shall see, one can use the zones to address issues of over-segmentation, and the zone segmentation is naturally hierarchical.

The algorithm labels the lattice vertices and the disambiguation points, in decreasing value order of the extended $\delta$. The algorithm does not require multiple passes, but we present it in this fashion for ease of understanding.