5.1 Criticality graph and zone of influence definitions

Definition 21 For purposes of defining our criticality graph (see figures 19 through 24), if $\tau$ is not in the range of the extended $\delta$ define $O_p ( \tau )$ be the connected component of $\bf {X}_{f}^{\geq \tau }$ that contains p, where p is a point or degenerate set. If $\tau$ is in the range of the extended $\delta$ , (equal to a data reading or disambiguation point), define $O_p (\tau )$ to be $\cup_{c > \tau } O_p (c)$ (note that this second case is a technicality since our axioms have not defined $\bf {X}_{f}^{\geq \tau }$ for values precisely equal to a data reading).

Definition 22 The vertices of the graph are the criticalities (labeled by the associated critical value), with parent pointer edges between criticalities defined as follows:

Let p be a criticality and let q be a saddle or minimum criticality with value f(q)< f(p), such that the following properties uniquely hold:

1) for all $\tau$ between f(p) and f(q), $q \notin O_p ( \tau )$ , and there does not exist a criticality r, with f(p) > f(r) > f(q), such that $r \in O_p ( \tau )$ .

2) For each $\epsilon > 0$ , $q \in O_p ( f(q) - \epsilon )$ .

Then q is the parent of p, and is there is a directed edge from p to q.

In rare cases, we must ``break ties'' arbitrarily.

If there is more than one criticality that satisfies the above two properties (and thus has the same value f(q)), let us call this collection Q. Arbitrarily order Q, with enumeration q_i, $0 \leq i \leq m$ , select q₀ as parent of p, and set q_i+1 as the parent of q_i, for each $0 \leq i < m$ .

Note that the second (tie breaking) case is just for the rare situation in which one object merges with several others at precisely the same value.

The criticality graph, together with the associated zones, represent all possible topologically distinct level set objects (components of $\bf {X}_{f}^{\geq \tau }$ , ranging over all $\tau > 0$ ) very succinctly. As we decrease the threshold from a critical value, with associated criticality p, the object containing p grows and deforms. When another critical value, with associated criticality q, is reached, the object changes in topology. We then make q the parent of p.

The zone of a criticality is the volume swept by the boundary manifolds of a level set component, as the threshold $\tau$ is varied over the interval of values for which this component remains topologically invariant. See figure 19 through figure 24 for an example of the criticality graph and associated zones, for a simple dataset.