8.2 Properties of the criticality graph and zones.

Zones are topologically equivalent families of iso-surfaces

Theorem 3   Let p be a criticality with parent q.

1) For each $\tau$ satisfying $f(p) > \tau > f(q)$, $B (O_p ( \tau ))$ is contained in $\zeta (p)$.

2) For any pair $f(p) > \tau_1 > \tau_2 > f(q)$, $B (O_p ( \tau_1 ))$ and $B (O_p ( \tau_2 ))$ are homeomorphic.

3) For all $\tau$, if O is any component of $\bf {X}_{f}^{\geq \tau}$ not containing p, then B (O) and $\zeta (p)$ are disjoint.

4) Moreover, for any $\tau$ such that $\tau > f(p)$ or $\tau < f(q)$, $B(\bf {X}_{f}^{\geq \tau})$ and $\zeta (p)$ are disjoint.

Proof: A point $r \in \zeta (p)$ if and only if f(q) < f(r) < f(p) and $r \in O_p ( f(q) )$. For $f(p) > \tau > f(q)$, $B (O_p ( \tau ))$ consists of points r with $f(r) = \tau$, and by monotonicity of $\bf {X}_{f}^{\geq \tau}$ each such $r \in O_p ( f(q) )$, establishing claim 1. By the definitions of the criticality graph and the zones, $\zeta (p)$ contains no critical point with value between f(p) and f(q); this establishes claim 2. Claim 3 follows from the connectivity of $O_p(\tau)$ and the fact that boundary manifolds are pairwise disjoint by axiom 1. Claim 4 follows immediately from the proof of claim 1.

This suggests a powerful way to organize the data readings, and means that we can simplify the data in a zone without losing any essential topological information. The criticality graph, with the zones associated with each vertex, hierarchically organize the entire dataset (as opposed to [2]).

Since the zone $\zeta (p)$ of criticality p represents a collection of topologically identical level set components (objects), we can thus trace an object's evolution, as the threshold is varied, using the criticality graph. To summarize:

Corollary 8.1  

For the criticality graph, the following hold:

1

Each node in the graph represents a topologically distinct object. Each leaf node (in-degree 0) of the graph is a maximum.

2

A vertex with multiple children represents a saddle point, saddle set, or minimum set, and the merging of several object boundaries at the critical value.

3

Each saddle with one child represents an increase in the topological complexity of the object associated with the child (increase in genus of a boundary manifold), as the threshold is decreased though the saddle value.

4

Each local minimum on a path represents either a decrease in topological complexity of the boundary of one of the child objects and, in the case of a minimum set with multiple children, represents the merging of objects (islands in the minimum set), as the threshold is decreased through minimum's value.

5

The root (sink) of the graph (out-degree 0) represents the single object containing all data reading points for any threshold $\tau$ less than all data reading values.

6

The topological changes of a sample object, as threshold $\tau$ decreases, can be traced from creation to death by starting from its leaf and following the parent pointers to the root of the graph (when it merges into the single object encompassing the entire volume dataset $\delta$).

Proof: The results follow immediately from theorems 2 and 3, and the definition of the criticality graph.