5.2 Properties of the criticality graph and zones.

Zones are topologically equivalent families of iso-surface bounded objects; zone components contain 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 $\zeta (p)$ and each component M of $B (O_p ( \tau ))$ is contained in a single component of $\zeta (p)$.

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) 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 $r \in O_p ( f(q) )$, so that that $B (O_p ( \tau ))$ is contained in $\zeta (p)$. The connectivity of M insures that it is contained in a single component of $\zeta (p)$, 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.

Observe that, by the continuity of f and the monotonicity of the level sets as the threshold is decreased, one can show that $\zeta (p)$ will have one component for each boundary iso-surface of $O_p ( \tau )$, for $f(q) < \tau < f(p)$.

The theorem 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 5.1  

The following hold:

1

Each node in the graph represents a topologically distinct family of objects (over all thresholds). Each object is associated with a node in the graph.

2

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

3

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 non-zero data reading values.

4

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 $\delta$).

5

The zone of a criticality p completely contains the iso-surface boundaries of each member of the topologically equivalent family of objects represented to p.

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