4.1 Definition of the topological zones

The intuition behind the criticality tree and zones is quite simple (see figure 3). The nodes are the criticalities. As the threshold is decreased from the value of a critical point p, the object containing that criticality grows and deforms, until it comes to contain another criticality q, which in some manner changes the topology of the contours bounding the object. The criticality q is made the parent of p in the criticality tree. The topological zone of criticality p is the volume thus swept by the topological boundary of this object, and each connected component of this zone is the volume swept by an individual boundary component of the object. We define the zones by a set difference so that no assumption that the boundary manifolds vary continuously is required (they will not if the data contains isosets).

Let $O_p ( \tau )$ be the object containing p at threshold . For equal to a data value define to be the closure of $\cup_{c > \tau } O_p (c)$ (note that this second case is a technicality since our axioms have not defined for values precisely equal to a data reading).

Definition 9  

The vertices of the criticality tree are the criticalities with a parent pointer edge from a criticality p with value x_p to criticality q, with value x_q, if as the threshold is decreased q is the first criticality that becomes part of the same object as p, i.e., $q \in O_p( x_q - \epsilon )$ and there is no criticality r (with value x_r) such that x_q < x_r < x_p and $r \in O_p( x_q - \epsilon )$. For a given criticality p with parent q, the topological zone of p, denoted $\zeta (p)$, is the set difference O_p ( f(q)) - O_p ( f(p) ). The toplogical zone components are just the connected copmponents of . Note that when we do not assume data uniqueness, a given object may merge with several criticalities simultaneously, and we just arbitrarily order the parentage of these to break the ties.