3.1 Critical points for volume data satisfying data uniqueness.

Let us assume that satisfies data uniqueness. A critical value c will be a value at which the topology of the contours of f change as the threshold is decreased through c; each such change will have an associated critical point p for which f(p) = c. The critical points for simplicial data are the same as in [4], where a linear interpolant is assumed, but regular data adds another type of saddle not equal to a data reading (4 below).

Let . Let N be the set of points in V adjacent to p. Let N_H (resp. N_L) be the Highs in N (resp. Lows in N) with respect to the value of p. Compute the connected components of N_H and N_L, with respect to the cells that share p, excluding p. For example, two Highs are in the same component of N_H if they are path connected in the cells sharing p, by a path of adjacent Highs that does not include p.

1

If N = N_L (hence p has a larger value than all its neighbors) then p is a local maximum critical point and f(p) a local maximum critical value.

2

If N = N_H (hence p has a smaller value than all its neighbors) then p is a local minimum critical point and f(p) a local minimum critical value.

3

If either N_H or N_L consist of more than one connected component then p is saddle critical point and f(p) is a saddle critical value.

4

Let p be the disambiguation point of a cell face F. If the Highs of F are not part of the same connected set of Highs within the cells sharing F for all thresholds greater than disambiguation value c ($c+\epsilon$) then p is saddle critical point and c = f(p) is a saddle critical value (see figure 2).

Theorem 2  

Let f be admissable and satisfy data uniqueness. If is a critical value, then for all sufficiently small $\epsilon > 0$, $B(\bf {X}_{f}^{\geq \tau - \epsilon })$ and $B(\bf {X}_{f}^{\geq \tau + \epsilon })$ are not homeomorphic. Conversely, if $[\tau_1 , \tau_2 ]$ is a critical value-free interval then $B(\bf {X}_{f}^{\geq \tau_1 })$ and $B(\bf {X}_{f}^{\geq \tau_2 })$ are homeomorphic.

Proof:

Topology changes to can occur in several ways (and, of course, several changes can occur simultaneously). First of all a new contour (boundary manifold) can be created as is decreased through a value c. It is easy to see from property 1 of admissable functions that this can happen if and only if c is a local maximum critical value, as the associated critical point p comprises a single component of Highs for $\tau = c - \epsilon$. Similarly, a contour can vanish at c, this can happen if and only if c is a local minimum critical value, where p comprises a single component of Lows for $\tau = c + \epsilon$.

Two or more separate contours can merge at value c. From property 1 this means that there exists two separate components of Highs, at , that are merged into one component of Highs at . This can happen if and only if c is a saddle critical value of type 3 (p is a data point that becomes High at c), or c is a saddle valuet of type 4 (and changes the connectivity of the Highs in a 4-hit face).

One or several contours can split at c. From admissability property 1 this will happen only when a component of Lows is separated by a change in connectivity at c. This can happen if and only if c is a saddle critical value of type 3 or 4.

Finally a contour can change in genus at c (a handle is formed or vanishes). A handle is formed if there are (at least) two locally separate portions of the same contour in a neighborhood $\nu$ of a point p at , so that they are merged, but do not comprise all of , at . By admissablity, this means that two locally separate components of Highs merge at c, and thus c must be a saddle critical value (and p a saddle point) of type 3 or 4 above. Symmetrically, if the genus of a contour decreases, then a locally connected set of Lows must be split at a saddle criticality of type 3 or 4. Q.E.D.