3.2 Critical isosets when data does not satisfy uniqueness.

If the data does not satisfy data uniqueness then the properties of admissable functions imply that connected sets of data points (as Highs) of identical value c will be enclosed in isovolumes at thresholds $c-\epsilon$. As the threshold is decreased through c, an object can merge with the isovolume containing these points. Complex topology changes can occur at the value c of these sets, termed isosets. Let S be an isoset with value c and N, N_H, and N_L be defined as above. For an admissable function, in addition to the point criticalities, the critical isosets are:

1

If N_H is empty, then S is a local maximum isoset and c a maximum critical value.

2

If N_H = N, then S is a local minimum isoset and c a minimum critical value.

3

If N_H is non-empty and either N_H or N_L consists of more than 1 connected component then S is a saddle isoset, and c is a saddle critical value.

4

If N_H and N_L each consist of a single nonempty component, then S is called a regular isoset. This set can cause a genus change in an object O, when it merges with O if it adds a new handle. This occurs if there exists a loop (closed path) in N_L and a loop in S so that the loops and interlocked.

Note that a new object of arbitrary topological complexity (with possibly multiple boundary manifolds) is created at a maximum isoset, one or several manifolds vanish and multiple objects can merge at a minimum isoset, and extremely complex merges, joins, and genus changes can occur at saddle isosets. However only a genus change can occur at a regular isoset. For an example of a critical regular isoset, consider a donut with flat bottom and sides (the isovolume of the isoset) that is merged with a solid cube (the object) so that it rests flat on the top of the cube. In this case no genus change occurs as the boundary is still homeomorphic to a sphere, but if it is merged so that it is glued to the top of the cube resting on its side, a new handle is formed. We leave as an open problem developing an efficient combinatorial algorithm for recognizing critical regular isosets in all dimensions.

With addition of the critical isosets we can drop the data uniqueness requirements,

Theorem 3  

Let f be admissable. If is a critical value, then for all sufficiently small , and are not homeomorphic. Conversely, if is a critical value-free interval then and are homeomorphic.

Proof: The proof follows the proof of theorem 2, except that now an additional type of genus change can occur: one that is caused neither by the merging of two locally separate components of Highs or the local separatation of a component of Lows, but rather by the merging with a contour of an isoset of Highs, the boundary of which adds at least one handle to the contour. Q.E.D.