Unexpected behavior from Merge and Contour Tree (FTM)

[ngorski9]

Unexpected behavior from Merge and Contour Tree (FTM)

[julien-tierny]

Hi,
thanks for your message.

I do confirm that not all the critical points of a scalar field are captured by the Reeb graph (or the contour tree and its variants, including the merge tree). By definition, the Reeb graph is obtained by contracting connected components of level sets to points. Thus, branching (i.e. vertices) in the Reeb graph only occurs when the number of connected components of level sets change.

However, it is possible to cross a critical value without changing the number of connected components. For example, in 3D, a level set component can "loop" back to itself and change its genus when passing a 1-saddle. In this example, the number of connected components does not change and the said saddle will therefore not be captured by the Reeb graph.

The original paper by G. Reeb (in French) describes all the possible configurations. You can find a translation of this result in my habilitation thesis (page 55, property 12): https://julien-tierny.github.io/stuff/papers/tierny_habilitation.pdf

I hope this helps.

Best regards,