Persistence Diagram calculation failing to detect some features?

[hajo...@colorado.edu]

Hello TTKers!

I have a potential energy function defined on a torus and am trying to perform some analysis using TTK in Paraview.

The critical point calculations and the Morse-Smale complex come out great and as expected (though the critical points perturb a little bit for the Morse-Smale complex).

The Persistence Diagram appears to be missing a few bars though. Specifically, there are two bars which should correspond to the nontrivial loops and one which should correspond to the nontrivial void which do not appear in the diagram.

I also tried to detect these features using a torus and the Elevation filter for the scalars but there I can only get a single bar, for the connected component. So, I'm presuming that this is outside TTK's calculations, but want to be sure that it's not a mistake on my part. :)

Thanks for the great work and your time!

-Howie

[Julien Tierny]

hi,

thanks for your message.

the persistence diagram module (in default mode) computes the diagrams for H_0 and H_{d-1} where d is the dimension of your domain (in your case 2, if I'm not mistaken. if you want the inside void to appear, you'll need to consider a volume data set).

this computation is done with the join and split trees of the function (see https://graphics.stanford.edu/courses/cs468-03-winter/Papers/carr-contour-soda.pdf and https://julien-tierny.github.io/stuff/papers/gueunet_tpds19.pdf) and indeed this computation does not track pairs of H_1 with infinite persistence (these can be inferred easily by the euler characteristic of the domain, without even considering the data defined on it). note also that for H_0, ttk is using the following convention:
the connected component created in the global minimum (in principle with infinite persistence) is always considered to be destroyed at the global maximum. that special pair is considered in both diagrams (H_0 and H_{d-1}) in order not to loose the main feature of the data when doing further analysis (such as distances or barycenters for instance).

I hope this helps.

cheers,
--
Dr Julien Tierny
CNRS Researcher
Sorbonne Universite
http://lip6.fr/Julien.Tierny