Questions about the Persistent homology and persistence diagrams

[E53klasky]

Hello there, I have recently been studying the effects of lossy compression and how it affects the topology of contours. I was wondering whether these persistence diagrams can be used to determine changes in the topology of a given contour?

[hschreiber]

A persistence diagram records changes in the homology, so I guess, it does what you want. Can you give us a bit more context to your problem (theoretically, but also practically, like the format of your data etc.)?

[E53klasky]

Thanks for the response. I am working with CFD data generated with Xcompact3D. The data are defined on a structured Cartesian mesh. While some simulations are fully 2D and others are 3D, my current analysis is always performed on 2D slices of the field.

For example, if a contour at a given isovalue gains or loses components, develops or removes holes, or experiences contour merging/splitting after compression, can these changes be inferred from differences in the persistence diagrams?

Any references or suggestions for analyzing compression-induced topological changes in this setting would be greatly appreciated.

[hschreiber]

If you are using structured meshes, are you faces rectangles or a more complicated objects? If you have something very grid-like, it could simplify some of my points below.

The biggest hurdle I see in your description, is that the compression does not have to induce an inclusion (in either direction) from the data before and after, no? Or do you have any guarantees on the maps between your spaces/meshes? If you can guarantee that to go from your mesh to the compressed version of it you only add or remove squares/edges/vertices or you have the possibility to have triangles instead of squares (i.e. a simplicial complex) and you have a mix of simplex additions and contraction of two vertices into one, then you can directly apply persistence homology to it and this will give you information of how many components and holes appear and disappear. If you don't have those guarantees, you will need to trick a bit by using the union of both spaces to induces inclusion maps. To summarize, the possibilities:

1. M --- only additions ---> compr(M) or M <--- only removals --- compr(M) : use standard persistence (that is what you can find in python Gudhi)
2. M --- additions and removals ---> compr(M) : use zigzag persistence (the only implementation for general zigzag persistence and not just simplicial (as you don't seem to have a simplicial complex) that I know is the one of gudhi. But only in C++, there is no python interface yet. If you can get somehow something simplicial, there is fast zigzag.)
3. simplicial(M) --- additions and vertex contractions ---> compr(M) (or the inverse process) : simplicial map persistence (I am not aware of a python implementation yet, but there is some C++ code here)
4. M --- additions ---> M U compr(M) <--- removals --- compr(M) : zigzag persistence again.

For tracking merge and splits of your contours, you cannot directly use the persistence diagrams as they will not tell you which cycle splits or merge, it just tells you that something appears or disappears. If you needed to track the merge and splits of components, not holes, I would have suggested you using merge trees as it is quite easy to know when two trees (aka components) have to merge (you will easily find video about it). The splitting is way more tricky, but I think I saw something about it, I just don't remember what/where. You can perhaps play with merges in the dual space.
But for contours, you need to track the merging and splitting of the 1-dimensional cycles and the criteria is less clear. I have a vague memory of a direct relation between the 0-homology of a d-dimensional space and the (d-1)-homology of the dual of that space (so here we need d = 2). So, you can search in that direction as 0-homology corresponds to counting components, I just don't want to guarantee it.

[hschreiber]

I forgot to point out that you will not need to look at "differences between persistence diagrams" as you said, but at just one diagram: you add everything of M at time 0 and every changes done to get compr(M) at time 1 (if you do a second compression, you put them at time 2 etc.) and from there you will get a persistence diagram with 3 types of points:

• (0, inf): cycles which where existing in M and are still existing in compr(M)
• (0, 1): cycles which where existing in M, but disappeared in compr(M)
• (1, inf): cycles appearing in compr(M) and therefore where not in M

You just have to count how many you have of each in each dimension and you know how the cycles changed.

Except if I completely misunderstood your problem...