| Abstract: |
| The classical bottleneck stability theorem tells us that two functions which are close in the sup norm will yield persistence diagrams which are close in the bottleneck distance; that $C^0$ perturbations may only produce small changes in the lifespans of generators. There is no guarantee, however, on the number of generators which may be produced by such a perturbation. Indeed, millions of such generators may arise when studying noisy datasets, presenting an obstacle to both (1) the actual computation of persistent homology (PH) of large 3D images, and (2) any analysis of PH which incorporates the number of generators.
As such, it is often necessary to denoise the data before computing its PH. In this talk, we analyze the PH of synthetic 3D images of porous media in the presence of spatially uncorrelated noise, and perform a comparative analysis of various topological measures (e.g. bottleneck distance, Wasserstein distance, persistence statistics, persistence images, etc.) to assess their robustness to the noising and denoising process. |
|