Persistant homology - Point data sets from images

Question

I have been reading about topological data analysis techniques and specifically about Persistent Homology. The examples I have seen so far use point clouds as the data sets. But what if we have, say, a set of images? How do we take our images and go to point clouds?

Answer 1

One way to take an image to a point cloud is if you want to study the statistics of the small local patches in that image. For example, you can study $3 \times 3$ pixel patches by cutting a black-and-white image up into many $3 \times 3$ pixel patches, and then considering each patch as a vector in 9-dimensional space (as each patch is a list of 9 different pixel intensity values). The papers "On the Local Behavior of Spaces of Natural Images" (https://link.springer.com/article/10.1007/s11263-007-0056-x) and "On the Nonlinear Statistics of Range Image Patches" (https://www.math.colostate.edu/~adams/research/OnTheNonlinearStatisticsOfRangeImagePatches.pdf) take this approach, for example. To try out example computations along these lines, see the image patch datasets at https://github.com/ds4m/topological-data-analysis/wiki or at Section 6 of https://www.math.colostate.edu/~adams/research/javaplex_tutorial.pdf.

It is important to note that you can instead transform an image into persistence barcodes (not passing through