Applied topology 9: Spaces of 3x3 natural image patches

[@pavan8632]

Can’t say enough about how fun you’ve made learning applied topology, a huge thanks and I earnestly look forward to watching all your lessons!

[@HenryAdamsMath]

Thanks so much Pavan, I appreciate it!

[@vijaysrirambhatla3874]

is there any software to generate barcode form a 3d image ?

[@HenryAdamsMath]

If you have a single 3d grayscale image, you could produce a barcode by computing the sublevelset persistent homology of that image. From the software packages at www.math.colostate.edu/~adams/advising/appliedTopologySoftware/, several packages that should allow you to do that are GUDHI and Perseus, for example (working with cubical complexes for this would be convenient). If you have a single 3d color image, then you could either produce a grayscale image from that, or produce a sublevelset persistence diagram for each channel (red, blue, green). If you have a collection of 3d images, then you could think of those as a point cloud dataset in high-dimensional space, and compute the point cloud (say Vietoris-Rips) persistent homology using for example GUDHI or Ripser.

[@umbertolupo5609]

In addition to Henry's excellent answer, I would also like to point out that the giotto-tda package (giotto-ai.github.io/gtda-docs/0.4.0/library.html) contains a number of utilities for generating grayscale (2D or 3D) images for processing with cubical persistence. GUDHI's cubical persistence is wrapped in giotto-tda so you also have access to sublevel set persistence via cubical complexes. ripser is also wrapped (and very slightly improved) so you can process point clouds (and abstract weighted graphs), again as Henry mentions.

[@vijaysrirambhatla3874]

@@HenryAdamsMath Thanks a lot.

[@alexandramalyugina4311]

Hi Henry! Thank you, a good video and a paper too! I have a technical question - how to calculate contrast norm for say 7x7 patches? In this case we need 49x49 matrix and calculate xMx^t, I can't really find any info on that, could you pls point out to some references? Cheers!

[@HenryAdamsMath]

Thanks Alexandra! Yes, it is possible to create a 49x49 matrix used to calculate the contrast norm for 7x7 patches, which is analogous to the 9x9 matrix used to calculate the contrast norm for 3x3 patches contained on page 88 of "The nonlinear statistics of high-contrast patches in natural images" by Lee, Pedersen, and Mumford. Indeed, I wrote code to compute this 49x49 matrix for my paper www.math.colostate.edu/~adams/research/OnTheNonlinearStatisticsOfRangeImagePatches.pdf with Gunnar. Please send me an email and I can share this 49x49 matrix and the code for creating it. Best, Henry