This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.


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Filtration is a sequence of "nested" sets, topological spaces, cell complexes, cubical complexes etc: $$K¹ ↪ K² ↪ K³ ↪ K⁴ ↪ … ↪ K^s,$$ where the arrows are the inclusions.

Below is an example of a gray scale image and its three thresholded versions. These are binary images that can be understood as cell complexes (see Cell decomposition of images). Here white is empty.

“Lena”: original “Lena”: T = 50 “Lena”: T = 100

For completeness sake one can add a fully white image in the beginning and fully black in the end.

Another example:

Another source of filtrations comes from "enlarging" a point cloud that results in a Vietoris-Rips complex:

For topological analysis of filtrations see Persistent homology.

There can also be multi-parameter filtrations, see Color images.