# Projekt: Geometrie 1

### Thema: Topological data analysis for image segmentation

Jana Vráblíková

Topology studies the properties of objects that are preserved under continuous deformation, such as stretching, twisting or bending. There is an old joke that a coffee cup and a doughnut are the same thing to a topologist! While in many situations we want to distinguish objects with finer metrics, in many cases the topological features give us the key to analysing a given object.

A circle is clearly different from an infinity sign, even to a topologist, but if they are represented by points sampled from the object, can we still tell them apart? What if there is also some noise in the data? Can we say anything about the topology of the object if it is only made up of isolated points?

Can this be useful in the task of image segmentation? How do we divide a digital image into several segments based on common features such as color or texture and identify their boundaries? Let us find out!