Topological Data Analysis [1, 2, 3, 4, 5, 6] is a sound family of techniques that is gaining an increasing importance for the interactive analysis and visualization of data in imaging and machine learning applications. Given the increasing complexity and size of current collections of acquired or simulated data-sets (2D, 3D and nD), these approaches aim at helping users understand the complexity of their data by providing insights about its topological and geometric structure.

In low dimensions (typically 2 or 3), Topological Data Analysis enables users to rapidly extract, interact with, measure and classify geometrical features defined by level sets or integral lines. Thanks to simplification mechanisms based on Persistent Homology, such algorithms additionally construct multi-scale topological representations of the data, that enable users to perform robust analyses and comparisons despite the presence of noise. The soundness, efficiency and robustness of this class of approaches made it increasingly popular in the last few years in a variety of 2D and 3D imaging analysis applications. In higher dimensions, these techniques have recently been adapted to form the basis of new clustering algorithms.

The purpose of this course is to introduce the main concepts of the recent field of Topological Data Analysis and illustrate their use in imaging (scientific visualization) and machine learning applications, both from a mathematical and practical point of view [7, 8].