The last decade saw an enormous boost in the field of computational topology: methods and
concepts from algebraic and differential topology, formerly confined to the realm of pure …

Topological Data Analysis (TDA) is a recent and fast growing field providing a set of new
topological and geometric tools to infer relevant features for possibly complex data. This …

" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

Segmentation algorithms are prone to make topological errors on fine-scale struc-tures, eg,
broken connections. We propose a novel method that learns to segment with correct …

Many data sets can be viewed as a noisy sampling of an underlying space, and tools from
topological data analysis can characterize this structure for the purpose of knowledge …

Accurate segmentation of topological tubular structures, such as blood vessels and roads, is
crucial in various fields, ensuring accuracy and efficiency in downstream tasks. However …

Detecting periodicity in large scale data remains a challenge. While efforts have been made
to identify best of breed algorithms, relatively little research has gone into integrating these …

We define a new topological summary for data that we call the persistence landscape. Since
this summary lies in a vector space, it is easy to combine with tools from statistics and …

Inferring topological and geometrical information from data can offer an alternative
perspective in machine learning problems. Methods from topological data analysis, eg …

Persistence theory emerged in the early 2000s as a new theory in the area of applied and
computational topology. This book provides a broad and modern view of the subject …