" 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 …

The Reeb graph is a construction which originated in Morse theory to study a real-valued
function defined on a topological space. More recently, it has been used in various …

When a category CC satisfies certain conditions, we define the notion of rank invariant for
arbitrary poset-indexed functors F: P → CF: P→ C from a category theory perspective. This …

We propose a metric for Reeb graphs, called the functional distortion distance. Under this
distance, the Reeb graph is stable against small changes of input functions. At the same …

Given a continuous function f: X → R f: X→ R and a cover II of its image by intervals, the
Mapper is the nerve of a refinement of the pullback cover f^-1 (I) f-1 (I). Despite its success in …

Recovering hidden structure from complex and noisy non-linear data is one of the most
fundamental problems in machine learning and statistical inference. While such data is often …

A special family of non-trivial loops on a surface called handle and tunnel loops associates
closely to geometric features of" handles" and" tunnels" respectively in a 3D model. The …

The Reeb space, which generalizes the notion of a Reeb graph, is one of the few tools in
topological data analysis and visualization suitable for the study of multivariate scientific …

We present a deterministic algorithm to compute the Reeb graph of a PL real-valued
function on a simplicial complex in O (m log m) time, where m is the size of the 2-skeleton …

Contour Trees and Reeb Graphs are firmly embedded in scientific visualization for analysing
univariate (scalar) fields. We generalize this analysis to multivariate fields with a data …