In this paper we consider two topological transforms that are popular in applied topology: the
Persistent Homology Transform and the Euler Characteristic Transform. Both of these …

This paper introduces decorated merge trees (DMTs) as a novel invariant for persistent
spaces. DMTs combine both π 0 and H n information into a single data structure that …

In this paper we study functions on the interval that have the same persistent homology,
which is what we mean by the fiber of the persistence map. By imposing an equivalence …

We use persistent homology in order to define a family of fractal dimensions, denoted dim
PH i (μ)\dim _ PH^ i (μ) for each homological dimension i≥ 0, assigned to a probability …

A recurring task in mathematics, statistics, and computer science is understanding the
connectivity information, or equivalently, the topological properties of a given object. For …

Currently, medical images represent basis of clinical diagnosis and source of telehealth and
teleconsultation processes. Exchange of these images can be subject to several challenges …

Cellular sheaves are a discrete model for the theory of sheaves on cell complexes. They
carry a canonical cochain complex computing their cohomology. This thesis develops the …

The field of mathematical morphology offers well-studied techniques for image processing
and is applicable for studies ranging from materials science to ecological pattern formation …

This dissertation is about building fundamental techniques for comparing data via a
geometric and topological data analysis method called multiparameter persistent homology …

We hypothesized that persistent lifetime (PLT) images could represent tumor imaging traits,
locations, and persistent contrasts of topological components (connected and hole …