The Euler calculus—an integral calculus based on Euler characteristic as a valuation on
constructible functions—is shown to be an incisive tool for answering questions about …

In this survey, we review the literature on inverse problems in topological persistence theory.
The first half of the survey is concerned with the question of surjectivity, ie the existence of …

Shape plays a fundamental role in biology. Traditional phenotypic analysis methods
measure some features but fail to measure the information embedded in shape …

The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding
the size and prominence of topological features at varying scales, the PD provides important …

Background: In recent geometric design, many effective toolkits for geometric modeling and
optimization have been proposed and applied in practical cases, while effective and efficient …

Topological data analysis (TDA) is a powerful tool to solve classification problems using the
underlying shape of data. Furthermore, TDA is useful in shape reconstruction of compact …

Persistent homology is a tool that can be employed to summarize the shape of data by
quantifying homological features. When the data is an object in $\mathbb {R}^ d $, the …

Shape recognition and classification is a problem with a wide variety of applications. Several
recent works have demonstrated that topological descriptors can be used as summaries of …

Persistent homology has been an important tool in topological and geometrical data
analysis to study the shape of data. However, its ability to differentiate between various …

Topological data analysis and, more specifically, persistent homology have received
significant attention as a method of describing the shape of complex data. Persistent …