• Corpus ID: 245650637

A Faithful Discretization of the Augmented Persistent Homology Transform

  title={A Faithful Discretization of the Augmented Persistent Homology Transform},
  author={Brittany Terese Fasy and Samuel Micka and David L. Millman and Anna Schenfisch and Lucia Williams},
The persistent homology transform (PHT) represents a shape with a multiset of persistence diagrams parameterized by the sphere of directions in the ambient space. In this work, we describe a finite set of diagrams that discretize the PHT such that it faithfully represents the underlying shape. We provide a discretization that is exponential in the dimension of the shape (making it practical for low dimensional shapes immersed in high dimensional spaces). Moreover, we show that this… 

Figures from this paper

Quantifying barley morphology using the Euler Characteristic Transform

This work focuses first on quantifying the morphology of barley spikes and seeds using topological descriptors based on the Euler characteristic, and then successfully train a support vector machine to classify 28 different varieties of barley based solely on the shape of their grains.

Efficient Graph Reconstruction and Representation Using Augmented Persistence Diagrams

An improved algorithm for graph— and, more generally, one-skeleton—reconstruction is presented and the improvement comes in reconstructing the edges, where the binary search employed takes advantage of the fact that the edges can be ordered radially with respect to a reference plane.



Learning Simplicial Complexes from Persistence Diagrams

This paper presents an algorithm for reconstructing plane graphs K=(V,E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams.

Reconstructing Embedded Graphs from Persistence Diagrams

Challenges in Reconstructing Shapes from Euler Characteristic Curves

This abstract explores the use of a finite number of Euler Characteristic Curves (ECC) to reconstruct plane graphs and shows that plane graphs without degree two vertices can be reconstructed using a finiteNumber of ECCs.

Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition

The method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data, and is not dependent on any particular clustering algorithm, i.e. any clustering algorithms may be used with Mapper.

Persistent Homology Transform for Modeling Shapes and Surfaces

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence

How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms

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 transforms are of

Persistent homology and Euler integral transforms

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 injectivity and

The Structure and Stability of Persistence Modules

This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such

Constructing Shape Spaces from a Topological Perspective

A generic construction scheme is presented and it is demonstrated how to apply this scheme when shape is interpreted as the differences that remain after factoring out translation, scaling and rotation, and the utility is demonstrated on the problem of distinguishing segmented hippocampi from normal controls vs. patients with Alzheimer’s disease.

Inverse Problems in Topological Persistence

Throughout, the tools and theorems that underlie advances in topological persistence theory are highlighted, and the reader's attention is directed to open problems, both theoretical and applied.