Paul Bendich

Computing Robustness and Persistence for Images

We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchical algorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, the dual complexes are geometrically realized in R3 and can thus be used to construct level and interlevel sets. We apply these tools to study 3-dimensional images of plant root systems.

Persistent homology analysis of brain artery trees

New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.

Inferring Local Homology from Sampled Stratified Spaces

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

Topological and statistical behavior classifiers for tracking applications

This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced. The track appraisal uses elementary topological data analysis coupled with basic machine-learning techniques, and it adjusts the traditional kinematic data association likelihood (i.e., track score) using an established formulation for feature-aided data association. The proposed method is tested and demonstrated on synthetic vehicular data representing an urban traffic scene generated by the Simulation of Urban Mobility package. The vehicles in the scene exhibit different driving behaviors. The proposed method distinguishes those behaviors and shows improved data association decisions relative to a conventional, kinematic MHT.

Persistent Intersection Homology

The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper incorporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersection homology gives useful information about the relationship between an embedded stratified space and its singularities. We give an algorithm for the computation of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcomplexes, and we prove its correctness. We also derive, from Poincaré Duality, some structural results about persistent intersection homology.

The robustness of level sets

We define the robustness of a level set homology class of a function f : X → R as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X = R3 has ramifications in medical imaging and scientific visualization.

Improving homology estimates with random walks

This experimental paper makes the case for a new approach to the use of persistent homology in the study of shape and feature in datasets. By introducing ideas from diffusion geometry and random walks, we discover that homological features can be enhanced and more effectively extracted from spaces that are sampled densely and evenly, and with a small amount of noise. This study paves the way for a more theoretical analysis of how random walk metrics affect persistence diagrams, and provides evidence that combining topological data analysis with techniques inspired by diffusion geometry holds great promise for new analyses of a wide variety of datasets.

Feature-aided multiple hypothesis tracking using topological and statistical behavior classifiers

This paper introduces a method to integrate target behavior into the multiple hypothesis tracker (MHT) likelihood ratio. In particular, a periodic track appraisal based on behavior is introduced that uses elementary topological data analysis coupled with basic machine learning techniques. The track appraisal adjusts the traditional kinematic data association likelihood (i.e., track score) using an established formulation for classification-aided data association. The proposed method is tested and demonstrated on synthetic vehicular data representing an urban traffic scene generated by the Simulation of Urban Mobility package. The vehicles in the scene exhibit different driving behaviors. The proposed method distinguishes those behaviors and shows improved data association decisions relative to a conventional, kinematic MHT.

A point calculus for interlevel set homology

Highlights? Topology of inter-level sets of real-valued functions via Persistent Homology. ? Point diagrams to quantitatively measure topological features of inter-level sets. ? Combinatorial reasoning for dealing with families of inter-level sets. The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a real-valued function. Our account of the subject is expository, devoid of proofs, and written for non-experts in algebraic topology.

Persistent homology under non-uniform error

Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.