Related papers: Critical Points to Determine Persistence Homology
The growing size of point clouds enlarges consumptions of storage, transmission, and computation of 3D scenes. Raw data is redundant, noisy, and non-uniform. Therefore, simplifying point clouds for achieving compact, clean, and uniform…
Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…
Point clouds are widely used representations of 3D data, but determining the visibility of points from a given viewpoint remains a challenging problem due to their sparse nature and lack of explicit connectivity. Traditional methods, such…
The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops,…
The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis.…
Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic,…
While several convolution-like operators have recently been proposed for extracting features out of point clouds, down-sampling an unordered point cloud in a deep neural network has not been rigorously studied. Existing methods down-sample…
In computer vision, keypoint detection is a fundamental task, with applications spanning from robotics to image retrieval; however, existing learning-based methods suffer from scale dependency and lack flexibility. This paper introduces a…
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher-dimensional topological…
We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological…
Most existing 3D geometry copy detection research focused on 3D watermarking, which first embeds ``watermarks'' and then detects the added watermarks. However, this kind of methods is non-straightforward and may be less robust to attacks…
Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we…
Given a finite set in a metric space, the topological analysis generalizes hierarchical clustering using a 1-parameter family of homology groups to quantify connectivity in all dimensions. The connectivity is compactly described by the…
Point clouds are essential for object modeling and play a critical role in assisting driving tasks for autonomous vehicles (AVs). However, the significant volume of data generated by AVs creates challenges for storage, bandwidth, and…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent…
This technical report introduces a novel approach to efficient computation in homological algebra over fields, with particular emphasis on computing the persistent homology of a filtered topological cell complex. The algorithms here…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…