Topological pattern recognition for point cloud data

The Vietoris-Rips filtration for an n-point metric space is a sequence of large simplicial complexes adding a topological structure to the otherwise disconnected space. The persistent homology is a key tool in topological data analysis and studies topological features of data that persist over many scales. The fastest algorithm for computing persistent homology of a filtration has time O(M (u) + u 2 log 2 u), where u is the number of updates (additions or deletions of simplices), M (u) = O(u 2.376 ) is the time for multiplication of u × u matrices.

ArXiv, 2021

We introduce a linear dimensionality reduction technique preserving topological features via persistent homology. The method is designed to find linear projection L which preserves the persistent diagram of a point cloud X via simulated annealing. The projection L induces a set of canonical simplicial maps from the Rips (or Čech) filtration of X to that of LX. In addition to the distance between persistent diagrams, the projection induces a map between filtrations, called filtration homomorphism. Using the filtration homomorphism, one can measure the difference between shapes of two filtrations directly comparing simplicial complexes with respect to quasi-isomorphism μquasi-iso or strong homotopy equivalence μequiv. These μquasi-iso and μequiv measures how much portion of corresponding simplicial complexes is quasi-isomorphic or homotopy equivalence respectively. We validate the effectiveness of our framework with simple examples.

Advances in Applied Mathematics

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data-for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when given data points split into well-separated clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a long time in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimal spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

2008 19th International Conference on Pattern Recognition, 2008

In this paper, we design linear time algorithms to recognize and determine topological invariants such as the genus and homology groups in 3D. These properties can be used to identify patterns in 3D image recognition. This has many and is expected to have more applications in 3D medical image analysis. Our method is based on cubical images with direct adjacency, also called (6,26)-connectivity images in discrete geometry. According to the fact that there are only six types of local surface points in 3D and a discrete version of the well-known Gauss-Bonnett Theorem in differential geometry, we first determine the genus of a closed 2Dconnected component (a closed digital surface). Then, we use Alexander duality to obtain the homology groups of a 3D object in 3D space. This idea can be extended to general simplicial decomposed manifolds or cell complexes in 3D.

2016

Every moment of our daily life belongs to the new era of “Big Data”. We continuously produce, at an unpredictable rate, a huge amount of heterogeneous and distributed data. The classical techniques developed for knowledge discovery seem to be unsuitable for extracting information hidden in these volumes of data. Therefore, there is the need to design new computational techniques. In this paper we focus on a set of algorithms inspired by algebraic topology that are known as Topological Data Analysis (TDA). We briefly introduce the principal techniques for building topological spaces from data and how these can be studied by persistent homology. Several case studies, collected within the TOPDRIM (Topology driven methods for complex systems) FP7FET project, are used to illustrate the applicability of these techniques on different data sources and domains.

Proceedings 2001 International Conference on Image Processing (Cat. No.01CH37205), 2001

There are a number of tasks in low level vision and image processing that involve computing certain topological characteristics of objects in a given image including connectivity and the number of holes. In this note, we combine a new method combinatorial topology to compute the number of connected components and holes of objects in a given image, and fast segmentation methods to extract the objects.

2007

We present a computational method for extracting simple descriptions of high dimensional data sets in the form of simplicial complexes. Our method, called Mapper, is based on the idea of partial clustering of the data guided by a set of functions defined on the data. The proposed method is not dependent on any particular clustering algorithm, i.e. any clustering algorithm may be used with Mapper. We implement this method and present a few sample applications in which simple descriptions of the data present important information about its structure.

Computers & Graphics: X

Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for oneparameter persistent homology have been established to connect persistent homology with machine learning techniques with applicability on shape analysis, recognition and classification. We contribute a kernel construction for multi-parameter persistence by integrating a one-parameter kernel weighted along straight lines. We prove that our kernel is stable and efficiently computable, which establishes a theoretical connection between topological data analysis and machine learning for multivariate data analysis.

2010

The ability to perform not only global matching but also partial matching is investigated in computer vision and computer graphics in order to evaluate the performance of shape descriptors. In my talk I will consider the persistent homology shape descriptor, and I will illustrate some results about persistence diagrams of occluded shapes and partial shapes. The main tool is a Mayer-Vietoris formula for persistent homology. Theoretical results indicate that persistence diagrams are able to detect a partial matching between shapes by showing a common subset of points both in the one-dimensional and the multi-dimensional setting. Experiments will be presented which outline the potential of the proposed approach in recognition tasks in the presence of partial information.

ArXiv, 2018

In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-D persistence modules obtained from families of point clouds. Results include observations of how differences in geometric structure of point clouds affect the matching distance between persistence modules. We offer these results as a starting point for the investigation of more complex data.

Environmetrics

This paper presents a new clustering algorithm for space-time data based on the concepts of topological data analysis and in particular, persistent homology. Employing persistent homology-a flexible mathematical tool from algebraic topology used to extract topological information from data-in unsupervised learning is an uncommon and a novel approach. A notable aspect of this methodology consists in analyzing data at multiple resolutions which allows to distinguish true features from noise based on the extent of their persistence. We evaluate the performance of our algorithm on synthetic data and compare it to other well-known clustering algorithms such as K-means, hierarchical clustering and DBSCAN. We illustrate its application in the context of a case study of water quality in the Chesapeake Bay.

Computer Graphics Forum, 2015

Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1-dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n).

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. Despite recent progress, the computation of PH remains a wide open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of applied mathematicians and computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

Inverse Problems

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning. 2010 Mathematics Subject Classification. 55N99.

2014

In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been implemented as extensions of the Kenzo system and have shown a good performance when applied on some actual images extracted from a public dataset.