🍩 Database of Original & Non-Theoretical Uses of Topology

(found 9 matches in 0.002361s)

  1. Topological Data Analysis: Concepts, Computation, and Applications in Chemical Engineering (2021)

    Alexander D. Smith, Paweł Dłotko, Victor M. Zavala
    Abstract A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g., convolutional neural nets, Fourier series). Topological Data Analysis (TDA) is a field of mathematics that analyzes data from a fundamentally different perspective. TDA represents datasets as geometric objects and provides dimensionality reduction techniques that project such objects onto low-dimensional descriptors. The key properties of these descriptors (also known as topological features) are that they provide multiscale information and that they are stable under perturbations (e.g., noise, translation, and rotation). In this work, we review the key mathematical concepts and methods of TDA and present different applications in chemical engineering.

  2. Identification of Key Features Using Topological Data Analysis for Accurate Prediction of Manufacturing System Outputs (2017)

    Wei Guo, Ashis G. Banerjee
    Abstract Topological data analysis (TDA) has emerged as one of the most promising approaches to extract insights from high-dimensional data of varying types such as images, point clouds, and meshes, in an unsupervised manner. To the best of our knowledge, here, we provide the first successful application of TDA in the manufacturing systems domain. We apply a widely used TDA method, known as the Mapper algorithm, on two benchmark data sets for chemical process yield prediction and semiconductor wafer fault detection, respectively. The algorithm yields topological networks that capture the intrinsic clusters and connections among the clusters present in the data sets, which are difficult to detect using traditional methods. We select key process variables or features that impact the system outcomes by analyzing the network shapes. We then use predictive models to evaluate the impact of the selected features. Results show that the models achieve at least the same level of high prediction accuracy as with all the process variables, thereby, providing a way to carry out process monitoring and control in a more cost-effective manner.

  3. Evasion Paths in Mobile Sensor Networks (2015)

    Henry Adams, Gunnar Carlsson
    Abstract Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In ‘Coordinate-free coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path.

  4. Coverage Criterion in Sensor Networks Stable Under Perturbation (2014)

    Yasuaki Hiraoka, Genki Kusano
    Abstract To the coverage problem of sensor networks, V. de Silva and R. Ghrist (2007) developed several approaches based on (persistent) homology theory. Their criteria for the coverage are formulated on the Rips complexes constructed by the sensors, in which their locations are supposed to be fixed. However, the sensors are in general affected by perturbations (e.g., natural phenomena), and hence the stability of the coverage criteria should be also discussed. In this paper, we present a coverage theorem stable under perturbation. Furthermore, we also introduce a method of eliminating redundant cover after perturbation. The coverage theorem is derived by extending the Rips interleaving theorem studied by F. Chazal, V. de Silva, and S. Oudot (2013) into an appropriate relative version.

  5. A Topology-Based Object Representation for Clasping, Latching and Hooking (2013)

    J. A. Stork, F. T. Pokorny, D. Kragic
    Abstract We present a loop-based topological object representation for objects with holes. The representation is used to model object parts suitable for grasping, e.g. handles, and it incorporates local volume information about these. Furthermore, we present a grasp synthesis framework that utilizes this representation for synthesizing caging grasps that are robust under measurement noise. The approach is complementary to a local contact-based force-closure analysis as it depends on global topological features of the object. We perform an extensive evaluation with four robotic hands on synthetic data. Additionally, we provide real world experiments using a Kinect sensor on two robotic platforms: a Schunk dexterous hand attached to a Kuka robot arm as well as a Nao humanoid robot. In the case of the Nao platform, we provide initial experiments showing that our approach can be used to plan whole arm hooking as well as caging grasps involving only one hand.

  6. Applications of Persistent Homology to Time Varying Systems (2013)

    Elizabeth Munch
    Abstract \textlessp\textgreaterThis dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of time varying persistence diagrams, called vineyards when they were introduced by Cohen-Steiner, Edelsbrunner, and Morozov.\textless/p\textgreater\textlessp\textgreaterWe will first show that with a good choice of metric, these vineyards are stable for small perturbations of their associated point clouds. We will also define a new mean for a set of persistence diagrams based on the work of Mileyko et al. which, unlike the previously defined mean, is continuous for geodesic vineyards. \textless/p\textgreater\textlessp\textgreaterNext, we study the sensor network problem posed by Ghrist and de Silva, and their application of persistent homology to understand when a set of sensors covers a given region. Giving each of these sensors a probability of failure over time, we show that an exact computation of the probability of failure of the whole system is NP-hard, but give an algorithm which can predict failure in the case of a monitored system.\textless/p\textgreater\textlessp\textgreaterFinally, we apply these methods to an automated system which can cluster agents moving in aerial images by their behaviors. We build a data structure for storing and querying the information in real-time, and define behavior vectors which quantify behaviors of interest. This clustering by behavior can be used to find groups of interest, for which we can also quantify behaviors in order to determine whether the group is working together to achieve a common goal, and we speculate that this work can be extended to improving tracking algorithms as well as behavioral predictors.\textless/p\textgreater

  7. Coverage in Sensor Networks via Persistent Homology (2007)

    Vin de Silva, Robert Ghrist
    Abstract We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris–Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications.

  8. Coordinate-Free Coverage in Sensor Networks With Controlled Boundaries via Homology (2006)

    V. de Silva, R. Ghrist
    Abstract Tools from computational homology are introduced to verify coverage in an idealized sensor network. These methods are unique in that, while they are coordinate-free and assume no localization or orientation capabilities for the nodes, there are also no probabilistic assumptions. The key ingredient is the theory of homology from algebraic topology. The robustness of these tools is demonstrated by adapting them to a variety of settings, including static planar coverage, 3-D barrier coverage, and time-dependent sweeping coverage. Results are also given on hole repair, error tolerance, optimal coverage, and variable radii. An overview of implementation is given.

  9. Blind Swarms for Coverage in 2-D (2005)

    V. D. Silva, R. Ghrist, A. Muhammad
    Abstract We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a “blind” swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic topology. I. COVERAGE PROBLEMS Many of the potential applications of robot swarms require information about coverage in a given domain. For example, using a swarm of robot sensors for surveillance and security applications carries with it the charge to maximize, or, preferably, guarantee coverage. Such applications include networks of security cameras, mine field sweeping via networked robots [18], and oceanographic sampling [4]. In these contexts, each robot has some coverage domain, and one wishes to know about the union of these coverage domains. Such problems are also crucial in applications not involving robots directly, e.g., communication networks. As a preliminary analysis, we consider the static “field” coverage problem, in which robots are assumed stationary and the goal is to verify blanket coverage of a given domain. There is a large literature on this subject; see, e.g., [7], [1], [16]. In addition, there are variants on these problems involving “barrier” coverage to separate regions. Dynamic or “sweeping” coverage [3] is a common and challenging task with applications ranging from security to vacuuming. Although a sensor network composed of robots will have dynamic capabilities, we restrict attention in this brief paper to the static case in order to lay the groundwork for future inquiry. There are two primary approaches to static coverage problems in the literature. The first uses computational geometry tools applied to exact node coordinates. This typically involves ‘ruler-and-compass’ style geometry [10] or Delaunay triangulations of the domain [16], [14], [20]. Such approaches are very rigid with regards to inputs: one must know exact node coordinates and one must know the geometry of the domain precisely to determine the Delaunay complex. To alleviate the former requirement, many authors have turned to probabilistic tools. For example, in [13], the author assumes a randomly and uniformly distributed collection of nodes in a domain with a fixed geometry and proves expected area coverage. Other approaches [15], [19] give percolationtype results about coverage and network integrity for randomly distributed nodes. The drawback of these methods is the need for strong assumptions about the exact shape of the domain, as well as the need for a uniform distribution of nodes. In the sensor networks community, there is a compelling interest (and corresponding burgeoning literature) in determining properties of a network in which the nodes do not possess coordinate data. One example of a coordinate-free approach is in [17], which gives a heuristic method for geographic routing without coordinate data: among the large literature arising from this paper, we note in particular the mathematical analysis of this approach in [11]. To our knowledge, noone has treated the coverage problem in a coordinate-free setting. In this note, we introduce a new set of tools for answering coverage problems in robotics and sensor networks with minimal assumptions about domain geometry and node localization. We provide a sufficiency criterion for coverage. We do not answer the problem of how the nodes should be placed in order to maximize coverage, nor the minimum number of such nodes necessary; neither do we address how to reallocate nodes to fill coverage holes.