2,688 research outputs found
Positive Alexander Duality for Pursuit and Evasion
Considered is a class of pursuit-evasion games, in which an evader tries to
avoid detection. Such games can be formulated as the search for sections to the
complement of a coverage region in a Euclidean space over a timeline. Prior
results give homological criteria for evasion in the general case that are not
necessary and sufficient. This paper provides a necessary and sufficient
positive cohomological criterion for evasion in a general case. The principal
tools are (1) a refinement of the Cech cohomology of a coverage region with a
positive cone encoding spatial orientation, (2) a refinement of the Borel-Moore
homology of the coverage gaps with a positive cone encoding time orientation,
and (3) a positive variant of Alexander Duality. Positive cohomology decomposes
as the global sections of a sheaf of local positive cohomology over the time
axis; we show how this decomposition makes positive cohomology computable as a
linear program.Comment: 19 pages, 6 figures; improvements made throughout: e.g. positive
(co)homology generalized to arbitrary degrees; Positive Alexander Duality
generalized from homological degrees 0,1; Morse and smoothness conditions
generalized; illustrations of positive homology added. minor corrections in
proofs, notation, organization, and language made throughout. variant of
Borel-Moore homology now use
Cellular Sheaves And Cosheaves For Distributed Topological Data Analysis
This dissertation proposes cellular sheaf theory as a method for decomposing data analysis problems. We present novel approaches to problems in pursuit and evasion games and topological data analysis, where cellular sheaves and cosheaves are used to extract global information from data distributed with respect to time, boolean constraints, spatial location, and density. The main contribution of this dissertation lies in the enrichment of a fundamental tool in topological data analysis, called persistent homology, through cellular sheaf theory. We present a distributed computation mechanism of persistent homology using cellular cosheaves. Our construction is an extension of the generalized Mayer-Vietoris principle to filtered spaces obtained via a sequence of spectral sequences. We discuss a general framework in which the distribution scheme can be adapted according to a user-specific property of interest. The resulting persistent homology reflects properties of the topological features, allowing the user to perform refined data analysis. Finally, we apply our construction to perform a multi-scale analysis to detect features of varying sizes that are overlooked by standard persistent homology
Topological tracking of connected components in image sequences
Persistent homology provides information about the lifetime of homology
classes along a filtration of cell complexes. Persistence barcode is a graphi-
cal representation of such information. A filtration might be determined by
time in a set of spatiotemporal data, but classical methods for computing
persistent homology do not respect the fact that we can not move back-
wards in time. In this paper, taking as input a time-varying sequence of
two-dimensional (2D) binary digital images, we develop an algorithm for en-
coding, in the so-called spatiotemporal barcode, lifetime of connected compo-
nents (of either the foreground or background) that are moving in the image
sequence over time (this information may not coincide with the one provided
by the persistence barcode). This way, given a connected component at a
specific time in the sequence, we can track the component backwards in time
until the moment it was born, by what we call a spatiotemporal path. The
main contribution of this paper with respect to our previous works lies in a
new algorithm that computes spatiotemporal paths directly, valid for both
foreground and background and developed in a general context, setting the
ground for a future extension for tracking higher dimensional topological
features in nD binary digital image sequences.Ministerio de EconomĂa y Competitividad MTM2015-67072-
Optimal steering for kinematic vehicles with applications to spatially distributed agents
The recent technological advances in the field of autonomous vehicles have resulted in a growing impetus for researchers to improve the current framework of mission planning and execution within both the military and civilian contexts. Many recent efforts towards this direction emphasize the importance of replacing the so-called monolithic paradigm, where a mission is planned, monitored, and controlled by a unique global decision maker, with a network centric paradigm, where the same mission related tasks are performed by networks of interacting decision makers (autonomous vehicles). The interest in applications involving teams of autonomous vehicles is expected to significantly grow in the near future as new paradigms for their use are constantly being proposed for a diverse spectrum of real world applications.
One promising approach to extend available techniques for addressing problems involving a single autonomous vehicle to those involving teams of autonomous vehicles is to use the concept of Voronoi diagram as a means for reducing the complexity of the multi-vehicle problem. In particular, the Voronoi diagram provides a spatial partition of the environment the team of vehicles operate in, where each element of this partition is associated with a unique vehicle from the team. The partition induces, in turn, a graph abstraction of the operating space that is in a one-to-one correspondence with the network abstraction of the team of autonomous vehicles; a fact that can provide both conceptual and analytical advantages during mission planning and execution. In this dissertation, we propose the use of a new class of Voronoi-like partitioning schemes with respect to state-dependent proximity (pseudo-) metrics rather than the Euclidean distance or other generalized distance functions, which are typically used in the literature. An important nuance here is that, in contrast to the Euclidean distance, state-dependent metrics can succinctly capture system theoretic features of each vehicle from the team (e.g., vehicle kinematics), as well as the environment-vehicle interactions, which are induced, for example, by local winds/currents. We subsequently illustrate how the proposed concept of state-dependent Voronoi-like partition can induce local control schemes for problems involving networks of spatially distributed autonomous vehicles by examining different application scenarios.PhDCommittee Chair: Tsiotras Panagiotis; Committee Member: Egerstedt Magnus; Committee Member: Feron Eric; Committee Member: Haddad Wassim; Committee Member: Shamma Jef
Sensor-Based Topological Coverage And Mapping Algorithms For Resource-Constrained Robot Swarms
Coverage is widely known in the field of sensor networks as the task of deploying sensors to completely cover an environment with the union of the sensor footprints. Related to coverage is the task of exploration that includes guiding mobile robots, equipped with sensors, to map an unknown environment (mapping) or clear a known environment (searching and pursuit- evasion problem) with their sensors. This is an essential task for robot swarms in many robotic applications including environmental monitoring, sensor deployment, mine clearing, search-and-rescue, and intrusion detection. Utilizing a large team of robots not only improves the completion time of such tasks, but also improve the scalability of the applications while increasing the robustness to systems’ failure.
Despite extensive research on coverage, mapping, and exploration problems, many challenges remain to be solved, especially in swarms where robots have limited computational and sensing capabilities. The majority of approaches used to solve the coverage problem rely on metric information, such as the pose of the robots and the position of obstacles. These geometric approaches are not suitable for large scale swarms due to high computational complexity and sensitivity to noise. This dissertation focuses on algorithms that, using tools from algebraic topology and bearing-based control, solve the coverage related problem with a swarm of resource-constrained robots.
First, this dissertation presents an algorithm for deploying mobile robots to attain a hole-less sensor coverage of an unknown environment, where each robot is only capable of measuring the bearing angles to the other robots within its sensing region and the obstacles that it touches. Next, using the same sensing model, a topological map of an environment can be obtained using graph-based search techniques even when there is an insufficient number of robots to attain full coverage of the environment. We then introduce the landmark complex representation and present an exploration algorithm that not only is complete when the landmarks are sufficiently dense but also scales well with any swarm size. Finally, we derive a multi-pursuers and multi-evaders planning algorithm, which detects all possible evaders and clears complex environments
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