Numerical Analysis of Target Enumeration via Euler Characteristic Integrals: 2 Dimensional Disk Supports

In this paper, we look at the types of errors that are introduced when using the Euler characteristic integral approach to count the number of targets in a discrete sensor field. In contrast to other theoretical analyses, this paper examines discrete sensor fields rather than continuous ones. The probabilities of first and second order errors are worked out combinatorially, and a general formula is found which is discovered to be proportional to much higher order errors. Asymptotic results are also derived. Thus, this work gives us the bias of the Euler characteristic integral as a point estimator for the number of targets in a sensor field, and a general understanding of how the Euler characteristic integral fails, or more often succeeds.Comment: 20 pages, 12 figures, 2 table

Target Enumeration via Euler Characteristic Integrals

We solve the problem of counting the total number of observable targets (e.g., persons, vehicles, landmarks) in a region using local counts performed by a network of sensors, each of which measures the number of targets nearby but neither their identities nor any positional information. We formulate and solve several such problems based on the types of sensors and mobility of the targets. The main contribution of this paper is the adaptation of a topological sheaf integration theory — integration with respect to Euler characteristic — to yield complete solutions to these problems

Evasion Paths in Mobile Sensor Networks

Suppose that ball-shaped sensors wander in a bounded domain. A sensor doesn't 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 deSilva 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

Euler-Bessel and Euler-Fourier Transforms

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible \zed-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena

Self-stabilising target counting in wireless sensor networks using Euler integration

Target counting is an established challenge for sensor networks: given a set of sensors that can count (but not identify) targets, how many targets are there? The problem is complicated because of the need to disambiguate duplicate observations of the same target by different sensors. A number of approaches have been proposed in the literature, and in this paper we take an existing technique based on Euler integration and develop a fully-distributed, self-stabilising solution. We derive our algorithm within the field calculus from the centralised presentation of the underlying integration technique, and analyse the precision of the counting through simulation of several network configurations.Postprin

Optimizing source and receiver placement in multistatic sonar networks to monitor fixed targets

17 USC 105 interim-entered record; under review.The article of record as published may be found at https://doi.org/10.1016/j.ejor.2018.02.006Multistatic sonar networks consisting of non-collocated sources and receivers are a promising develop ment in sonar systems, but they present distinct mathematical challenges compared to the monostatic case in which each source is collocated with a receiver. This paper is the first to consider the optimal placement of both sources and receivers to monitor a given set of target locations. Prior publications have only considered optimal placement of one type of sensor, given a fixed placement of the other type. We first develop two integer linear programs capable of optimally placing both sources and receivers within a discrete set of locations. Although these models are capable of placing both sources and receivers to any degree of optimality desired by the user, their computation times may be unacceptably long for some applications. To address this issue, we then develop a two-step heuristic process, Adapt-LOC, that quickly selects positions for both sources and receivers, but with no guarantee of optimality. Based on this, we also create an iterative approach, Iter-LOC, which leads to a locally optimal placement of both sources and receivers, at the cost of larger computation times relative to Adapt-LOC. Finally, we perform compu tational experiments demonstrating that the newly developed algorithms constitute a powerful portfolio of tools, enabling the user to slect an appropriate level of solution quality, given the available time to perform computations. Our experiments include three real-world case studies.Office of Naval Research

Workshop on Smart Sensors - Instrumentation and Measurement: Program

On 18-19 February, the School of Engineering successfully ran a two-day workshop on Smart Sensors - Instrumentation and Measurement. Associate Professor Rainer Künnemeyer organised the event on behalf of the IEEE Instrumentation and Measurement Society, New Zealand Chapter. Over 60 delegates attended and appreciated the 34 presentations which covered a wide range of topics related to sensors, sensor networks and instrumentation. There was substantial interest and support from local industry and crown research institutes

Bounds on Multiple Sensor Fusion

We consider the problem of fusing measurements from multiple sensors, where the sensing regions overlap and data are non-negative---possibly resulting from a count of indistinguishable discrete entities. Because of overlaps, it is, in general, impossible to fuse this information to arrive at an accurate estimate of the overall amount or count of material present in the union of the sensing regions. Here we study the range of overall values consistent with the data. Posed as a linear programming problem, this leads to interesting questions associated with the geometry of the sensor regions, specifically, the arrangement of their non-empty intersections. We define a computational tool called the fusion polytope and derive a condition for this to be in the positive orthant thus simplifying calculations. We show that, in two dimensions, inflated tiling schemes based on rectangular regions fail to satisfy this condition, whereas inflated tiling schemes based on hexagons do.Comment: 23 page

Optimizing source and receiver placement in multistatic sonar

17 USC 105 interim-entered record; under review.Multistatic sonar networks consisting of non-collocated sources and receivers are a promising development in sonar systems, but they present distinct mathematical challenges compared to the monostatic case in which each source is collocated with a receiver. This paper is the first to consider the optimal placement of both sources and receivers to monitor a given set of target locations. Prior publications have only considered optimal placement of one type of sensor, given a fixed placement of the other type. We first develop two integer linear programs capable of optimally placing both sources and receivers within a discrete set of locations. Although these models are capable of placing both sources and receivers to any degree of optimality desired by the user, their computation times may be unacceptably long for some applications. To address this issue, we then develop a two-step heuristic process, Adapt-LOC, that quickly selects positions for both sources and receivers, but with no guarantee of optimality. Based on this, we also create an iterative approach, Iter-LOC, which leads to a locally optimal placement of both sources and receivers, at the cost of larger computation times relative to Adapt-LOC. Finally, we perform computational experiments demonstrating that the newly developed algorithms constitute a powerful portfolio of tools, enabling the user to slect an appropriate level of solution quality, given the available time to perform computations. Our experiments include three real-world case studies.Dr. Craparo is funded by the Office of Naval Research

The value of remote sensing techniques in supporting effective extrapolation across multiple marine spatial scales

The reporting of ecological phenomena and environmental status routinely required point observations, collected with traditional sampling approaches to be extrapolated to larger reporting scales. This process encompasses difficulties that can quickly entrain significant errors. Remote sensing techniques offer insights and exceptional spatial coverage for observing the marine environment. This review provides guidance on (i) the structures and discontinuities inherent within the extrapolative process, (ii) how to extrapolate effectively across multiple spatial scales, and (iii) remote sensing techniques and data sets that can facilitate this process. This evaluation illustrates that remote sensing techniques are a critical component in extrapolation and likely to underpin the production of high-quality assessments of ecological phenomena and the regional reporting of environmental status. Ultimately, is it hoped that this guidance will aid the production of robust and consistent extrapolations that also make full use of the techniques and data sets that expedite this process