This paper describes the coupling of Bayesian learning methods with realistic statistical models for randomly scattered signals. Such a formulation enables efficient learning of signal properties observed at sensors in urban and other complex environments. It also provides a realistic assessment of the uncertainties in the sensed signal characteristics, which is useful for calculating target class probabilities in automated target recognition. In the Bayesian formulation, the physics-based model for the random signal corresponds to the likelihood function, whereas the distribution for the uncertain signal parameters corresponds to the prior. Single and multivariate distributions for randomly scattered signals (as appropriate to single- and multiple-receiver problems, respectively) are reviewed, and it is suggested that the log-normal and gamma distributions are the most useful due to their physical applicability and the availability of Bayesian conjugate priors, which enable efficient refinement of the signal hyperparameters. Realistic simulations for sound propagation are employed to illustrate the Bayesian processing. The processing is found to be robust to mismatches between the simulated signal distributions and the assumed forms of the likelihood functions.
Proc. SPIE. 10200, Signal Processing, Sensor/Information Fusion, and Target Recognition XXVI
KEYWORDS: Signal to noise ratio, Statistical analysis, Scattering, Sensors, Interference (communication), Receivers, Sensor performance, Signal processing, Turbulence, Atmospheric propagation, Signal detection, Probability theory, Environmental sensing
The receiver operating characteristic (ROC curve), which is a plot of the probability of detection as a function of the probability of false alarm, plays a key role in the classical analysis of detector performance. However, meaningful characterization of the ROC curve is challenging when practically important complications such as variations in source emissions, environmental impacts on the signal propagation, uncertainties in the sensor response, and multiple sources of interference are considered. In this paper, a relatively simple but realistic model for scattered signals is employed to explore how parametric uncertainties impact the ROC curve. In particular, we show that parametric uncertainties in the mean signal and noise power substantially raise the tails of the distributions; since receiver operation with a very low probability of false alarm and a high probability of detection is normally desired, these tails lead to severely degraded performance. Because full a priori knowledge of such parametric uncertainties is rarely available in practice, analyses must typically be based on a finite sample of environmental states, which only partially characterize the range of parameter variations. We show how this effect can lead to misleading assessments of system performance. For the cases considered, approximately 64 or more statistically independent samples of the uncertain parameters are needed to accurately predict the probabilities of detection and false alarm. A connection is also described between selection of suitable distributions for the uncertain parameters, and Bayesian adaptive methods for inferring the parameters.
Effective use of passive and active sensors for surveillance, security, and intelligence must consider terrain and
atmospheric effects on the sensor performance. Several years ago, U.S. Army ERDC undertook development of software
for modeling environmental effects on target signatures, signal propagation, and battlefield sensors for many signal
modalities (e.g., optical, acoustic, seismic, magnetic, radio-frequency, chemical, biological, and nuclear). Since its
inception, the software, called Environmental Awareness for Sensor and Emitter Employment (EASEE), has matured
and evolved significantly for simulating a broad spectrum of signal-transmission and sensing scenarios. The underlying
software design involves a flexible, object-oriented approach to the various stages of signal modeling from emission
through processing into inferences. A sensor placement algorithm has also been built in for optimizing sensor selections
and placements based on specification of sensor supply limitations, coverage priorities, and wireless sensor
communication requirements. Some recent and ongoing enhancements are described, including modeling of active
sensing scenarios and signal reflections, directivity of signal emissions and sensors, improved handling of signal feature
dependencies, extensions to realistically model additional signal modalities such as infrared and RF, and XML-based
communication with other calculation and display engines.
Determination of an optimal configuration (numbers, types, and locations) of a sensor network is an important practical
problem. In most applications, complex signal propagation effects and inhomogeneous coverage preferences lead to an
optimal solution that is highly irregular and nonintuitive. The general optimization problem can be strictly formulated as
a binary linear programming problem. Due to the combinatorial nature of this problem, however, its strict solution
requires significant computational resources (NP-complete class of complexity) and is unobtainable for large spatial
grids of candidate sensor locations. For this reason, a greedy algorithm for approximate solution was recently introduced
[S. N. Vecherin, D. K. Wilson, and C. L. Pettit, "Optimal sensor placement with terrain-based constraints and signal
propagation effects," Unattended Ground, Sea, and Air Sensor Technologies and Applications XI, SPIE Proc. Vol. 7333,
paper 73330S (2009)]. Here further extensions to the developed algorithm are presented to include such practical needs
and constraints as sensor availability, coverage by multiple sensors, and wireless communication of the sensor
information. Both communication and detection are considered in a probabilistic framework. Communication signal and
signature propagation effects are taken into account when calculating probabilities of communication and detection.
Comparison of approximate and strict solutions on reduced-size problems suggests that the approximate algorithm yields quick and good solutions, which thus justifies using that algorithm for full-size problems. Examples of three-dimensional outdoor sensor placement are provided using a terrain-based software analysis tool.
This paper presents an algorithm for optimal sensor placement that allows one to find the number, types, and locations of
sensors satisfying inhomogeneous coverage requirements and minimizing a specified cost function. The cost function
can reflect the actual cost of sensors or other disincentives, e.g., the number of sensors, vulnerability, or emplacement
costs of the sensors. The sensors are characterized in terms of a probability of detection, which takes into account
signature propagation effects, such as geometrical spreading and inhomogeneous attenuation. The proposed approach
incorporates many realistic requirements, e.g., existence of high-value objects, obstacles, forbidden emplacement areas,
and perimeter protection. For large spatial grids, the strict optimal solution is, in general, difficult to calculate. A fast
algorithm for finding a suboptimal but nonetheless highly satisfactory solution is developed. The developed algorithm is
compared against a heuristic algorithm that places sensors one-by-one in the most poorly covered spots. Numerical
simulations suggest that the algorithm for a suboptimal solution always outperforms the heuristic algorithm. Software for
optimal sensor placement is presented and discussed.
The combinatorial nature of sensor placement optimization has motivated the use of heuristic algorithms to
avoid the high computational costs of finding global optima by focusing instead on satisfactory local optima.
Transition of an optimization strategy from research to practice should involve a detailed inquiry into the dependence
of its results on the representation of realistic scenarios. A sampling method was used to examine
how the specification of a sensor placement problem for optimization affects the statistical properties of various
ensembles of optimum networks produced by a heuristic algorithm. Features sampled in each ensemble were
the resolution of the grid used for computing network coverage, the range of each sensor, and the dimensions
of obstacles to line-of-sight sensing. The candidate placement grid also was sampled to examine the consequences
of being unable to place sensors at a subset of a regularly spaced grid. The objective function was the
number of sensors required to exceed a probability of detection threshold throughout the coverage area. The
relative importance of variability in each parameter was found to depend on the widths and baseline values
of the assumed variability ranges. Important length scale ratios were identified for ensuring the feasibility and
integrity of the optimization process.
As reliance upon advanced networked sensors increases, expert decision support tools (DSTs) are needed to recommend
appropriate mixes of sensors and placements that will maximize their effectiveness. These tools should predict effects on
sensor performance of the many complexities of the environment, such as terrain conditions, the atmospheric state, and
background noise and clutter. However, the information available for such inputs is often incomplete and imprecise. To
avoid drawing unwarranted conclusions from DSTs, the calculations should reflect a realistic degree of uncertainty in the
inputs. In this paper, a Bayesian probabilistic framework is developed that provides sensor performance predictions
given explicit uncertainties in the weather forecast, terrain state, and tactical scenario. A likelihood function for the
signature propagation model parameters is specified based on the forecast and additional local information that may be
supplied by the user. The framework also includes a likelihood function for the signal/noise features as a function of the
propagation model parameters and tactical scenario. Example calculations illustrate the significant impact of uncertainty
in optimal sensor selection and DST predictions.