Exposure of variable speed targets through a sensor field Thomas Clouqueur, Parameswaran Ramanathan, Kewal K. Saluja University of Wisconsin-Madison 3634 Engineering Hall 1415 Engineering Drive Madison, WI 53706 Phone: 608-265-3804 Fax: 608-262-1267 Email:
[email protected]
Abstract – In order to monitor a region for target intrusion, sensors can be deployed to perform collaborative target detection. To adequately deploy sensors, metrics need to be define to measure the ability of a given deployment to successfully complete this task. Recently, the measure of exposure was defined to evaluate the coverage of the region by a sensor network. Exposure is a versatile metric that is expected to provide a pertinent measure of the region coverage for various scenarios. This paper develops a method to find the minimum exposure of a target traversing the region at variable speed. It also introduces target activities different from region traversal for which the coverage by the sensors remains not well defined. Finally, the problem of coverage in the presence of obstacles in the region is identified. Exposure is expected to provide a good coverage measure for these various problems. Keywords: Collaborative target detection, deployment, exposure, sensor networks, information fusion.
1
Introduction
Sensor networks is a fast emerging technology that is expected to answer the need for interfacing systems with the environment and perform various applications related to the environment [7]. A sensor network is a set of nodes with capabilities to sense the environment, process the information they collect and communicate with each other. Sensor networks are rapidly evolving with the recent advances in computing hardware and software, and in networking. They allow multiple nodes to sense, process, and communicate information as the environment changes. Applications of sensor networks are found in fields as diverse as defense, biomedicine, biology or meteorology [12]. They can be used for surveillance, monitoring of pollution,
traffic, agriculture or civil infrastructures. This paper investigates problems related to the deployment of sensors to detect targets in a region of interest. The basic premise of the detection application is that sensors are deployed over a region and are required to determine if a target carries out some activity in that region. It is assumed that the sensors are provided with an infrastructure hardware and software for collaboration to perform region-wide detection based on their local measurements. The problem is to determine how well a given deployment of sensors is performing its task in the region it monitors. Different deployments can then be compared to determine how many sensors and which sensor locations are most appropriate for detecting a given target activity in a given region. Recently, path exposure was defined as a general metric to evaluate the quality of deployments [2]. The exposure of a path is the probability to detect a target traveling along that path. Exposure can be defined for variety of paths that capture target activity. Paths across the region correspond to target traversing activities, while paths reducing to a single point correspond to target idling activities. The quality of a deployment is measured by the minimum exposure when considering all possible behaviors expected from the target. Exposure is a versatile measure that can account for various target activities. However, only targets of constant speed have been studied in literature so far. This paper demonstrates how to determine the minimum exposure of a target traversing a region at a variable speed. The paper is organized as follows. Section 2 presents the model considered for the sensor network and the target. Section 3 presents work previously conducted in deployment that is pertinent to this study. In particular, the measure of exposure is explained. Section 4 introduces the problem of variable speed target traversing and presents
2.2.1
a solution based on a multi speed approach. Section 5 presents simulation results that demonstrate the solution proposed and show some trends of the least exposed paths. Section 6 presents deployment problems for which the measure of exposure can be used. Different target activities and the presence of obstacles in the sensor field are considered. The paper concludes with section 7.
2
Energy
A target at location u emits a signal which is measured by the sensors deployed at locations si , i = 1; : : : ; n. The strength of the signal emitted by the target decays as a polynomial of the distance. If the decay coefficient is k, the signal energy of a target at location u measured by a sensor at location si is given by
Si (u) = jju ?Ks jjk
Model development
(1)
i
where K is the energy emitted by the target and jju ? si jj is the geometric distance between the target and the sensor. Depending on the environment the value k typically ranges from 2.0 to 5.0 [8]. Energy measurements at a sensor are usually corrupted by noise. If Ni denotes the noise energy at sensor i during a particular measurement, then the total energy measured at sensor i, when the target is at location u, is
The network considered in this paper is assumed to be composed of a set of nodes connected to sensors. In this section, a model is developed for each sensor node and for the collaboration among nodes performing target detection. A model is also developed for the target that is characterized by the signal it emits and the activity it carries out in the region.
2.1
Model of sensor network for target detection
Ei(u) = Si (u) + Ni = jju ?Ks jjk + Ni : i
Sensor nodes with different sensing modalities are deployed over a region R to perform target detection. Sensors measure signals at a given sampling rate to produce time series data that are processed by the nodes. The nodes fuse the information obtained from every sensor according to the sensor type and location to provide an answer to a detection query. The nodes are assumed to have ability to communicate reliably with each other. The study assumes that the sensor nodes obtain an energy measurement every T seconds and this energy accounts for the behavior of the target during time T . Acquiring this energy requires preprocessing of the time series and possibly fusing the data from different sensors at each node. The detection algorithm consists of an exchange and a fusion of the energy values produced by the different nodes to obtain an answer to the detection query. This paper does not deal with finding algorithms for target detection, but it is assumed that the characteristics, i.e. the detection probability and false alarm probability of the scheme are known. Note that, if the sensors exchange their time series instead of energies, a more accurate answer can be obtained. However that requires high communication bandwidth which is often not available in a sensor network.
2.2
2.2.2
(2)
Target activity
Sensors
Target emitting signal Path to traverse
Figure 1. Unauthorized traversing A target can follow different trajectories in the region depending on its objective when intruding the region. In this study, a target is assumed to traverse the region from the west side to the east side. This activity is referred to as unauthorized west-east traversing as shown in Figure 1. Note that the problems of determining the least exposed path for west-east and east-west traversal are identical. Also, any method to solve this problem can be used to find the least exposed path for north-south or south-north traversal. Any trajectory from the west side to the east side of the region can be modeled as a path that is a time series of target positions. For every target position in a path, a signal
Target model
A target is characterized by the energy of the signal it emits. That signal is measured by the sensors after propagating from the target to the sensor. Also, a target is characterized by the activity that it carries out in the monitored region and this activity is modeled by paths. 2
energy can be associated using equation 2 that models the propagation of the signal in the environment. The value of parameter K can vary at each point to represent the energy emitted by the target in different points of the path. Modifying K can account for many different activities of the target. In particular, the value of K is expected to change with the speed of the target since a target moving at high speed is expected to be “louder” than a target not moving or moving at low speed. For some applications, the target emits communication signals that are measurable by the sensors. In such cases, varying K can also account for times where the target is communicating with other foreign devices that could be located outside of the region. As mentioned in the previous subsection, sensor networks considered in this study are assumed to produce detection results every T seconds. In general, several detection attempts are associated to a target following path P . The target is considered to be detected by the sensor network if it is detected during any of the detection attempts associated with the path.
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on the path the target follows. However, it is not obvious as to which path is the least exposed path. Indeed, some paths are short but traverse parts of the region with high sensor density, other paths are longer but traverse parts of the region with smaller sensor density. In [9], the authors relate the exposure of a path P to the distance from P to the sensors and studied two types of paths of interest. The maximal breach path is defined as a path where its closest distance to any of the sensors is as large as possible, which is the least exposed path. And the maximal support path is defined as a path where its farthest distance from the closest sensor is as small as possible, which is the most exposed path. The authors also describe algorithms for efficiently determining the maximal breach and maximal support path for a given sensor field using Voronoi diagrams and Delaunay triangulation. In [10], exposure of a path P is defined as the total energy that the sensors will gather from the target moving through the field following P . The smaller this energy the lesser the likelihood of detecting the target and the coverage provided by a deployment is measured by the minimum exposure when considering all possible paths. The exposure is found by summing energy over time when following a given path with a speed that can vary. An algorithm for determining a path with the least exposure in this sense is developed in [10] with the assumption that the speed of the target is constant. The algorithms presented in [9, 10] assume that all the sensors will collaborate in performing detection. A similar algorithm for finding minimum exposure is developed in [11] when assuming that detection is run locally by only part of all the sensors. In [2], exposure of a path P is defined as the net probability of detecting a target that traverses the field using P . Thus, exposure is a direct measure of the ease of detecting a target traversing the region monitored and the minimum exposure is a direct measure of the quality of a given deployment of sensors. The authors propose a method to determine the least exposed path through a region with a given deployment assuming that the target is traveling at a constant speed. The authors also proposed a strategy for randomly deploying sensors, sequentially, until a desired exposure level is reached. Alternative to the above approach that have been investigated are deployments that “fully cover” the region of interest instead of minimizing the exposure of path through the region. For example, the problem can be stated as to reach a desired average coverage as well as maximizing the coverage of the most vulnerable points of the region. In [6], authors assume that the probability of a sensor detecting a target decreases exponentially with the distance between the target and the sensor, and this probability can also be altered by the possible presence of obstacle between the target and the sensor. Under these assumptions, they propose algo-
Previous work
Recent studies in the area of sensor network target detection have attempted to define and evaluate the performance of a given sensor deployment and present methods for determining sensor placement to effectively cover a region of interest.
Paths B A
Sensors
Figure 2. Three paths from A to B through a sensor field To evaluate the coverage of a region by a set of sensors performing target detection, the notion of exposure was introduced in [2, 9, 10, 11]. The exposure of a given path in the sensor field measures the detectability of a target following the path. An example with three different paths through a sensor field is shown in Figure 2. The detectability of a target traveling from point A to point B will vary depending 3
form expressions for D(u) can be found in [4]. If G(P ) denotes the net probability of not detecting a target as it travels along path P , then,
rithms for placing sensors on grid points as to maximize the average or minimum detection probability of a target that could be located at any grid point. An extension of that study developed similar algorithm assuming that a subset of the sensors can fail [5]. In [1], authors assume that each sensor has a sensing range in which it is guaranteed to detect any target. They develop an algorithm based on integer linear programming that minimizes the cost of sensors when requiring complete coverage of the sensor field. They do not solely consider deployment for target detection but also for target localization. Finally, another study aiming at full region coverage assumes that the sensors have the ability to move [13]. It develops an algorithm for relocating the sensors through sensor mobility after being randomly deployed in the region. Assuming a given target detection probability function of distance between the target and the sensor, the algorithm is based on the concept of virtual forces that cause the sensors to move until they reach a steady state.
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log
X
u2P
log(1
? D(u))du;
Since the exposure of P is (1 ? G(P )), the problem is to find the path which minimizes (1 ? G(P )) or equivalently the path that minimizes j log G(P )j (Note that, G(P ) lies between 0 and 1 and thus log G(P ) is negative). In general, the path P that minimizes j log G(P )j can be of fairly arbitrary form. The solution proposed in [2] divides the region into a fine grid and the target is assumed to move along this grid. The problem then is to find the path P on this grid that minimizes j log G(P )j. For the target not to be detected at any point u(i) 2 P , it must not be detected at any point u when moving between two adjacent grid points of P . Therefore, path P is divided as a chain of grid segments. Let v1 and v2 be two adjacent points on the grid. Let l denote the line segment between v1 and v2. Also, let ml denote the probability of not detecting a target traveling between v1 and v2 on the line segment l. Then, from the discussion above,
Minimum exposure for unauthorized traversing at variable speed
This section presents a formulation and a method to assess the minimum exposure given a sensor deployment when the target is assumed to traverse the region from west to east with variable speed. First, the single speed method from [2] is presented and the problems arising from the variable speed assumption are analyzed. Then, a method to solve the variable speed problem is proposed.
4.1
G(P ) =
log
ml =
X u2l
log(1
? D(u))du;
(3)
The probability ml can be evaluated by finding the detection probability D(u) at each point u 2 l. Again, ml lies between 0 and 1 and, therefore, log ml is negative. To find the least exposed path, a non-negative weight equal to j log ml j is assigned to each segment l on this grid. Thus, finding the least exposed path from the west side to the east side of the region consists in finding the path from the west side to the east side with minimum total weight. This can be done using Dijkstra’s shortest path algorithm for example. The minimum exposure for the sensor deployment is 1 ? 10?w , where w is the total weight of the least exposed path.
Constant speed problem
Let P denote a path in the sensor field. A target that follows P is not detected if and only if it is not detected at any time while it is on that path. Detection attempts by the sensor network occur at a fixed frequency (every T seconds), and each detection attempt i is associated with a point u(i) 2 P . Assuming that the target travels at a constant and known speed, finding the points u(i) is straightforward. The detection attempt i is based on energy measured over a period of time T during which the target is moving from u(i ? 1) to u(i). Therefore, the detection probability associated with each point u(i) reflects the measurements performed during time T . Considering the path, the net probability of not detecting a target traversing the field using P is the product of the probabilities of no detection at each point u(i) 2 P . These “point detection” probabilities depend on the algorithm used for distributed detection. Algorithms presented in [3], namely value or decision fusion, or other algorithms can be used, and the detection probability when a target is at position u is denoted as D(u). Closed
4.2
Variable speed problem
Assume that the target can travel with variable speed v with vmin v vmax . Traveling at a faster speed reduces the amount of time the target spends crossing the region and therefore reduces the number of times the sensors have to detect the target. If the energy emitted by the target is independent of the velocity of the target, it can be argued that the least exposed path is obtained when the target travels at maximum speed. However, the energy emitted by the target is expected to be an increasing function of its speed. Thus, as the energy emitted by the target increases, the probability that the sensors detect the target increases and the exposure 4
detection probability D(u) depends on the energy emitted by the target on the segment, and therefore on the target speed. Note that it is assumed that the target travels in one direction only (east, west, north or south) during a period T . This assumption simplifies the problem by reducing the number of edges in the graph. Such a grid with p1 = 1 and p2 = 2 is illustrated in Figure 3. In this graph, all the grid points are connected to their four distance one neighbors and their four distance two neighbors, provided that these neighbors are within the region boundaries. Each edge in the graph is given a weight that is a function of the probability to detect the target traveling along that edge with speed v0 for edges of size 1 and speed 2:v0 for edges of size 2. Once the graph is constructed, the least exposed path is found using the same approach as in the constant speed problem: the west and east periphery grid points are connected to fictitious points A and B and the least exposed path is found using Dijkstra’s shortest path algorithm between points A and B. The higher the number of speeds, I , considered, the higher the complexity of building the graph and finding the shortest path.
of the target increases. Therefore, there is a tradeoff when increasing the speed of the target and this tradeoff depends on the energy model as a function of the speed and the sensors topology. In general, the speed along the least exposed path is not expected to be constant. In order to find the least exposed path across the region, the variable speed problem is transformed into a multiple speed problem. The target is assumed to be only traveling at discrete speeds vi , i = 1; ; I . Note that this approximation can be made arbitrarily fine by increasing the number of speeds, I , to achieve desired accuracy. For speed vi , the region can be divided into a grid with grid size gi = vi :T so that each segment corresponds to a detection attempt by the sensors. Note that cruder approximations of paths can be found with reduced amount of computations by using grid sizes gi = m:vi :T corresponding to m detection attempts. The least exposed path for a target traveling at constant speed vi can be found using the grid of size gi using the method described in section 4.1. However, these grids can not be used to find the least exposed path for a target traveling at variable speed. This is because the different grids do not line up in general and therefore they do not allow for the target to change speed.
5
Demonstration
This section demonstrates the algorithm developed to find the least exposed path through a region where sensors are deployed and when targets can travel at multiple speeds. A simulator implementing the algorithm described in the previous section was developed to solve the problem for variable number of sensors, speeds and target energies. The simulator assumes that value fusion is used for collaboration among the sensors and the threshold of value fusion can be chosen to obtain a desired false alarm probability for each detection attempt [4]. For the simulation performed, it was assumed that the detection attempts occur every T = 2 seconds and the false alarm probability is chosen so that the expected number of false alarms is one per hour. The energy emitted by the target, i.e. coefficient K in Equation 1, was assumed proportional to the square root of the speed. This assumption used in the experiment can be easily replaced if one knows the characteristics of the target that is expected to cross the region. An example of least exposed path through a region of size 20 20 with 25 sensors deployed is shown in Figure 4. Both the shape and the speed profile of the least exposed path are presented. In that example, the target is assumed to have three different possible speeds, v1 = 1:v0, v2 = 2:v0 and v3 = 4:v0 , and the energy parameters are set to K = 20 (maximum energy) and k = 2 (decay coefficient). The minimum exposure was found to be 36%. The example in Figure 4 exhibits a general trend of least exposed paths that was observed throughout simulations: the least exposed path corresponds to high speeds when crossing parts of the region with low sensor density
Grid point
Edge for speed 2
Edge for speed 1
Figure 3. Grid with edges for two speeds target, p1 = 1 and p2 = 2. The solution consists in constructing a graph with all vertices on a fine grid and the edges connect vertices that are gi; 1 i I apart from each other. Let the speeds considered be vi = pi:v0 where pi are integers and v0 is a unit speed. Note that this is possible in general by making v0 small and v0 defines the granularity of speeds for all practical purposes. Let p0 be the greatest common divider of pi ; 1 i I and let the grid size be g = pi :v0 T , so that all the points of the grid can be reached (except certain points close to the region boundaries). Each grid point is connected to other grid points that are pi away and each edge is given a weight of value j log(1 ? D(u))j where the 5
nodes lep west−east
region monitored
Figure 5. Minimum exposure for variable and constant speed.
deployment region 5
speed factor p
4
nodes lep west−east
3
2
1
0 0
2
4
6
8
10
12
14
segment number
Figure 4. Shape and speed profile of the least exposed path (lep) through a region with 25 sensors and for speeds p1 = 1, p2 = 2 and p3 = 4.
region monitored
deployment region
Figure 6. Least exposed path (lep) for constant speed p = 2.
and low speed when crossing parts of the region with high sensor density. The proximity of sensors to the path make high speed targets (i.e. high energy) easy to detect.
6 To compare the multi-speed results with constant speed results, the minimum exposure was found for the same sensor deployment as in Figure 4 but when assuming the target can only travel at a constant speed v1 = v0 , or v2 = 2:v0 or v3 = 4:v0. The minimum exposures found are respectively 38%, 49% and 64%, as shown in the graph of Figure 5. It is expected that the minimum exposure for constant speed targets is greater than for multiple speeds because constant speed paths are a special case of the multiple speed problem. Another observation is that the least exposed path can be substantially different in shape when considering variable or constant speed. Figure 6 shows the least exposed path for a target traveling at constant speed v2 = 2:v0 which is different from the least exposed path for variable speed shown in Figure 4.
Future problems
The method presented in this paper allows to quantify the region coverage provided by a given sensor deployment for targets traversing the region with variable speed. This coverage is measured using the minimum exposure of the paths across the region. This metric is a powerful and versatile metric with the potential to provide a coverage measure for target activities other than unauthorized traversing. For example, an intruding target may not want to traverse the region but only reach a special point of interest in the region following a certain path and then turn around and leave the region following the same path. Such target activity is referred to as unauthorized reaching and is illustrated in Figure 7. Some methods to find least exposed paths for unauthorized traversing cannot be carried over to the unautho6
rized reaching problem. For example, the method described in [9] for which least exposed paths lie on the Voronoi diagram will not work unless the point to be reached by the target lies on that diagram. However, it is expected that the minimum exposure as defined in this paper will give an appropriate measure of the region coverage for unauthorized reaching activity. Indeed, the minimum exposure can be found by evaluating the detection probability of every path joining the point of interest to the edge of the region. Finding the path with minimum exposure can be done efficiently by considering paths lying on a grid, assigning weight to every edge and searching for the minimum weight path.
increase the ability for the sensors to detect targets. The exposure defined in this paper is expected to be able to measure the region coverage in the presence of obstacles. That requires modifying the energy model to model the obstacle absorption and modifying the edges connecting the grid points in the region to avoid target going across obstacles.
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Conclusion
This paper addresses the problem of sensor deployment in a region to be monitored for target traversing. Assuming a mechanism for sensor collaboration to perform target detection exists, a method to evaluate the quality of a given deployment is proposed and analyzed. The metric used is the minimum exposure when considering all the paths through the region, the best deployment being the one that maximizes the exposure of the least exposed path in the region. The paper proposes a method for computing the minimum exposure when allowing the target to cross the region with a variable speed. The solution consists of assuming that the target can travel at discrete speeds and model the target energy level as a function of the speed of the target. The method was demonstrated through an example and it was observed under the simulation assumptions that the target is less likely to be detected if traveling fast (respectively slow) through parts of the region that have low (respectively high) sensor density. Finally, the paper identifies several future problems for which the minimum exposure is expected to provide a solution. The versatility of exposure needs to be demonstrated for various target activities and problems arising from the presence of obstacles in the region monitored need to be investigated.
Sensors
Point of interest Path to reach point
Unauthorized reaching Figure 7. Unauthorized reaching Another example is when the target idles in the region monitored and carries some activity in which it generates a signal detectable by the sensors. Such target does not have to move into or out of the region maybe because it was present in the region before sensors were deployed. Such target activity is referred to as unauthorized idling. Authors in [6] define the coverage of the region for unauthorized idling using the average or minimum detection probability over a grid. A similar coverage measure can be defined using exposure when reducing paths to single points. The exposure obtained for idling targets can be compared to the exposure of unauthorized traversing or reaching in cases where various target activities are expected in the region. Also, combinations of activities can be considered using exposure, for example if a target idles at some points while traversing the region. Another problem that applies to any target activity is to measure the region coverage in the presence of obstacles in the region [6]. Obstacles impact the propagation of signals emitted by the target and therefore decrease the ability for the sensors to detect targets. On the other hand, obstacles add constraints on the target trajectory, assuming that the target cannot go across an obstacle. These constraints
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