Information-Based Adaptive Sensor Management for Sensor Networks

Transcription

1 2011 American Contro Conference on O'Farre Street, San Francisco, CA, USA June 29 - Juy 01, 2011 Information-Based Adaptive Sensor Management for Sensor Networks Karen L. Jenkins and David A. Castañón Abstract Consider the probem of controing a network of surveiance sensors that are capabe of seecting which areas to observe and which modes to observe these areas. In this paper, we study the probem of controing the observations of these sensors adaptivey in order to cassify accuratey a coection of objects using information on their observed features. Our proposed approach is modeing objects as tempates of 3-D features, and modeing sensors as observing features of individua objects, subject to degradation by noise, obscuration, missed detections and background cutter. We expoit a statistica framework based on random sets simiar to those used in mutitarget tracking to mode the statistica reationship between observed features and object types to compute informationtheoretic estimates of the probabiity of error in cassification. We present a nove approach for computation of these distances between distributions of random sets using k-best assignment agorithms. These estimates are combined with rea time information to generate predictions of the information vaue of individua measurements for sensor management. Using these predictions, we deveop assignment agorithms to compute sensor management strategies to minimize this bound. The resuting sensor management agorithms are capabe of soving probems invoving a arge numbers of objects in rea-time. We show simuations of the resuting agorithms for cassifying 3- dimensiona objects from 2-dimensiona noisy projections that iustrate how the agorithms seect compementary views to overcome obscuration and provide accurate cassification. Our resuts show that our information-based sensor management agorithms achieve comparabe cassification accuracy to adaptive simuation-based approaches that evauate the vaue of information, whie requiring neary five orders of magnitude ess computation. I. INTRODUCTION Inteigent sensors of mutipe modaities are increasingy avaiabe in diverse appications, ranging from buiding security and defense to transportation and medicine. In turn, this has created a need for automated processing and reduction of sensor information, and the opportunity for controing the nature of information coected so that the resuting systems can accompish their task faster. One of the key tasks that these systems perform is is object recognition, such as the identification of exposives inside uggage, determining the type of vehice in surveiance, or identifying the persons in a given ocation. These are compex tasks due to the arge number of variations in ambient iumination as we as object appearance and pose in the sensed information. Robust object recognition is often based on extracting features of objects provide discrimination information, and which can be reiaby observed under the range of sensed conditions. The This work was supported by AFOSR grants FA and by ODDR&E MURI Grants FA and FA D. Castañón is with the Eectrica and Computer Engineering Department at Boston University. K. Jenkins is with Livevo, Inc set of observabe features of an object by a specific sensor depends on factors such as environmenta conditions, reative sensor-object position and orientation, positions of other objects and sensing modaity. Thus, accurate recognition often requires fusion of information from mutipe sensors using different views to obtain a sufficient coection of discriminating features on each object. In this paper, we consider the probem of adaptive management of a team of distributed sensors in order to cassify accuratey a set of spatiay distributed objects by observing errored subsets of their features. These sensors can be oriented dynamicay to coect information on seected objects from different points of view in order to compement best the avaiabe information to achieve rapid cassification. Sensor management has received increased attention in recent years, as summarized in the recent book [1]. Most of this work focuses on finding and tracking objects, and not on identifying them [2] [5]. A common approach in sensor management is to use information theory metrics as the basis for evauating aternative sensing actions [6] [11]. Aternative approaches use expected task performance such as as tracking error or Bayesian cassification costs, eading to stochastic contro formuations and corresponding agorithms [12] [16]. The work in [17] compares taskdriven sensor management schemes with information-driven schemes, and determines that the performance difference between the two casses of approaches was sma. The probem of sensor management for cassification objectives was studied in [18] using information theoretic criteria. Aternative approaches based on stochastic contro techniques and partiay observed Markov decision processes were deveoped in [12], [15], [19]. In these papers, the sensor mode for cassification assumes that sensors observe a conditionay independent estimate of object type, which is a highy unreaistic assumption. In actuaity, processing of information generated by sensors with a simiar point of view wi resut in highy correated tentative cassification decisions because they wi be based on the same set of features. Instead, a more accurate mode woud represent sensor measurements as providing conditionay independent information about object features, and determine how observation of additiona features yieds improved object cassification decisions. In our previous work [20], we proposed a mode for feature-based sensor management representing objecdts as spatiay reated coections of features, characterized by object type and pose. Such modes have been proposed for object recognition by severa authors [21] [23]. In this mode, sensors obtain noisy observations of subsets of object /11/$ AACC 4934

2 features, where the observed subset of features depend on object type, and reative sensor/object pose, and incudes missed features due to obscuration as we as missed detections, and cutter features, and are modeed as random point sets [24] derived from the object modes. The resuts of [20] used as the objective for coected information the reduction in the expected probabiity of cassification error. Rea-time evauation of these metrics requires the use of time-consuming simuations of future measurements. [20] deveoped a nove approach that combines off-ine computation of a priori vaue of measurements using Bhattacharyya distances between modes of random sets of point features, and rea-time estimates of object type and pose generated from past measurements, to generate a prediction of the vaue of information that a sensor coud coect on an object. This vaue was used in assignment agorithms to compute sensor management strategies to maximize the expected information coected. In this paper, we extend the resuts of [20] to incorporate more accurate estimates of the reduction in probabiity of cassification error. In particuar, we deveop estimates based on Chernoff coefficients instead of Bhattacharyya distances, which can ead to the use of more accurate Chernoff bounds on the expected probabiity of error between two random sets. We derive a new set of bounds on the expected reduction in probabiity of error based on the Chernoff coefficients, and integrate them into a rea-time adaptive sensor management agorithm that scaes we to scenarios with arge numbers of objects. We aso describe an efficient way for computation of Chernoff coefficients that avoids a combinatoria enumeration of hypotheses using k-best data association agorithms and importance samping techniques. We evauate the proposed sensor management agorithms using data generated from synthetic 3-D modes of object casses. We aso evauate the agorithms on synthetic aperture radar data generated under DARPA s MSTAR program and simuate sensors as extracting features from 2-D projections of these modes. We compare the performance of our reatime sensor management agorithm with other informationtheory approaches that use measurement simuations. Our rea-time agorithms achieve comparabe cassification accuracy, whie requiring neary three orders of magnitude ess computation. Our resuts estabish the feasibiity of a practica, scaabe and accurate approach for the rea-time management of a team of sensors. An overview of the remainder of this paper is as foows: Section II, presents an overview of the mathematica formuation of feature-based sensor management using random set observations introduced in [20]. Section III describes two approaches for adaptive sensor management based on information-theory criteria, incuding our soution for combining off-ine bounds with on-ine estimates. Section IV discusses simuation resuts that compare the two agorithms in terms of cassification performance and computation time. Section V discusses our resuts and presents directions for future work. Fig. 1: Exampe of a three-dimensiona mode II. PROBLEM FORMULATION Assume that there are M known ocations that contain objects, with centers at ocations z m R 3. Each ocation m contains an object of type c m {1,..., k}, that is oriented with a pose consisting of azimuth ange θm az and eevation (or simiary, incination) ange θm. e We assume that the center ocations z m are known, and that objects are stationary. The unknown state of an object at ocation m is (c m, θ az m, θ e m) and the overa unknown system state is X (x 1,..., x M ) Assume for the purposes of inferencing that the set of possibe azimuth and eevation orientations is finite, and that the states are independent random variabes across ocation, with margina prior distributions specified by p m ( ), m 1,..., M. Associated with each object type k 1,..., K is a coection of features, described by their spatia ocations in a reative coordinate frame centered at the object centroid. Extending our formuation to adding additiona information to features such as type is straightforward (e.g. [25]). Exampes of such features can incude extreme points of modes such as corners, as iustrated in Fig. 1. Denote the reative ocations of features for object type k as M k {M1 k, M2 k,..., Mn k k }, where Mi k R 3. Given an object state at known ocation m, the mode M cm specifies unambiguousy the number of features and their 3-D positions through appication of the appropriate rotation matrices R z (θm az ), R y (θm), e where R z (θ) represent a cockwise rotation through an ange θ about the z axis and and R y (θ) is a cockwise rotation through an ange θ about the y axis. The true feature ocations F m of object m with state are generated by transating and rotating mode features from cass c m as: F m i z m o +R z (θ az m ) R y (θ e m) M cm i, i 1,..., n cm (1) In this manner, three-dimensiona objects can be generated as coections of points, spatiay distributed about the known ocation of the object centroid. Sensors coect measurements on object ocations to estimate the states X. A singe sensor observes ony one 4935

3 ocation at a time and generates a set of projected noisy feature positions for that ocation, that misses obscured features, and incudes random missed feature detections as cutter features. The statistica mode for the observation is a random finite point set [24]. The measurement mode is described constructivey as foows: Denote the random set corresponding to measurement of ocation m by sensor j as mj. For each feature ocation Mi k for object type k, we assume that there is a visibiity map Ii k (θ, δ) which is a function of reative azimuth θ and eevation ange δ to a sensor ocation, which takes vaue 1 whenever this feature is visibe from a reative direction corresponding to (θ, δ), and 0 otherwise because of sef-obscuration. Without oss of generaity, assume that the sensor is in the far fied, so this obscuration depends ony on the reative ange and not on the range to the sensor. Thus, given sensor ocation s j and object state at ocation m, the reative azimuth and orientation anges can be computed, and the subset of visibe features for object type c m are readiy identified. The sensor s ocation s j defines a ine-of-sight vector to the ocation z m, and a projection pane perpendicuar to this vector. The sensor measures the projections of the features which are visibe according to the visibiity map to the measurement pane. If feature i is visibe, then its projected ocation is mj i (F m i s j) (zm sj)(f m i s j) T (z m s j) z m s j 2 These projected ocations are represented as twodimensiona ocations in a sensor-centered coordinate frame. Misdetection of visibe features is modeed by a Bernoui detection process which is independent across features, measurements and sensors, with probabiity of detection p D. Let d i {0, 1} denote the indicator that feature i for object of type c m is both visibe and detected when viewed from sensor j, for i 1,..., n cm. Detected projected ocations as measured imprecisey, with Gaussian additive errors, independent across features and measurements, yieding noisy measurements for Ỹ mj mj i i + v mji where v mji N(0, σmj 2 ) has covariance that depends on the sensor-object ocation geometry and is a 2-dimensiona random error in the projection measurement pane of sensor j. The coection of these detected noisy features for sensor j of ocation m are denoted as mj d, defined as mj d {Ỹ mj i d i 1, i 1,..., n cm } Cutter features in the measurement sets are modeed by a homogeneous spatia Poisson point process with tota intensity λ distributed uniformy over the sensor s fied of view, independent across sensors and measurement times. Denote this random set of points c mj of ocation m by sensor j is defined as mj mj. Then, the observation c mj The resuting observation mj is a random set of projected ocations [24], [26], [27]. d The above mode defines the ikeihood p( mj ). Let I m (t) denote a the random set observations of ocation m coected for a times prior to and incuding time t, and the sensor actions u j (m) s that coected the observations. The sufficient statistic or information state summarizing the observations at ocation m is the conditiona probabiity p( I m (t)). Let I(t) N m1i m (t). Given the independence assumptions, it is straightforward to show that the sufficient statistic for the overa state X at time t is given by M p(x I(t)) Π(t + 1) p( I m(t)) m1 The evoution of the information state is governed by Bayes rue, given the new observation mj (t) at time t, as p( I m (t)) p( mj (t) )p( I m (t 1)) x m p( mj (t) x m)p(x m I m (t 1)) This inference invoves averaging over possibe data associations from the random set ocations to the features of object m under state [26], [27]. With the above setup, adaptive sensor management probem seects sensing actions u j (t) {1,..., M} for each sensor, sensors j 1,..., J identifying the ocation at for each time t that each sensor wi observe, based on knowedge of the information state Π(t). Sensor actions are constrained so that a sensor can ony observe ocations which are within its fied of regard. In this paper, we pursue the approach of [20] to use surrogate performance measures based on information theory as the basis for seection of sensor actions, as discussed in the next section. III. INFORMATION-BASED ADAPTIVE SENSOR MANAGEMENT The adaptive sensor management schemes we consider are based on generating stage-by-stage optima poicies. At time t, the sensor management system must seect a sensing action u i (t) for each sensor i, that are functions of the information state Π(t). We define utiity functions R(j, m) associated with the vaue of measuring ocation m with sensor j at time t, which depend on the current information state Π(t) and the random set observation modes described in Section II. Given these utiity functions, computed for each sensor-ocation pair, the actions at time t wi be seected to maximize the tota coected utiity by soving an inequaity-constrained inear assignment probem of the form max m1 M R j,ma j,m subject to (2) M a j,m 1, j 1,..., N m1 a j,m 1, m 1,..., M a j,m {0, 1}, j 1,... N, m 1,..., M 4936

4 where a j,m 1 if sensor i is assigned to measure ocation m at time t, and 0 otherwise. The soution of this probem transates to decision actions at time t, as u j (t) m if a j,m 1. Note that this formuation restricts the actions of sensors so that no two sensors can observe the same ocation at the same time. This restriction coud be reaxed, but the resuting probem woud require a combinatoria enumeration of the predicted performance of combinations of sensors for each ocation, significanty increasing the computation. Given the presence of mutipe observation times, the oss in performance by this restriction is minima, as sensors that shoud observe a ocation that is aso currenty observed by other sensors can simpy deay their observation to another time. The main issue in our approach is the choice of surrogate utiity function that decomposes additivey over sensor actions, as required by the optimization approach. One aternative is the discrimination gain, or equivaenty, the expected reduction in sampe conditiona entropy of the underying conditiona probabiity distributions of the object states at each time [4], [10], [18]. The sampe conditiona entropy of the sufficient statistic Π(t) as H[Π(t))] M p( I m(t 1)) og p( I m(t 1) m1 M H m[p( I m(t))] (3) m1 Note that we are not averaging over I(t) as woud be required for the standard definition of conditiona entropy, since H[Π(t)] is adapted to the sufficient statistic Π(t). As in [18], the expected discrimination gain at time t of a coection of sensing actions U(t) is the sum of the expected sampe conditiona entropy reduction for each action. For action u j (t) m, the discrimination gain from ocation m is D j,m(p( I m(t 1))) H m[p( I m(t 1))] [ E mj (t) Hm[p( I m(t 1), u j(t) m, mj (t))] ] (4) For sensor management, we define the stage t utiity functions R(j, m) D j,m (p( I m (t 1))). Note that the overa cost (3) is additive in the individua ocation discrimination gains. Direct computation of the expectation in (4) using our random set mode is difficut, so the functions R(j, m) are typicay estimated through simuation of the random sets mj (t) for each ocation m and sensor j. A different metric for sensor management is reduction in the expected probabiity of object cassification error based on maximum a posteriori cassification for each ocation. For ocation m, given the information state Π(t) at state t, the probabiity of making a cassification error using the maximum a posteriori cassifier before coecting additiona measurements is R m (t) 1 max p( I m (t 1)) Simiary, after sensor j coects a measurement mj (t), the posterior probabiity of cassification error is R m (t, mj (t)) 1 max p( I m (t 1), mj (t)) The expected reduction in probabiity of error from measuring ocation m with sensor j is R j,m E mj [max p( I m (t 1), mj (t))] max p( I m (t 1)) (5) By summing over the set of possibe sensing actions, the utiity functions R j,m in (5) maximize the expected reduction in the number of cassification errors over the set of ocations. Again, direct computation of this objective requires averaging over the random sets mj (t) for each sensor-ocation pair using Monte Caro techniques. The approach proposed in [20] is to use bounds on the probabiity of error to estimate the expected reduction above. In particuar, these bounds do not require on-ine simuation of random sets, and use quantities based on informationtheoretic distances that can be computed off-ine. We extend the resuts of [20] to use bounds on the probabiities of error in confusing potentia states, x m in terms of Chernoff coefficients [28], as described next. Assume we have two distributions for the same random variabe, given by p( x) and p( x ). The Chernoff coefficient for parameter 0 < α < 1 between these distributions is defined as ρ α x,x p( x) α p( x ) 1 α d (6) The specia symmetric case for α 1/2 is the Bhattacharyya coefficient, which was used in [20], and is denoted without superscript as ρ x,x p( x)p( x )d (7) The Chernoff coefficients in (6) depend on the distribution of the random set measurements conditioned on the discrete states, x m, and not on the rea-time information I m (t). Thus, these coefficients can be computed off-ine, which we wi expoit for fast rea-time sensor management. The Chernoff coefficient provides a bound on the probabiity of cassification error in making a decision between a pair of hypotheses and x m. If the prior probabiities on the casses are p( ), p(x m), and the maximum a posteriori (MAP) rue is used to seect one of the two vaues or x m, the average probabiity of error can be bounded by P err(, x p( xm)p (xm) m) E [min(, p( x m)p (x m) )] P ( ) P ( ) min(p()p( ), p(x m)p( x m))d p() α p(x m) 1 α ρ α,x m Note that the above bound combines the avaiabe information given by the prior probabiities, together with the off-ine computation of the Chernoff coefficient. We can compute this (8) 4937

5 bound for mutipe vaues of the Chernoff coefficients α, and choose the minimum as the best bound on the probabiity of error. The above bound on the average probabiity of error in seecting between two hypotheses can be extended to compute a bound for the probabiity of error when there are many discrete states {Xm, 1..., Xm N } assuming that the estimate is seected by the MAP rue. Theorem 1: Assume there are N possibe hypothesis for ocation m, with prior probabiities p( ), and that we observe a measurement with known conditiona probabiities P ( ), 1,..., N. Then, for any α (0, 1), the expected probabiity of error for the maximum ikeihood decision rue, P err is bounded by P err p( j) α p( j ) 1 α ρ α j,j The proof appears in the appendix. Theorem 1 gives a bound on the probabiity of cassification error for a muti-hypothesis probem, in terms of the Chernoff coefficients between pairs of measurement ikeihood modes. For sensor management, we want to use this bound to approximate the expected reduction in probabiity of error in (5). The expected reduction R(j, m) is R(j, m) R m (t) R m (t, mj (t)) The expected probabiity of error after coecting a measurement from sensor j to ocation m, denoted by R m (t, mj (t)) is bounded above from Theorem 1, by R m (t, mj ) p( k I m (t 1)) α k1 k k+1 p( k I m (t 1)) 1 α ρ α k,k We use this bound to define the utiity function R(j, m) R m (t) p( k I m (t 1)) α k1 k k+1 p( k I m (t 1)) 1 α ρ α k,k (9) which is an estimate of the reduction in expected probabiity of error in estimating the state given a new random set measurement mj (t) by sensor j. Note in particuar that this utiity is easiy computed given the information states Π(t) and the precomputed Chernoff coefficients. A. Computation of Chernoff Coefficients Computing Chernoff coefficients requires integration over random sets, a time-consuming task to perform exacty. Instead of (6), we propose to compute the Chernoff coefficient between two possibe states, x m using importance samping, as: ρ α x [p( mj )] αp( mj m,x m p( mj x x m)d mj (10) m) We use sampes Ỹ mj, 1,..., L generated from the distribution p( mj x m), aong with the ikeihood modes of Section II to evauate ρ xm,x m to yied the resut: Lemma 1: The estimator ˆρ α,x m using Monte Caro techniques L [p(ỹ mj ) p(ỹ mj x m) 1 is an unbiased estimator of ρ α,x m The proof is straightforward from the properties of independent sampes. The computation of Chernoff coefficients from Monte Caro sampes Ỹ mj invoves the computation of ikeihoods p( mj ). Note that mj is a random set, therefore computation of this ikeihood requires summation over a possibe data associations of points in mj with the predicted visibe features for sensor j from the modes at ocation m with state [24]. From our mode in Section II, these ikeihoods become: p( mj ) M ] α p( mj, M)p(M ) (11) where M is a data association hypothesis that expains uniquey the correspondence of points in mj to cutter points and visibe features for sensor j for an object in state at ocation m. The detais of this computation can be found in [23], and other simiar references. The number of terms in the summation in (11) grows exponentiay with the number of features. Furthermore, most of the terms p( mj, M) tend to be vanishingy sma, so the sum is dominated by few terms. A common approach in mutiobject tracking is to compute ony the argest term in the sum, using a generaized ikeihood method, for which efficient soutions are avaiabe [23]. The idea is to approximate M max argmax M p( mj, M)p(M ) p( mj ) p( mj, M max ) However, this generaized ikeihood method can be very inaccurate. Instead, we use a K-best approximation, where we compute the probabiities of the K most-ikey matchings from the simuated measurements to the mode for state to approximate the sum in (11). The best K matches can be found efficienty in time that scaes poynomiay with the number of points in the random sets using variations of assignment agorithms [29]. IV. EXPERIMENTS The experiments in this section consist of scenarios with 7 sensors and 20 objects corresponding to two different casses of cars, shown in Figure 2. Sensors correspond to cameras that image objects that are within a sensing radius, but are constrained to be capabe of viewing just one object at a time. In our simuations, we start with the a priori information that each car type is equay ikey. Each object is presented at an unknown azimuth ange which is uniformy distributed between 0 and 2π, with respect to the system reference axis. We mode a non-fat terrain by a distribution on object 4938

6 eevation ange, which is uniformy distributed on π/6 to +π/6. Object tempates are three-dimensiona modes of cars, with features defined as the ocations of corner points. Objects are opaque, so features can be obscured based on observation orientation. Visibe features are projected noisiy onto the sensor viewing pane. Measurements of object features are conditionay independent and Gaussian distributed with a measurement noise variance of σ m 0.5. On average, each sensor has 4 objects within its sensing radius. Features are detected with a probabiity p D 0.9 and fase aarms are generated at a rate of λ 1 uniformy on the measured image Entropy Reduction Bhattacharyya Distance Chernoff Distance Fig. 3: Comparison between Discrimination Gain and our agorithms using Bhattacharyya and Chernoff coefficients Fig. 2: Two object modes at the same pose Figure 3 shows the performance of our rea-time approach to sensor management, in terms of average probabiity of error versus number of decision times. The resuts compare an on-ine discrimination gain maximization agorithm with our rea-time agorithms using two different approaches at computing off-ine bounds: the Bhattacharyya indices proposed in [20], and the minimum bound obtained by seecting the optima index for Chernoff coefficient α in the bound of (6). The performance curves represent Monte Caro averages in cassification error over 100 simuations, where each simuation varied the ocation and orientation of the objects present as we as the measurement errors. Note that the performances of the agorithm using Bhattacharyya indices and Chernoff indices are neary identica, which suggests that the choice of best sensor/object geometry is not very sensitive to the quaity of the bound used. The main point to note is that the performance achieved by our onine/off-ine approach is comparabe to that achieved by the discrimination gain agorithm, but reducing computation in our MATLAB impementations by three orders of magnitude. V. CONCLUSIONS In this paper, we presented agorithms for computing adaptive sensor management strategies for a group of sensors seeking to cassify a set of spatiay distributed objects. Our approach used a nove mode based on random set observations in terms of coections of extracted features from measured imagery, which are ange-dependent. We presented a nove approach that is scaabe and suitabe for arge numbers of objects based on off-ine computation of Chernoff coefficients between measurement ikeihoods under pairs of hypotheses, together with rea-time estimates of conditiona probabiities of the state hypotheses given past measurements. We aso proposed a new agorithm for computation of the off-ine Chernoff coefficients using a k-best assignment agorithm. Our simuations show that our offine/on-ine agorithms achieve performance comparabe to that of agorithms that use rea-time Monte Caro predictions of performance, whie reducing computation by three orders of magnitude. There are many directions in which this work can be extended. The resuts can be readiy extended to moving objects and moving sensors where sensor-object geometry woud be changing dynamicay. In addition, our simpe feature mode can be extended to add categorica feature types in addition to feature ocations. Another extension woud aow mode uncertainty, so that the tempates coud have variations over where features are ocated. These directions remain as subjects for future investigation. APPENDIX Proof of Theorem 1: Given hypotheses x {1,..., N} with prior probabiities p(x), and observations with conditiona distributions p( x), the maximum aposteriori estimate { } p( x)p(x) ˆx max x {1,...,N} p( ) and the probabiity of error in making this decision is: P err p(j)p(ˆx j j) Define A j to denote the event that the cassification decision ˆ j. This corresponds to a coection of observations for which p(x j ) p(x j ) for a j j. Then, P err p(j)p( A j x j) j j We can assume the events A j are disjoint by using a tiebreaking rue for the maximum a posteriori estimate. Thus, p( A j x j) j j j 1,j j p(a j x j) 4939

7 so we can write P err K p(j) p(j) j 1,j j j 1,j j p(a j x j) p(ˆx j x j) from the definition of A j. We can rearrange symmetric terms in the summation to obtain: P err [p(ˆx j x j)p(x j)+ p(ˆx j x j )p(x j )] We define j to denote those vaues of observation for which event A j is true, that is, the cassification decision is as ˆx j. Then, P err j p(j)p( x j)d + p(xj )p( x j )d j min { p(j)p( j), p(j )p( j ) } d j j min { p(j)p( j), p(j )p( j ) } d p(j) α p( j) α p(j ) 1 α p( j ) 1 α d where the first equaity foows because p(j)p( j) p(j )p( j ) under j and the ast inequaity hods for 0 < α < 1. This gives us the resut: P err p(j) α p(j ) 1 α p(j) α p(j ) 1 α ρ α j,j REFERENCES p( j) α p( j ) 1 α d [1] A. Hero, D. A. Castañón, D. Cochran, and K. Kastea, Foundations and Appications of Sensor Management. Springer, [2]. Li, L. W. Krakow, E. K. P. Chong, and K. N. 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