A Statistical Approach for Target Counting in Sensor-Based Surveillance Systems

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1 Proceedings IEEE INFOCOM A Statistical Approac for Target Counting in Sensor-Based Surveillance Systems Dengyuan Wu, Decang Cen,aiXing, Xiuzen Ceng Department of Computer Science, Te George Wasington University, Wasington, DC 5, USA Division of Epidemiology and Biostatistics, Uniformed Services University of te Healt Sciences, MD 84, USA Department of Computer Science, University of Science and Tecnology of Cina, P. R. Cina {andrewwu,ceng}@gwu.edu, dcen@usus.mil, kxing@ustc.edu.cn Abstract Target counting in sensor-based surveillance systems is an interesting task tat potentially could ave many important applications in practice. In suc a system, eac sensor outputs te number of targets in its sensing region, and te problem is ow one can combine all te reported numbers from sensors to provide an estimate of te total number of targets present in te entire monitored area. Te main callenge of te problem is ow to andle different sensors outputs tat contain some counts of te same targets falling into te overlapped area from tese sensors sensing regions. Tis paper introduces a statistical approac to estimate te target count in suc a surveillance system. Our approac avoids direct andling of te overlapping issue by adopting statistical metods. First, depending on weter or not certain prior knowledge is available regarding te target distribution, te procedure in minimizing te residual sum of squares or kernel regression is used to estimate te distribution of targets. Ten te estimated count of te total targets is obtained by te metod of likeliood estimation based on a sequence of binomial distributions tat are derived from a sampling procedure. Comparisons based on simulations sow tat our proposed counting approac outperform te state of art counting algoritms. Extensive simulations also sow tat our proposed approac is very fast and very promising in estimating te target count in sensor-based surveillance systems. I. INTRODUCTION Wireless sensor networks ave been widely used to monitor various activities in many different types of environments [] [6]. One of important monitoring tasks is te estimation of te total number of targets witin a region at a specific time. For example, one may want to estimate te size of a group of sea birds in a certain seasore area at specified times for a given day. A sensor network can be deployed for tis purpose to gater relevant data and make inferences periodically. Altoug it is often difficult to obtain te exact count, target counting algoritms wit a ig counting accuracy are always sougt. In general, tere are two types of errors tat lead to inaccurate counting: miss-detection and double-counting [7]. A target may fail to be detected by a sensor even if te target is in te sensor s sensing region. Tis is often due to imprecision of te ardware. A target in te overlapped area of different sensors sensing regions may be detected by more tan one sensor. Tis can result in double counting, were a target may be counted for at least two times in te final estimate. Double counting tends to occur as sensors are more densely deployed and more targets are present. Usually, te issue of miss-detection can be andled by using ardwarebased detection algoritms or building more sensitive sensors. However, double-counting is normally resolved or mitigated by analyzing information obtained from sensors. In tis paper, we propose a simple but effective statistical approac to estimate te total number of targets in a sensor-based surveillance system, given te count of targets reported by eac sensor witin its sensing region. Te issue of miss-detection is not considered ere. However, te issue of overlapping or double-counting as been avoided wit our metod. In our approac, we first rebuild te target distribution in te monitored area using regression tecniques. Parametric or non-parametric metods are used to estimate te target distribution, depending on weter or not certain prior knowledge is available regarding te distribution. Ten a sampling procedure is used to randomly select subsets of sensors, were sensors from te same subset do not ave overlapped sensing regions. Finally, te total count of targets is estimated troug binomial distributions and te metod of maximum likeliood estimation. Based on our knowledge, te work presented in tis paper is te first tat employs statistical metods to effectively deal wit te double counting problem in wireless counting sensor network. Te first and te most relevant researc work tat deals wit double counting in te context of sensing is [7]. Te counting metod in [7], largely based on te probability teory, mainly works for te scenario were ( targets are uniformly distributed, ( te number of targets present in te monitored area is not large, and (3 two sensing regions from two sensors do not overlap eavily. Compared to te state-of-te-art, our paper as te following novel contributions. Regression tecniques (minimizing te residual sum of squares and kernel regression, sampling, and te metod of likeliood estimation ave been integrated into an approac for target counting. It is te first time tat a completely statistical approac is introduced to estimate target counts in te field of wireless sensor networks. Our approac avoids direct andling of te issue of double counting or overlapping of sensing regions. 3 Our approac works for any number of targets present in te monitored area //$3. IEEE 6

2 4 Te proposed counting approac works for any distribution of targets. Targets can come from a specific distribution, or tey can form clusters. 5 Te proposed counting approac can provide real time estimates of target counts. 6 Te proposed approac outperforms te state-of-te-art counting algoritms in terms of counting accuracies. Te rest of te paper is organized as follows. In Section II, we review te most related researc in target counting. Our network model and oter background information are introduced in Section III. In section IV we discuss ow to estimate te distribution of targets by using parametric and non-parametric statistical metods. Te sampling procedure used to select subsets of sensors and te metod of likeliood estimation used to estimate te target count are provided in Section V. Simulation results are reported in Section VI. And tis paper is concluded wit a discussion on future researc in Section VII. II. RELATED WORS Researc works on target counting are rougly categorized into four types corresponding to minimalistic, complex, binary, and energy-based modeling of wireless sensor network. Te works in [7] [9] are representatives of te researc based on te minimalistic model, were eac sensor outputs te number of distinct targets in its sensing region. Te work in [7] estimates te target count by probability teory. Te autors claim tat under te assumption of uniform targets, te number of targets falling into a specific region follows a Poisson distribution. So tey first partition te monitored area into non-overlapped subareas. Ten te expected value based on tese subareas and te Poisson modeling is used to estimate te number of targets in te monitored area. Tis approac tat aims at resolving te double counting issue as several disadvantages. It is impossible to provide an estimate for a big target count witin a reasonable period of time. Wen sensing regions overlap eavily, not only te counting accuracy degrades but also te amount of time in computation increases dramatically. Paper [8] proposes a target counting algoritm to estimate te range of te number of targets in te monitored area. Te range is based on te worst case and tus migt be too wide to be applicable in real applications. Paper [9] starts from topology integration and uses te expected value as an estimate of te number of targets. Since tere are no experimental or simulation results provided, it is difficult to evaluate te proposed metod. In te setting of complex sensor models, [] also addresses te double counting problems. However, in addition to te number of targets witin te sensing region of a sensor, more information suc as distances or angles is needed in teir algoritms. Binary-sensing based approaces [] [3] estimate te number of targets by assuming tat a sensor reports a value if one or more targets are detected in its sensing region and oterwise. Te success of tese algoritms relies on te assumption of sparse targets. Energy-based approaces [4] [6] estimate te count of targets by using te unit target energy volume and te estimated energy volume of te energy landscape. Tese algoritms assume te same energy level and energy decay model for all individual targets. Oter algoritms in counting targets exist in te literature. For example, [7] relies on probability teory to estimate te number of targets, but te algoritm works only for sparse targets. And in [8], a system is developed to measure te veicle count in a road network. Regression is te process of constructing a curve, or matematical function, wic is closest to a series of data points under a predefined metric. In statistics, regression analysis involves te analysis between a single dependent variable and a few independent variables. In wireless sensor networks, polynomial regression as been used in [4] to recover an approximate energy landscape. In [9], polynomial regression is used to compress te data transported in wireless sensor network communications. In our work, we use te regression tecnique to estimate te target distribution over te twodimensional monitored area. In tis paper we propose a statistical approac for target counting under a minimalistic model. Wile most existing counting algoritms focus on estimating te number of targets in eac overlapped area of sensing regions of sensors, our approac treats te total number of targets in te system as a parameter to be estimated by statistical metods. Wit no restriction on te distribution of targets and te number of monitored targets, our approac provides a fast and effective tool in target count estimation. III. BACGROUND INTRODUCTION We consider te network model were sensors are deployed in a grid or random uniform pattern in a monitored area tat is contained in te two-dimensional Euclidean space. Tese two deployment patterns represent two widely used scenarios in te literature. Suppose tere are N s sensors deployed in te field: s,s,,s Ns. Let x i =(x (i,x(i denote te location of sensor s i (i =,,,N s. Te set of locations of all sensors is represented by L S = {x,x,,x Ns }. Note tat sensor locations can be derived based on existing approaces suc as tose proposed in [], [] Te sensing region of a sensor s i, denoted by A(s i, is modeled as a circular region wit center x i and radius. TenallA(s i ave te same size a = π. Suppose at te time of investigation, sensor s i outputs a reading r i tat equals te number of targets residing in te sensor s sensing region A(s i.letr S = {r,r,,r Ns } denote te set of all suc readings from all sensors. Te entire monitored area A of te surveillance system can be defined as te union of te sensing regions of all sensors. Suppose N t targets are distributed in te monitored area A according to a specific distribution wit te probability density function (pdf f(x θ. Here, x represents a twodimensional vector (x,x and θ is te parameter or set of parameters associated wit te distribution. Examples of te target distribution include a uniform or bivariate normal 7

3 distribution. In tis paper, we assume a general form of te distribution of targets. Due to te use of different statistical approaces, we consider two separate cases: (a we ave some prior knowledge about f(x θ; and (b we do not ave any prior knowledge about f(x θ. Case (a does not necessarily mean tat we know completely about te distribution. For instance, we may know te distribution is bivariate normal but we do not know te exact parameter(s. Case (b is equivalent to te fact tat we know noting about te target distribution. Tis paper focuses on estimating te total number of targets (N t by using te sensors locations L S and sensors readings R S. Tis estimation task is callenging since a target could be caugt and reported by several sensors simultaneously. An intuitive metod to estimate N t migt be tat one simply adds te readings from all sensors and ten multiply te total by a factor γ small tan to eliminate te effect from repeatedly counting. Tis metod would fail frequently since it is usually difficult to find suc a factor γ tat could be influenced by a lot of oter factors suc as te size of te network, te target distribution, and te sensor deployment pattern, etc. Terefore, novel procedures are needed in order to obtain a reliable estimate of N t. In te next two sections, we will discuss ow to estimate te pdf f(x θ of te target s distribution for bot cases (a and (b by regression tecnique and ten discuss ow to estimate N t by te metod of maximum likeliood estimation. IV. TARGET POSITION PROBABILITY DENSITY DISTRIBUTION ESTIMATION In tis section, we will discuss in details ow to estimate te pdf f(x θ of targets distribution by using sensors locations L S and sensors readings R S. Our approac is based on te observation tat te expected count of targets in an unit area centered at x is approximately proportional to te value of f(x θ at x. Below is a derivation leading to tis observation. Suppose targets are independently distributed according to a distribution wit te pdf f(x θ. Consider te sensing region A(s i from sensor s i. It is seen tat a target falls into A(s i wit te probability p i = A(s f(x θ dx i dx. In general, for N t targets, te total number of targets, denoted by R i, tat reside A(s i follows te binomial distribution B(N t,p i. Terefore, te expected count of targets in tis sensing region is te following conditional expectation E(R i x i =N t f(x θ dx dx ( A(s i According to Calculus, wen f(x θ is a continuous function, we ave te following: A(s f(x θ dx i dx f(x i θ A(s dx i dx Set = f(x i θ a Y i = R i a ( We see tat Y i represents te count of targets per unit area around te sensor s i. From ( and (, we obtain te following E(Y i x i N t f(x i θ (3 Te approximation (3 is derived at sensors locations. However, it is easy to see tat (3 actually olds for any location x in te monitored area A, tatis, E(Y x N t f(x θ (4 were Y = R a for R denoting te count of targets in te circular region wit center x and radius. Terefore, te expected count of targets per unit area at x is approximately proportional to te value of f(x θ at x. Toug simple, tis relationsip between Y and f(x θ depicted in (4 is te basis for te following two subsections were we estimate f(x θ by using sensors location and readings. For simplicity, we treat Y as a continuous variable. A. Parametric Approac In tis subsection, we discuss ow to estimate te pdf f(x θ of te distribution of targets wen te form of f(x θ is known but θ is unknown. Tis corresponds to estimating f(x θ for te case (a stated in Section III. We will utilize (4 to obtain te estimate of θ. From te discussion proceeding (4, we know tat te following sequence is available as a given dataset for tis estimation problem: (x,y, (x,y,, (x Ns,y Ns, (5 were x i =(x (i,x(i is te location of sensor s i and y i = ri a is te observed count of targets per unit area around sensor s i. Using te idea in regression, we obtain an estimate ˆθ of θ by minimizing te residue sum of squares, i.e., by solving min θ (y i c f(x i θ, (6 were c is an unknown parameter. Tus, f(x ˆθ, denoted by ˆf(x, is an estimate of f(x θ. In tis paper, we used te Levenberg-Marquardt algoritm (LMA [] in Matlab 9a to find a solution ˆθ to tis minimization problem. LMA is a well known iterative procedure used to solve a wide variety of optimization problems. B. Non-parametric Approac Often in practice, te form of f(x θ is unknown. Tis is te case (b stated in Section III. We know tat (4 gives a simple relationsip between te conditional expectation E(Y x and f(x θ. Terefore, to estimate f(x θ, we only need to find and rescale an estimator of E(Y x. In tis subsection, we will discuss ow to estimate E(Y x by kernel regression, a non-parametric tecnique in estimating conditional expectation of a random variable [3]. Let g(y, x be te joint pdf of Y and X wit X denoting a random location in te monitored area A, g x (x te marginal 8

4 pdf of X, and g(y x te conditional pdf of Y given X = x = (x,x.weave E(Y x = E(Y X = x = yg(y xdy = yg(y,xdy g x(x If we use (5 to estimate densities g x (x and g(y, x by te tecnique of kernel density estimation, ten g x (x and yg(y, xdy can be estimated by and N s N s, (x x i, (x x i y i respectively, were by using a multiplicative kernel wit te same bandwidt (= =,, (x x i = ( x x (i ( x x (i for a univariate kernel (. Terefore, from (7 we obtain te following Nadaraya-Watson estimator of E(Y x : ( ( Ns x x (i x x (i y i ( Ns x x (i ( x x (i Ten te above function divided by its integral over te entire monitored area of te sensor filed will give one estimate ˆf(x of f(x θ. Tus ˆf(x =C ( x x (i Ns Ns ( x x (i ( x x (i ( x x (i y i (7 (8 were C is a constant. Note tat te distribution of sensors in our setting is known, i.e., sensors are uniformly distributed. In tis case, g x (x is a constant. Ten by (7, te following function divided by its integral over te entire monitored area will also give one estimator of f(x θ : Tat is, ˆf(x =C ( x x (i ( x x (i y i (9 ( x x (i ( x x (i y i ( for a constant C. In tis paper, we use te Gaussian kernel (u = π e u, as it is one of te widely used kernels. And we coose te bandwidt to be te radius of te sensing region of a sensor for simplicity. Note tat only L S and R S are needed in computing ˆf(x. V. TARGET COUNT ESTIMATION For sensors wose sensing regions do not overlap, te sum of teir readings is te exact count of targets falling into teir sensing regions. Tis section discusses ow to use tis observation and te estimated distribution of targets to estimate te total number of targets in te monitored area A by te metod of maximum likeliood estimation. A. Selecting Sensors wit Non-Overlapped Sensing Regions We begin wit a brief discussion on selecting sensors wit non-overlapped sensing regions. Two sensing regions A(s i and A(s j from two sensors s i and s j are overlapping if te intersection of A(s i and A(s j is not empty, i.e., A(s i A(s j. Clearly, A(s i and A(s j are overlapping if and only if te distance between s i and s j is less tan or equal to, were is te radius of a sensing region of a sensor. Tis simple rule can be used to randomly select a set of sensors tat do not contain two sensors wose sensing regions are overlapping. Suppose we repeat tis selection process for m times to obtain m suc sets S,S,,S m, were A(s i A(s j = for any two sensors from te same set S k ( k m. We can also add one more confinement in constructing tese sets of sensors by requiring tat eac S k is te largest in te sense tat te sensing region of any sensor outside S k overlap wit at least one sensing region of a sensor inside S k.inoter word, S k is te largest if it is impossible to add anoter sensor into S k wile keeping te non-overlapping status of te subset. Tere are several advantages wen using S k. For example, te integral of f(x θ over te union of sensing regions of all sensors in S k is te sum of te integrals of f(x θ over all individual sensing regions. As anoter example, we see easily tat te sum of te readings of all sensors in S k represents te total number of targets falling into te monitored areas of tese sensors. B. Target Count Estimation By te Metod of Maximum Likeliood Now we discuss ow to estimate te target count in te monitored area A. We will use te estimated pdf ˆf from Section IV, te sets S k constructed in Section V-A, and te metod of maximum likeliood. Te metod of maximum likeliood [4] is one of te most widely tecniques for obtaining estimates of parameters. Wit tis metod, te maximum likeliood estimator (MLE of te parameter(s θ is ˆθ at wic te likeliood function obtains its maximum as a function of θ. Suppose N t targets are independent and identically distributed in te monitored area A according to a distribution wit te pdf f(x θ. LetP i denote te probability tat a target falls into te area monitored by te sensors in te set S i ( i m. Ten using an estimate ˆf(x, we obtain an estimate ˆP i of P i as follows: ˆP i = ˆf(xdx A(s j,s j S i dx = ( ˆf(xdx s j S i A(s j dx 9

5 Let U i denote te number of targets tat fall into te area monitored by te sensors in te set S i. We see tat U i is a discrete random variable tat approximately follows te binomial distribution B(N t, ˆP i,were ˆP i is known and computed in ( but N t is unknown. Below we proceed to find an MLE for N t by using tese binomial variables U,U,,U m From te selection process of S,S,,S m, it follows tat te random variables U,U,,U m can be treated as independent. By te readings of sensors in S i, we obtain an observed value for U i,were = r j ( s j S i Terefore we ave te likeliood function m ( Nt l(n t = ˆP ui i ( ˆP i Nt ui (3 Te MLE of N t is obtained by maximizing te likeliood function. However, te traditional metod involving differentiation is not easy to be carried out due to te factorials. Terefore a different approac is needed to solve tis optimization problem. Since l(n t =for N t <max i { }, it is seen tat te MLE is an integer N t suc tat N t max{ } (4 i and l(n t l(n t, l(n t + < (5 l(n t Below we sow tat N t satisfying (4 and (5 is unique. In fact, it follows from (3 tat l(n t l(n t = m ( N t ˆP ( ˆP i i N t ˆP ( ˆP i i N t ( N t = m ( N t ( N t ( ˆP i = m N t N t ( ˆP i = m ( ˆP i m ui ( + N t Consider te function m g(w = ( ˆP m ( i + u i w (6 defined for w (max i { }, +. Clearly, g(w is continuous and strictly decreases in its domain. In addition, lim g(w = m ( ˆP i <, w + and as w approaces max i { } from te rigt g(w +. Terefore, it follows from (6 tat tere exists only one integer, denoted ˆN t, suc tat (4 and (5 old. Tis ˆN t is te MLE of N t. In practice, it is easy to find ˆN t by (4 and (5. Starting from max i { }, we compute te ratio l(n t m l(n t = N t ( N t u ˆP i (7 i for eac possible integer larger tan or equal to max i { }. (Assuming te ratio becomes + at max i { }. Te last integer wit wic te ratio is larger tan or equal to is ˆN t. Note tat only ( and ( are needed in computing te ratios. VI. PERFORMANCE EVALUATION In tis section, we mainly report our simulation results to demonstrate te strengt of our statistical approac for target counting in sensor-based surveillance systems. Te wireless sensor network under our study contains N s = sensors and N t =, 5,, 5, targets are present in te entire monitored area A. Sensors are deployed over te region of m m in two different patterns: te grid pattern and random uniform pattern. For grid sensor deployment, te center of eac m m grid cell as one sensor. For random uniform sensor deployment, eac m m grid cell contains one sensor wose location is random witin te grid. Te radius of te sensing region of a sensor is set to be 4.m(> m, to ensure tat te region of m m is fully covered by te sensing regions of sensors even if te sensors are randomly and uniformly deployed. Note tat sensing regions of neigboring sensors are eavily overlapped in our setting. For targets, we consider two scenarios: targets form one single cluster were only one distribution of te targets is available, and targets form multiple clusters were eac cluster corresponds to one distribution of te targets witin te cluster. Details about te simulation results are given in te following subsections. Before presenting our simulation results on target counting, we make a note regarding numerical computation of te probability ˆP i in (. Clearly, to obtain ˆP i, we only need to compute te integral ˆf(xdx A(s j dx in (. We do tis by te traditional approximation procedure. Specifically, we divide te region A(s j into m m cells, ten approximate te integral over eac cell. If te center x = (x,x of a cell lies inside A(s j, te integral over te cell is approximated by ˆf(x = ˆf(x. And if te center x is outside A(s j, te integral is simply approximated by. Ten ˆf(xdx A(s j dx is obtained by summing up tese approximated integrals over te cells. A. Single Cluster In tis part, we report te target counting accuracy wen targets form a single cluster over te monitored area. Bot uniform and normal distributions of targets are considered. We assume tat te nature ( uniform or normal of te distribution is known to te investigator aprioribut te exact form of te normal distribution is unknown. Terefore, if te distribution of te targets needs to be estimated, te parametric approac in Section IV-A sould be used. However, for a comparison,

6 Grid Sensors Random Uniform Sensors TABLE I RELATIVE ERRORS FOR COUNTING UNIFORM TARGETS. N t = N t =5 N t = N t = 5 N t = No Estimation -.% -.% -.%.% -.% Non-Parametric Estimation.% -.3%.3%.% -.% No Estimation -.3% -.% -.% -.% -.% Non-Parametric Estimation.4%.3% -.%.% -.% TABLE II RELATIVE ERRORS FOR COUNTING NORMALLY DISTRIBUTED TARGETS. N t = N t =5 N t = N t = 5 N t = Grid Sensors Parametric Estimation.5%.%.%.4%.% Non-Parametric Estimation 3.%.8% 3.% 3.% 3.% Random Uniform Sensors Parametric Estimation.4%.%.8%.%.4% Non-Parametric Estimation.6%.5%.%.7%.5% we also report te results wen te non-parametric approac in Section IV-B is used to estimate te distribution. To assess te accuracy of our approac in target counting, we run te simulation times, and a final value ofa performancemetric is reported. Details are given below. For uniformly distributed targets, no parametric estimation of te probability density function is needed in eac simulation. Specifically, we first generate N t (fixed number targets tat are uniformly distributed over te entire monitored area. Toug parametric estimation of te distribution of te targets is not needed, we record te corresponding pairs as listed in (5 and use Nadaraya-Watson estimator (8 to provide a non-parametric estimation of te distribution simply for a comparison. Ten we randomly select m = largest non-overlapped subsets S,S,,S, as described in Section V-A. Compute ˆP i in ( (wit te true f replacing ˆf if no estimation is required. And compute (te observed number of targets falling into te sensing regions of te sensors in te set S i in (. Finally, we use (7 to find te MLE ˆN t. After repeating te above process for times, we use te following Relative Error (RE based on runs as a performance metric to evaluate te counting accuracy of our approac: (i ˆN t RE = ˆN (i t N t N t (8 wit denoting te MLE of te target count at te it run. RE is te ratio of te difference between te averaged estimated count and te true count to te true count. RE can be positive (meaning overestimating or negative (meaning underestimating. Te simulation results are reported in Table I. For normally distributed targets, we estimate te distribution and te target count in eac simulation. Specifically, we first generate N t targets according to a bivariate normal distribution N (μ p,,wereμ p is fixed at te center of te monitored area and ( σ = ρσ σ ρσ σ σ wit σ and σ randomly cosen from [, ] and ρ randomly cosen from (,. We coose te interval [, ] to assure tat te generated targets are virtually witin te monitored area. Ten te corresponding pairs as listed in (5 are recorded, so tat (6 from te parametric approac can be used to estimate parameters μ p, σ, σ,andρ. For a comparison, Nadaraya- Watson estimator (8 from te non-parametric approac is also used to estimate te pdf of te normal distribution. And ten we randomly select largest non-overlapped subsets S,S,,S, as described in Section V-A. Compute ˆP i in ( and and use (7 to find te MLE ˆN t. After repeating te above process for times, te final values based on te performance metric RE in (8 is reported in Table II. From Tables I and II we observe te following. First, te relative error under grid sensor deployment is similar to tat under te random uniform sensor deployment. Toug it could be anticipated, tis observation indicates tat our approac is robust against spatial noise in sensor positions. Second, for uniformly distributed targets, it is difficult to see if te relative error from no estimation of te distribution is better tan tat from non-parametric estimation. However, for normally distributed targets, te relative error from te parametric estimation of te target distribution is smaller tan tat resulting from te non-parametric estimation. Tis observation sows tat te prior knowledge in te target distribution can elp us better rebuild te distribution and tus better estimate te target count. Te difference between parametric and nonparametric metods is tat in parametric estimation, we only need to estimate te parameter(s of te distribution, wile in non-parametric estimation te 3-dimensional surface of te distribution as to be estimated. B. Multiple Clusters In tis subsection, we evaluate te performance of our counting metod on multiple clusters of targets. Tree cases are considered, were te distribution of targets over te entire monitored area A is muc more complicated tan a single distribution. Case (Figure (a involves piecewise uniformity of targets. Specifically, te monitored area is divided into four subareas wit approximately te same size. Targets witin eac subarea is uniformly distributed, and te ratio of te counts of targets in four subareas (in te order of lower left, upper left, lower rigt, and upper rigt is ::3:4. Clearly targets form four clusters. Case (Figure (b contains two clusters of targets cor- 3

7 (a Case (b Case (c Case 3 Fig.. Clustered Targets over te Monitored Area. TABLE III RELATIVE ERRORS FOR COUNTING TARGETS IN MULTIPLE CLUSTERS. N t = N t =5 N t = N t =5 N t = Case -.% -.5%.%.%.% Grid Sensors Case.5%.8%.8%.4%.% Case 3.%.%.%.3%.4% Case -.%.3% -.3%.% -.% Random Uniform Sensors Case 3.% 3.4% 3.4% 4.% 3.5% Case 3.9%.8%.9%.4%.% responding to one uniform distribution and one normal distribution. Specifically, te monitored area is divided into two alves wit approximately te same size. Over te lower alf targets are uniformly distributed. Over te upper alf targets are distributed according to a normal distribution as described in Section VI-A but wit some exception tat now μ is taken to be te center of te upper alf and σ and σ are cosen from [3, 8]. (Tis interval [3, 8] is cosen so tat all generated targets will virtually fall on te monitored area. We assume te ratio of te counts of targets in two alves is :. Case 3 (Figure (c is similar to case but deals wit normal distributions. Te monitored area is divided into four subareas wit approximately te same size. Targets witin eac subarea as a normal distribution centered at te subarea. All four normal distributions ave te same parameter setting as in case. Te ratio of te counts of targets in four subareas (in te order of lower left, upper left, lower rigt, and upper rigt is ::3:4. We assume no prior knowledge on any distributions of te targets in te above tree cases. Terefore, (8 in te nonparametric procedure of Section IV-B is applied to estimate te target distribution over te entire monitored area. Again largest non-overlapped subsets S,S,,S are randomly selected for te use of te maximum likeliood estimation. And for comparison, bot grid sensor and uniform sensor patterns are considered. Te relative errors based on runs of simulations are provided in Table III. A couple of observations follow immediately from te table. First, te results in te table sow tat te relative error under grid sensor deployment is similar to tat under te random uniform sensor deployment. Second, for any specific distribution pattern of te targets, tere is no muc variation on te relative error as te number of targets canges. C. Comparison wit Oter Algoritms Tis section provides a comparative study were we compare our counting approac wit te PC+ and PC- algoritms proposed in [4]. We use te same basic setting as in [4]: N s = sensors are deployed uniformly and randomly over te m m region. Te radius of te sensing region of a sensor is set to be 7m and m. (Note tat due to use of suc values of te radius, te monitored area may not cover te m m region entirely. Two types of distributions of targets are considered: uniform distribution over te monitored area and normal distribution as used in Section VI-A. And te true number of targets N t is set to be, 5,. We randomly select m = largest non-overlapped subsets S,S,,S and follow te steps in Section VI-A to conduct target counting wit our approac. Te results based on runs of simulations are sown in Figures (a, (b, 3(a, and 3(b. In te figures, NE, PA, and NPA represents our metod wen no estimation, parametric estimation, and non-parametric estimation of te target distribution is made, respectively. From tese figures, te following observations are immediate: All four metods acieve similar results for uniformly distributed targets wit te radius of te sensing region = 7. However, for all te oter tree cases, te performance of PC+ and PC- is worse tan tat of our approac. Te performance of PC+ and PC- on uniformly distributed targets is muc better tan tat on normally 3

8 5 NE NPA PC PC+ 5 NE NPA PC PC+ Relative Error in Percentage Relative Error in Percentage 5 5 True Count of Targets (a =7 5 5 True Count of Targets (b = Fig.. Comparison Results for Uniform Distributed Targets. 5 PA NPA PC PC+ 5 PA NPA PC PC+ Relative Error in Percentage Relative Error in Percentage 5 5 True Count of Targets (a =7 5 5 True Count of Targets (b = Fig. 3. Comparison Results for Normally Distributed Targets. distributed targets. Tis difference is not significant for our counting approac. Many oter simulations ave also been conducted to compare PC+ and PC- wit our approac. We ave found tat te performance of PC+ and PC- is good and comparable to ours if targets are uniformly distributed and sensing regions of sensors sligtly overlap. If te target distribution is not uniform, te performance of PC+ and PC- will degrade in general. If te sensing regions of sensors eavily overlap, te amount of time needed for PC+ and PC- to reac an estimate of target count would become uge and te estimated counts of targets by PC+ and PC- would look unrealistic. In addition, PC+ and PC- work slowly wen N t becomes large. Tese are te reasons tat we do not include large values (e.g., 4 of te radius of sensing regions and large number of targets (e.g., N t =5,, etc. in te above comparison. D. Furter Discussion In addition to te above simulations, we ave also conducted many oter simulations to evaluate te proposed counting approac. We summarize te major findings as follows. Grid sensor deployment is almost equivalent to random uniform sensor deployment in terms of relative errors of te proposed counting approac. Tis finding is expected and as been sown in Sections VI-A and VI-B. In general, if te distribution of targets can be obtained troug estimating te parameter(s, te parametric procedure sould be used instead of using te nonparametric procedure to estimate te entire distribution. An example of tis is provided in Section VI-A. 3 Te proposed approac works for a small number of targets as well as a large number of targets. Te relative error does not seem to cange significantly as te number of targets canges. See Tables I II, and III for a demonstration. 4 For te use of te maximum likeliood estimation, one can randomly select sets of sensors S,S,,S m, eac of wic do not contain two sensors wose sensing regions are overlapping. However, for a fixed m, largest suc sets sould be used, since largest sets contain more information on targets. 5 For non-parametric estimation of te target distribution, bot (8 and ( can be used. However, in our simulations, we ave found tat in general, (8 actually works better. One reason for tis is tat in our setting (8 usually provides a better estimation of te target s distribution. 6 Our proposed target counting approac works best if 33

9 targets are spread out over te entire monitored area. If targets are clustered in a small area, a good way to estimate te target count is applying our approac to an appropriate small area tat is larger tan or equal to te union of te sensing regions eac containing at least one target. 7 Our proposed approac in target counting is fast in practice. For example, in all te experiments in Sections VI-A VI-B and VI-C, it takes 3 seconds for eac run of simulation. VII. CONCLUSION AND FUTURE WOR Tis paper proposes a solid statistical approac for target counting in sensor-based surveillance systems. In tis approac, regression tecniques are firstusedtoestimatete distribution of target positions for two cases. If te parametric form of te distribution is known, te parameter(s is estimated by minimizing te residual sum of squares. If no prior information is available for te distribution, te kernel regression metod is used to estimate te distribution. Using te estimated distribution of te targets, we can estimate te probability tat a target falls into a specific subarea of te entire monitored area in te system. Te estimated count of te targets is ten obtained by te metod of likeliood estimation based on a sequence of binomial distributions derived from te estimated distribution of targets and a sampling procedure. A large number of simulations ave been performed and te results sow tat our approac is effective in estimating te actual number of targets present in te surveillance system. Simulations also sow tat our counting approac is in general superior to te most recent algoritms in te literature in terms of execution time and counting accuracies. Our future researc would lie in te following directions: i examining alternative procedures in estimating te target s distribution; ii analyzing te effect of noise in sensors reading on te estimated count of targets; iii seeking possibly better metrics and metods to measure te performance in target counting; iv investigating te effect of various bandwidts used in kernel regression on te counting accuracies; v examining in details te computational complexity of te proposed approac; and vi studying applications of target count estimation to real life detection problems. ACNOWLEDGMENT Tis work is partially supported by te NSF of te US under grants CNS-966, CNS , CNS-766, and CNS-7669, te NSF of Cina under grant 6767, Jiangsu Natural Science Foundation under grant B358, te National Basic Researc Program of Cina (973 Program under grant CB95, and RFDP under grant 8. REFERENCES [] M. Ding, D. Cen,. Xing, and X. 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