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modeled by a linear Gaussian central Kalman Markovian Kalman filtering transition filter. is applicable, density i.e. in case of wellseparated targets, assuming perfect detection, k|x k−1) = N xscheme and in absence k; F directly calculated: k|k−1 x k−1, Dto an arbitrary local posterior density is introduced in the following way. By p(x apply the proposed exploiting k|k−1 For the number sake x processing given that l|k = P Kalman l|k p(x apply the proposed scheme directly to an arbitrary number x l|k = P l|k P s −1 l|k x s k|x k−1) = N in [2], the result is equivalent to a central l|k fil In 2008 and 2009, first T2TF the product formula for Gaussians, we replace all of of simplicity, sensors. schemeInfor we 2010, arbitrary here the assume generalized conditions solution where was . measurement x k; F derived standard (3) k|k−1 x k−1, D k|k−1 . For the sake processing given that Kalman filter assumptions hold s=1 of false measurements. For notational simplicity let us assume all sensor covariances are known. instants of time, which is equivalent local covariances by a one: Kalman presented to a filtering in Kalman [2]. filter is applicable, i.e. in the P case −1 l|k S synchronized sensors produce measurements the same = S and of sensors. In 2010, the generalized solution was derived and s=1 of simplicity, we here assume conditions where standard all sensor covariances are known. To this end, a globalized of wellseparated Notational center, posterior combinations P s −1 l|k local Convex posterior combinations density is of (2) introduce this type Kalman presentedfiltering in [2]. is applicable, processing i.e. inallthe measurements case of wellseparated Notational targets, Preliminaries: assuming perfect Let in a fusion local Convex was density targets, Preliminaries: presented is ofintroduced this type are assuming instants of time t l, l =1,...,k denoted by Z l = {z s perfect Letin detection, all the fundamental time-varying following in s=1 way. almost and in absence S target By exploiting all data fusion l }S s=1. p(x k|Z k ) ∝ N x k; x s k|k The proposed methodology can be directly extended to asynchronous Military sensors. Communications The accumulation of and the sensor Information data Z l up Technology... same all P , the product applications formula (seefor e by Koch detection, all time-varying in [14] and and in[15]. absence target However, exploiting all of properties data was fusion the false measurements. of notinterest product possible applications For at formula to (see anotational given for e.g. time Gaussians, [16, Chapter simplicity t l be collected we replace 12]). let us assume by all S a local Note covariances Ps that k|k this type by aofglobalized T2TF (4) requ of properties false measurements. of interest For at anotational given time o apply the simplicity t proposed l be collected letscheme us assume by a directly local Sstate to Note covariances synchronized vector an that arbitrary this type x l, sensors whose by number aofglobalized T2TF requires produce posterior one: a decentralized measurements density conditioned x l|k = Pdecorre- lation, at the s=1 l|k on 288 lation, s −1 l|k x s l|k because . all sensors (3) observe Sstate synchronized vector x l, sensors whose produce posterior of sensors. measurements density conditioned In 2010, the the on generalized same all instants data solution because upoftotime the t to and including the time t k, typically the present l, current l =1,...,k time t k denoted is givenbyby Z time, is l the = {z Gaussian s = N x 1 S k; ˜x s a time series recursively defined by Z k = {Z k, Z k−1 k|k }. The S , 1 S l }S s=1. The N was derived all sensors and observe the same target. Therefore, s=1 the local tracks p(x k|Z S ˜P k x s l|k are not opti instants data up to the current time t presented k is given by the Gaussian [2]. ) ∝ k|k , N(5) x k; x N of time t l, l =1,...,k denoted by Z the local tracks x s l = {z s S proposed x l; x l|k , Pmethodology l|k with l|k are not optimal in a local sense, if (1) l x expectation can Convex be directly vector combinations xl|k extended and covariance of to this asynchronous matrix all time-varying Psensors. l|k . The The mechanical target s=1 accumulation alldynamics dataoffusion the of sensor applications the data system Z (see type are holds. fundamental However, inif almost l; x l|k , P l|k with expectation vector xl|k and covariance }S s=1. p(x holds. However, k|Z k ) ∝ N x if all of them k; x are s k|k all of them The proposed methodology can beNotational directly extended Preliminaries: to asynchronous matrix Psensors. l|k . The The mechanical accumulation dynamics , fused Ps k|k (4) according to (2) Let s=1 s=1 time series produced by the measurements of an individual l up are e.g. and [16, (3), Chaptera globally 12]). optimal estim properties of the of of interest sensor the data system aZ l given up are and (3), a globally optimal estimate is obtained. As shown to time modeled and t l including bebycollected a linear S filtering is appropriate sensor s ∈ {1,...,S} for tracking, only denoted the by covariance Zs k the time Gaussian t . The statistical matrices P lk can = be Ncalculated x k; ˜x s properties of an individual sensor measurement z locally for all sensors without exchanging sensor s k|k , ˜P k, typically Markovian the transition present time, density is k|k , = N (6) x 1 S ap(x k; time k|xseries k−1) = recursively N by a Note that this type of T2TF requires in a[2], decentralized the resultdecorre- lation, xby k−1, Zbecause l is described k D= k|k−1 {Zall k, . sensors Z For k−1 the }. observe The sake the processing same target. givenTherefore, k|x k−1) = N time t k, typically the present time, is in [2], the result is equivalent tomodeled and including by a linear the Gaussian Markovian transition density state vector x l, whose posterior density conditioned xon k; all p(x Fdefined k|k−1 = N equivalent x 1 S to a central measurement x k; F k|k−1 x that Kalman S fi s=1 s= data k−1, D up to k|k−1 . For the sake k; ˜x s a time series recursively defined the current time t k time is of given simplicity, by a probability density function p(z data, provided the measurement s series by produced theweGaussian here by the assume measurements conditionsofwhere an individual standard all sensor covariances are l |xl), also called sensor S sensor s ∈ {1,...,S} only is denoted where the by Zglobalized local parameters ˜x error covariance likelihood matrices function, of which each needsindividual to be known up tosensor a constantare known, k or if they can be s known N by Z k = {Z k, Z k−1 k|k }. The processing given that Kalman S filter assumptions , 1 S ˜P k|k , (5) the local tracks x s hold and s=1 l|k are not optimal in a local sense, if (1) time of simplicity, series produced we here by the assume measurements conditions x l; x l|k , P ofwhere an l|k with individual standard all sensor covariances are known. To this end, a globalized expectation Kalman vector xl|k filtering and covariance is applicable, S holds. i.e. However, in the k|k and covariance s . The case if all statistical of of wellseparated of them are local fused posterior according density = to is (2) Nintroduc sensor Kalman s ∈ filtering {1,...,S} is applicable, only is x k; ˜x matrix denoted i.e. P by in l|k . Zthe case of wellseparatedof targets, an individual assuming sensor perfect local posterior density is introduced in the following way. By The s k . The mechanical statisticaldynamics factor only: p(z reconstructed each node s l |xl) ∝ of s l (xl; properties the zs l ). of targets, the an individual system assuming = areN x sensor perfect and k; ˜P measurement k|k (3), ˜x s detection, area given globally zand by: s l isoptimal in described absence estimate exploiting is obtained. the product As shown s=1formula fo properties modeled measurement detection, by a linear zand s k|k in absence Gaussian Markovian by of afalse probability measurements. transition density density For function notational p(z Structure: This paper is organized sensor as follows. network. The next If the locally s l |xl), simplicity also called let us produced ˜x tracks s k|k = sensor assume likelihood function, which needs to be known up to a constant ˜P local covariances by a globalized p(x k|x k−1) = N exploiting the product formula for Gaussians, , ˜P k|k , (6) we replace all l is described s=1 in [2], the result is equivalent to a central measurement by of afalse probability measurements. density For function notational p(z s simplicity let us assume local covariances by a globalized one: l |xl), also called x k; Fsensor k|k P s where −1 k|k xs the globalized local k|k (7) param k|k−1 x k−1, where S synchronized D k|k−1 the globalized . For sensors the sake local produce parameters processing measurements ˜x S synchronized sensors produce measurements at the same given s k|k and that covariance the Kalman same filter assumptions hold and likelihood function, which needs ofsection to simplicity, be known states the we upproblem to here a constant assume addressed conditions this paper. In particular, x lk are sent at we some introduce arbitrary the productinstant representation of for time the fused to posterior a fusion node, they can be S factor instants only: ofwhere p(z time s fused −1 l |xl) t l, standard l =1,...,k s l (xl; zs l ). denoted by Z l = {z s ˜P k|k are given by: S instants of time t all sensor covariances arel }S known. s=1. To this end, a p(x k|Z k globalized l, l =1,...,k denoted by Z ) ∝ TheStructure: proposed methodology This kpaper iscan organized be directly as follows. extended ˜P according to (25), densityyielding which was the the key element density in [2] px for( exact k | Zsolution ) N ( xl; xlk | , Plk | ) k|k The = to Sasyn- chronous detection, sensors. and in˜x The s k|k absence accumulation of sensor data Z l up next P . According s −1 k|k . ˜x (8) section states the problem addressed in this paper. In particular, s=1 to s k|k = s=1 ˜P N x Kalman filtering is l = {z s S factor only: p(z s l applicable, }S s=1. l i.e. in the case of wellseparated organizedtargets, as follows. assuming The next perfect s −1 k|k k; The proposed methodology |xl) ∝ s l (xl; can zs l ). ˜P k|k are given by: p(x k|Z k ) ∝ N local x k; xposterior s k|k be directly extended to asynchronous sensors. The accumulation of the sensor data Z , Ps k|kdensity is introduced (4) in the following way. By Structure: This paper is k|k P exploiting the product formula for Gaussians, we replace all of T2TF. Based on the results of the cited preliminary paper, of false measurements. For notational l up = s=1 ˜P k|k P s −1 k|k S the approach of a globalized [6], it is likelihood not required we to simplicity introduce and including let us function is derived to update the product the assume time in sectionthe representation t III. Its global k, typically Note that track forthe the thepresent globalized fused at each posterior time, is covariance scan time ˜P = N x 1 to and including the time t local covariances by a globalized one: k|k does not ˜P depend on k; S k, typically the present time, is synchronized sensors produce measurements density a time which series was recursively the= same N x 1 S xs k|k (7) section states the problem addressed in this paper. In particular, S −1 we introduce the product representation for the fused posterior the key defined k; ˜x elementby impact on practical implementations is discussed in section in local [2] Z k s for = {Z sensor an k, exact Z k−1 index solution }. The k|k = S SP a time series recursively defined by Z s instants of time t s anymore. This two-stage prediction s=1 in order to obtain an k = {Z k, Z optimal k−1 k|k }. The S , 1 S ˜P k|k , (5) ˜P density which was the key element [2] for an l, exact l =1,...,k solution k|k = S P s −1 denoted result. of time Furthermore, T2TF. series by Z Based produced l = {z on s l }S s=1 k|k . S (8) time series produced by the measurements of an individual s=1. by it results is thenot s=1 measurements of necessary to send the fusion result x IV. We close the with a conclusion given in section V. (globalization the cited preliminary of an individual and application paper, of the evolution model) S of T2TF. Based on the resultsThe of the proposed cited preliminary methodologypaper, can be was asensor directly globalized s extended ∈ {1,...,S} to asynchronous sensors. The accumulation tracking likelihood only S p(x k|Z k ) ∝ N x k; x s function is denoted kk and P kk to any node. Therefore, necessary is derived by Z this to ins distributed k k|k . reveal section The statistical , Ps k|k (4) sensor s ∈ {1,...,S} only is denoted by Z a general III. Its Note that the globalized = covariance N x s k . The statistical = N x k; scheme for decorrelated tracks: II. FORMULATION OF impact properties of the sensor THE PROBLEM on of practical an data individual Z implementations l up k; ˜x s sensor measurement We obtain discussed z s s=1 l updates indescribed properties of an individual sensor measurement z s k|k , ˜P a globalized likelihood function is derived in section III. Its Note that the globalized covariance ˜P k|k does k|k not , depend on (6) of section the local sensor indexs=1 s anymore. impact on practical implementations to and including is discussed l is the indescribed time section t local track estimates using the global scheme is well suited for applications k, typically the local IV. by We a probability the sensor presentindex s=1 where close the reduced paper density time, sisanymore. This with function aor conclusion p(z covariance arbitrary s two-stage prediction l |xl), given also instead in = called communication of section N sensor x k; the local V. 1 S (globalization and application of one. where ˜x s In other the words, globalized we engage local param time In series this paper, recursively we address definedthe bylikelihood problem Z k = {Zof function, k, optimal Z k−1 k|k }. which T2TF Theneeds to be known up to a constant S , 1 S ˜P k|k , (5) IV. by We a probability close the paper densitywith function a conclusion p(z s l |xl), given alsoincalled section sensor V. (globalization where the globalized and application local parameters of the evolution ˜x s k|k andmodel) covariance was likelihood function, which needs to be known up to a constant s=1 necessary to reveal a general schem time a modified likelihood function in order to keep the tracks rates are to be taken at arbitrary series produced into instants by account. of the time. measurements The Asfactor discussed schematic only: of II. in p(z an FORMULATION [2], s individual l |xl) idea this ∝ s can l to (xl; the zs OF l THE decorrelated. ). ˜P k|k are given by: factor only: p(z s distributed PROBLEM S In this paper, Kalman We we derive filter obtain updates of local track sensor a closed formula for be achieved, s ∈ {1,...,S} if all measurement only is denoted errorby Structure: covariances Zs k . TheThis statistical are paper knownis organized as follows. = The N next x k; ˜x covariance s instead˜x of the local one. is illustrated in atFigure the sensor 3. s k|k In this paper, we address the thisproblem likelihood of function. optimal T2TF = ˜P k|k P s − k|k properties of an individual sites. To sensor this end, measurement section we states achieved z s k|k , ˜P l |xl) ∝ s l (xl; zs l ). necessary ˜P k|k are to given reveal by: a general scheme for decorrelated tracks: II. FORMULATION OF THE PROBLEM We obtain updates of local k|k , (6) Structure: This paper is organized as follows. The next lthe is problem described ˜x s k|k a product = addressed ˜P track estimates using the global covariance instead of the local k|k Pone. s −1 k|k In in xs k|k other words, we engage (7) this paper. Ins=1 section In this states paper, the we problem address addressed the problem in this paper. of optimal In particular, T2TF particular, a modified likelihood function in by at arbitrary instants of time. As discussed in [2], this can S representation a probability of density the functiondensity p(z a we s modified introduce of the state the product x l,l ≤ k: representation III. forGLOBALIZED the fused posterior l |xl), alsolikelihood called sensor function we introduce the product representation for the fused posterior where S in order −1 to keep the tracks at arbitrary instants of time. As discussed in [2], this can the globalized localLIKELIHOOD parameters decorrelated. ˜x FUNCTION In this FORpaper, we de likelihood function, which needs tobe decorrelated. density beachieved, known which upifIn to was all athis measurement constant the key element error in covariances [2] for are known s k|k and covariance ˜P paper, ˜P S DISTRIBUTED an exact solution k|k = S P density which was the key element in [2] for KALMAN this PROCESSING likelihood function. atthe sensor sites. To this end, we achieved a product s=1 factor only: p(z s an l p(x |xl) exact l|Z k of T2TF. )=c ∝ solution k|k = S we derive P s l (xl; zs l ). ˜P s −1 a closed k|k are k|k . formula for be achieved, if all measurement error covariances are known (8) this likelihood function. given by: at of the T2TF. sensor Based sites. on the To results this end, of the we cited achieved preliminary a product l|k paper, Nrepresentation x l; x s Based l|k , on the resultss=1 of the cited preliminary paper, Structure: This paper is organized Ps l|k of the posterior (1) representation density Firstof ofthe all, state we xintroduce l,l ≤ k: a new notation. III. GLOBALIZED The globalized a globalized as follows. likelihood The next function is derived in ˜x section s III. Its Note that the globalized LIKELIHO k|k covarianc a globalized likelihood of the posterior function density is derived of the in state section x l,l ≤ III. k: s=1 Its Note III. that GLOBALIZED the globalized LIKELIHOOD covariance section states the problem addressed local posterior ˜P for the = state ˜P k|k P s −1 k|k x k at xs k|k FUNCTION does notFOR depend on k|k (7) sensor s will be denoted by impact in this paper. on practical In particular, implementations Sensor S DISTRIBUTED KALMA discussed in section S the local −1 sensor index s anymore impact on practical implementations S is discussed in section the local sensor DISTRIBUTED index s KALMAN anymore. This PROCESSING two-stage prediction we introduce the product representation IV. We for close the p(x fused l|Z the k paper posterior )=c l|k with Nconclusion x l; x s given l|k , Ps in section l|k (1) V. (globalization First of all, we andintroduce application a new of IV. We close p(x l|Z the k paper )=c with a conclusion given in section V. (globalization and application of the evolution ˜P density which was the key element in [2] for an exact solution k|k model) = S wasP s −1 l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized s=1 local necessary k|k . (8) necessary to reveal a general scheme for decorrelated tracks: posterior to reveal for thea state general x k sche s=1 local posterior for the state x s=1 at s II. of T2TF. Based on the results of the cited preliminary II. FORMULATION paper, k at sensor s will be denoted by FORMULATION OF THE PROBLEM OF THE PROBLEM We obtain updates of local track We obtain updates of local track a globalized likelihood function is derived in section III. Its Note that Sensor estimates using the global Sensor the globalized covariance covariance instead of the local one In this paper, we address the problem of optimal T2TF ˜P k|k does not depend on covariance instead of the local one. In other words, we engage In this paper, we addressimpact the problem on practical of optimal implementations T2TF is discussed in section the local sensor index s anymore. aThis modified two-stage likelihood prediction function i at arbitrary instants of time. at a arbitrary modified instants likelihood of function time. Asindiscussed order toin keep [2], the thistracks can IV. As Wediscussed close thein paper [2], with this acan conclusion given in section V. (globalization and application of the decorrelated. evolution In model) this paper, was we d be achieved, if all measurement error covariances are known be decorrelated. achieved, ifInallthis measurement paper, we necessary error derive covariances a closed formula to reveal a are general known for scheme thisfor likelihood decorrelated function. tracks: at the sensor sites. To this end, we II. achieved at this thelikelihood sensor sites. function. To this end, we achieved a product FORMULATION a product OF THE PROBLEM We obtain updates of local track estimates using the global representation of the posterior density of the state x representation Fusion of the posterior covariance density of instead the state of the x l,l local ≤ k: one. In other III. words, GLOBALIZED we engageLIKELIH In this paper, we address l,l ≤ k: the problem III. GLOBALIZED LIKELIHOOD FUNCTION FOR of optimal T2TF Center S a modified DISTRIBUTED KALMA at S DISTRIBUTED KALMAN arbitrary instants of time. As discussed in [2], PROCESSING likelihood function in order to keep the tracks p(x l|Z k this can p(x )=c l|k decorrelated. N x l; x s l|k In , Ps this l|k paper, we (1) derive be achieved, if all measurement error covariances are known First a closed of all, formula we introduce for l|Z k )=c a ne l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized s=1 this likelihood function. s=1 at the sensor sites. To this end, local we achieved posterior for a product the state x local posterior for the state x k at k at sensor s will be denoted by representation Sensor of the posterior density of the state x l,l ≤ k: III. Sensor GLOBALIZED LIKELIHOOD FUNCTION FOR S DISTRIBUTED KALMAN PROCESSING p(x l|Z k )=c l|k N x l; x s l|k , Ps l|k (1) First of all, we introduce a new notation. The globalized s=1 local posterior for the state x k at sensor s will be denoted by Sensor Figure 3. Schematic illustration of a distributed Kalman filter. The sensor nodes process the data to local auxiliary tracking parameters. When communication is successful, these can be fused at the fusion center in order to obtain the estimated track. When applying exact Track-to-Track fusion, the result is equivalent to a central Kalman filter receiving all sensor data In the following the most common and state-of-the-art schemes for target tracking using multiple sensors are listed. The performance of all of them will be compared in the next section. These variable approaches can roughly be divided in the categories Measurement-to-Track Fusion (M2TF) and Track-to-Track Fusion (T2TF). • Central Kalman Filter (CKF): A Kalman filter at the fusion center is processing the measurements of all sensors. This scheme results in an optimal solution with respect to the mean squared error metric. • Single Kalman Filter (SKF): Each sensor node in the scenario performs M2TF using its local data. The tracking algorithm is a Kalman filter processing the linearised measurements. At each time step, the node sends its current estimate and estimation error covariance to the fusion center, which in turn
Chapter 3: Information Technology for Interoperability and Decision... 289 selects the track having the smallest trace of the covariance. This scheme is included in the evaluation as a benchmark of the worst performance achieved when no fusion is performed. • Naive Fusion: In this scheme, each sensor node performs its own local Kalman filter resulting in local optimal tracks, which are sent to the fusion center. In the fusion center the tracks are fused to a global estimate as if they were decorrelated. Given a set of local tracks {( x s | , s | )} S kk Pkk s 1 the fused parameters are obtained via the decorrelated fusion equations (24) and (25). As the local optimal tracks are correlated to each other if process noise is assumed, this fusion scheme ignores these cross-correlations. • Distributed Kalman Filter (DKF): Our approach to T2TF is the decorrelated DKF, which was proven to be exact under perfect data association conditions previously. It is based on a product representation for the global posterior density: S k s s k1 Z N( xk 1; xk 1| k1 , Pk 1| k1 s1 px ( | ) ). (26) The decorrelated DKF described in [6] uses a covariance globalisation step. For known posterior covariance matrices P s 1| 1 , the globalised prediction k k s 1, , S parameters are given by 1 s1 Pkk | 1 SF kk | 1 P k1| k1 Fkk | 1 Q kk | 1 , (27) s x SF P P x (28) s s1 s1 s . k| k1 k| k1 k1| k1 k1| k1 k1| k1 s In the scenario considered in this paper the posterior covariances of the remote sensors are known at each node. However, when the sensor measurement characteristic is geometry dependent, as with radar, the posterior covariances of the remote sensor nodes are not known as they are dependent on the target-sensor geometry, and there are no data transmissions between the sensors. In this case, it is possible to replace the exact remote covariances by approximations based on the local estimate and a radar model applied to known parameters of other sensors. • Distributed Kalman Filter with Feedback (DKF-FB): For the DKF with FB, we assume that the fusion center transmits the fused track to all sensors at each time step. With the global covariance Pk 1| k 1 for time available, the globalised prediction is given by the following lines. P S P F Q kk | 1 Fkk | 1 k1| k1 kk | 1 kk | 1 , x SF P P x s s s . k| k1 k| k1 k1| k1 k1| k1 k1| k1
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Contents Foreword .................
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Contents 5 Methodology for Gatherin
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8 Military Communications and Infor
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Building a Layered Enterprise Archi
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Chapter 1: Concepts and Solutions f
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Applying NAF for Performance Analys
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Generation of Nonlinear Feedback Sh
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Modern Usage of “Old” One-Time
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A Abut Fatih 161 Akcaoglu Ismail 11
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Military Communications and Information Technology: A Trusted ...

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