A Fast and Fault-Tolerant Convex Combination Fusion Algorithm under Unknown Cross-Correlation

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1 1th Internatonal Conference on Informaton Fuson Seattle, WA, USA, July 6-9, 9 A Fast and Fault-Tolerant Convex Combnaton Fuson Algorthm under Unknown Cross-Correlaton Ymn Wang School of Electroncs and Informaton Engneerng X an Jaotong Unversty X an, Shaanx 719, P.R.Chna erc.wang.xjtu@gmal.com Abstract Knowledge of the cross-correlaton of errors of local estmates s needed for many dstrbuted fuson algorthms. However, n a fully dstrbuted system or decentralzed network, the calculaton of crosscorrelaton between local estmates s qute nvolved and may be mpractcal. The Covarance Intersecton CI algorthm has been proposed under unknown correlaton. But the CI algorthm has hgh computatonal complexty because t requres optmzaton of a nonlnear cost functon. Ths paper presents a fast CI algorthm, and an alternatve optmzaton crteron wth a closed form soluton. Based on ths crteron, a fast and faulttolerant convex combnaton fuson algorthm s presented by ntroducng an adaptve parameter, whch can obtan robust estmate when estmates to be fused are nconsstent wth each other, and the degree of robustness of fuson result vares wth that of nconsstency between estmates to be fused. Keywords: Dstrbuted fuson, decentralzed network, convex combnaton, covarance ntersecton, chernoff nformaton, fault-tolerant. 1 Introducton Estmaton fuson, or data fuson for estmaton, s the problem of how to best utlze useful nformaton contaned n multple sets of data for the purpose of estmatng a quantty a parameter or process at a tme 1]. There are two basc fuson archtectures, centralzed and dstrbuted, dependng on whether raw data are used drectly for fuson or not. In centralzed fuson, all sensors send ther raw data to a fuson center, whch can produce globally optmal estmates. In dstrbuted fuson, each local sensor processes ts own raw data and sends the local estmate to other sensors or a fuson center to obtan global estmates. In dstrbuted fuson, two basc fuson archtectures are well known: herarchcal and fully dstrbuted 6], and sometmes, Research supported n part by Project 863 through grant 6AA1Z16 and ARO through grant W911NF X. Rong L Department of Electrcal Engneerng Unversty of New Orleans New Orleans, LA 718, U.S.A. xl@uno.edu the fully dstrbuted archtecture s referred to as decentralzed network. Generally, dstrbuted fuson has a less communcatonal burden and hgher survvablty, and t s more flexble and relable. However, dstrbuted fuson also has challengng problems not present n the centralzed fuson. One of the major ssues s dealng wth the cross-correlaton of errors of local estmates due to the common process nose, pror, correlaton among measurement noses across sensors, etc. Most of the exstng dstrbuted fuson algorthms see, e.g., 1]-8] accountng for ths problem try to explot the correlaton n the estmates. Consder a decentralzed network of N nodes whose connecton topology s completely arbtrary.e., t mght nclude loops and cycles and can change dynamcally. Each node has nformaton only about ts local connecton topology e.g., the number of nodes wth whch t drectly communcates and the type of data sent across each communcaton lnk 11]. In ths case, the calculaton of the cross-correlaton of local estmaton errors s qute nvolved and not practcal. 9]-11] proposed the Covarance Intersecton CI algorthm under unknown correlaton, whch bypasses ths problem, and used a convex combnaton of the local estmates to mnmze the trace or determnant of the mean-square error MSE matrx. 1] showed that the CI s equvalent to the log-lnear combnaton of two Gaussan functons, and dscussed the crteron that mnmzes the determnant of the fused MSE matrx and an alternatve crteron of mnmzatng Chernoff nformaton. However, mnmzaton of the determnant or Chernoff nformaton s computatonally complex. 1] presented a fast CI algorthm w.r.t. determnant mnmzaton. In some specal cases, the estmates to be fused are nconsstent wth each other e.g., f one poston estmate s far from another by more than a klometer, but ther respectve MSE matrces showthat eachestmate s accurate to wthn a meter, then there must be somethng wrong. In 1], covarance unon CU was ISIF 71

2 proposed to handle ths problem; the Mahalanobs dstance was suggested for detectng statstcal devatons between estmates; and a user-defned threshold was proposed to detect the nconsstency. A fault-tolerant mechansm combnaton of the CI and CU CI/CU was also presented. However, how to determne the thresholdsanopenproblem. In ths paper, a fast CI algorthm based on a proposed optmzaton crteron s presented, whch has a closed form soluton. Because tdoesnotrequreoptmzaton of a nonlnear cost functon, a large amount of computatonal tme can be saved. Snce the proposed crteron s to fnd a halfway pont between the dstrbutons of estmates, t wll be more robust when the estmates to be fused are nconsstent. Combnng ths crteron wth a proposed adaptve parameter, a fast and fault-tolerant convex combnaton fuson algorthm s presented. It can obtan robust fuson results when estmates to be fused are nconsstent, and the degree of robustness can vary adaptvely wth that of nconsstency between estmates. It s easy to be mplemented wthout any pre-defned threshold. Most mportantly, t does not need to deal wth any optmzaton problem, and t offers decentralzed fault tolerance meanwhle. The structure of ths paper s as follows. Secton presents a fast CI algorthm based on a proposed mnmzaton crteron. In secton 3 a fast and robust convex combnaton fuson algorthm under unknown correlaton s proposed. Illustratve smulaton examples are gven n Secton. Secton provdes conclusons. A New Fast CI Algorthm Suppose two estmates are to be fused, ˆx,=1,, wth MSE matrx P. The CI algorthm s gven by ˆx CI = P CI ωp 1 ˆx 1 +1 ωp ] ˆx 1 P CI = ωp1 +1 ωp ] If the parameter ω 1 s chosen to mnmze detp CI, t s called the general CI algorthm n ths paper. Then, under the determnant mnmzaton crteron, the fused estmate s ˆx,P. For the Gaussan case, the probablty densty functon pdf of the th local estmate s gven by: 1 p x= e 1/ πp x ˆxT P x ˆx 3 Then, the CI algorthm can be generalzed to a technque that selects a fused pdf that s a log-lnear combnaton of two orgnal pdfs 1]: p ω x= pω 1 x p1 ω x p ω 1 x p 1 ω x dx and mnmzaton of det P CI s equvalent to mnmzatng of the Shannon nformaton or contnuous entropy of the fused Gaussan functon: H p ω = p ω xlogp ω xdx = 1 log πn P ω ]+ k where k s dmenson, and P ω = P CI. An alternatve crteron s Chernoff nformaton mnmzaton: C p 1,p = mn ω 1 It s equvalent to log p ω 1 xp1 ω x dx 6 D D p ω p 1 =D p ω p 7 where D p A p B s the Kullback-Lebler K-L dstance from p A top B, and ω mnmzes Equaton 6. Then, t s natural to consder p ω x as the halfway pont snce the K-L dstance from p ω xto p 1 x s equal to that from p ω x top x. However, computaton of ths ω s complcated. A smple way to defne a halfway pont s to let ω =1/, whch s known as the Bhattacharyya dstance: B p 1,p = log p 1/ 1 xp 1/ x dx 8 Inspred by Equaton 7, an alternatve to defne the halfway pont p ω s: D D p 1 p ω =D p p ω 9 In contrast to the Chernoff nformaton crteron, ths problem has a closed form soluton: ω = D p 1 p D p 1 p +D p p 1 1 Ths determnes the parameter ω n the CI algorthm. Thus the new fast CI algorthm FCI s gven by: ˆx FCI = P FCI ω P1 ˆx 1 +1 ω P ] ˆx 11 P FCI = ω P1 +1 ω P ] 1 For the Gaussan dstrbutons, for,j =1,, D p p j = 1 ln P j P k +ˆx ˆx j T P j ˆx ˆx j +tr P P ] j 13 Note that the Chernoff nformaton D, whch s the halfway dstance between p 1 x andp x evaluated usng p ω x, s the best achevable exponent for the Bayesan probablty of error 13]. However, our proposed D, whch s the halfway dstance evaluated usng 7

3 8 6 6 FCI Fgure 1: Equal means FCI Fgure : Unequal means p 1 x andp x respectvely, does not have ths property n general. Ths appears make D more appealng, but our D s computatonally superor. For Gaussan dstrbutons, f pror dstrbutons have the same MSE matrces, that s, P 1 = P, the proposed equal dstance D and Chernoff nformaton D are dentcal, and ω = ω =1/. Fgures 1 and compare the fused estmates of each par of estmates calculated by and FCI, respectvely. Dotted ellpses are for the estmates to be fused. In Fgure 1, the means are equal and the fuson results are ˆx 1 =ˆx =ˆx =ˆx FCI = ] ] P 1 = P = P = ] P FCI = ] and for the estmate ˆx 1,P 1 concde, and the fuson results are: ] 1 ˆx 1 =ˆx = P 1 = P = ] 1 ˆx = P 1. = ] ˆx FCI = P.67 FCI = 33.3 The proposed crteron has several advantages: It has a closed form soluton, whch does not requre optmzaton of a nonlnear cost functon. It can save a large amount of computaton. As shown n Fgure, for the determnant mnmzaton crteron.e.,, f P <P j, j, the fused estmate wll be ˆx,P, although the two estmates are far from each other. Snce the determnant mnmzaton crteron, whch focuses on the uncertanty of the estmate, does not utlze the values of ˆx,theoptmalω s ndependent of the values of ˆx. As P ncreases from below P j to above P j, the fused estmate wll jump from ˆx,P to ˆx j,p j, whch s hghly undesrable. The proposed crterons to fnd a halfway pont between the dstrbutons of estmates, and both the uncertanty of the estmates and the dstance between estmates are taken nto account. Thus, when the estmates are nconsstent, the proposed crteron wll be more robust. Consderng the scalar case, n general, for fuson of unbased estmates, wll be a better choce snce t pcks the estmates wth a smaller varance, but for fuson of based estmates e.g., one s based, and the other s unbased but wth a larger varance, our proposed crteron would be better. For hgher dmensons, t s complcated and depends on specfc system desgn requrements and the bas of the estmates. CI can be generalzed to an arbtrary number of N updates usng the followng equatons 11]: ˆx CI = P CI N P CI = N =1 =1 ω P ˆx 1 ω P 1 N ω = 1 16 =1 By our proposed crteron, a generalzed constrant s gven by In Fgure, the means are not equal, P 1 <P. It can be shown that the ellpses for the fused estmate 73 D p j p ω Dp p j ω j =,,j =1,,...,N 17

4 Followng the generalzaton method of 1], the largest lnearly ndependent subset can be represented by D p +1 p ω Dp p +1 ω +1 =, =1,,...,N 1 18 Wth α = D p +1 p andβ = D p p +1, combnng the constrants 16 and 18 yelds the lnear system: α 1 β 1 ω 1 α β ω α N β N ω =. N ω N 1 19 Then, for the FCI algorthm, ω can be obtaned by where P 1 and P are the actual MSE matrces of ˆx 1 and ˆx, respectvely. For,j =1,, P j = E x ] x T j. The key to ths method s that δ s consdered as an overall belef of all estmates to be fused. If ˆx 1 =ˆx and P 1 = P, the two estmates are dentcal, obvously, δ =1. When the estmates dffer, especally f the dstance between estmates s large, the estmates are regarded as nconsstent wth each other n general. Gven that we have no dea whch one s more credble, a natural proposal s to scale-down the overall belef : δ<1. In nformaton theory, entropy s a measure of the uncertanty of a random varable. The relatve entropy or K-L dstance s a measure of the dstance between two dstrbutons. Then, we construct the adaptve parameter δ 1as ω = j=1 α N j k= β k N =1 j=1 α j N k= β k 3 A Fast and Fault-Tolerant Convex Combnaton Fuson Algorthm In some cases, the estmates to be fused are nconsstent wth each other. In 1], covarance unon CU was proposed to handle ths problem, and the Mahalanobs dstance was suggested for detectng statstcal devatons between estmates. Frst, a user-defned threshold was proposed to detect the nconsstency. If the threshold s exceeded, the estmates are regarded as nconsstent wth each other, mplyng that at least one of the estmates s not a consstent estmate defned n 1]. Then, the CU algorthm can be taken to resolve such nconsstency, whch s referred to as deconflcton. Also, a fault-tolerant mechansm as a combnaton of CI and CU CI/CU was presented. However, how to determne the threshold s an open problem. An adaptve parameter δ s ntroduced to solve ths problem. Suppose there are two estmates to be fused, ˆx, =1,, wth MSE matrx P. Consder the followng general convex combnaton fuson problem: ˆx CC = P CC ω1 P1 ˆx 1 + ω P ˆx 1 P CC = ω 1 P1 + ω P ω 1 + ω = δ 3 The estmaton error correspondng to estmate ˆx CC s x CC = P CC ω1 P1 x 1 + ω P x where x 1 and x are the estmaton error of ˆx 1 and ˆx, respectvely. Then the actual MSE matrx P CC s P CC = P CC ω 1 P 1 +ω 1 ω P P 1 P1 + ω 1 ω P1 P 1 P1 + ω P P P P 1 P ] PCC δ = H p 1 +H p H p 1 +Hp +J p 1,p 6 where H p stheentropyofp x, H p 1 +Hp s the overall uncertanty of p 1 x andp x, and J p 1,p s symmetrzed K-L dstance between two dstrbutons known as J-dvergence 16]: J p 1,p =D p 1 p +D p p 1 7 The δ of 6 s ntutvely appealng: gven the overall entropy H p 1 +H p, the larger the J-dvergence s, the less the overall belef δ s. Usually, δ s nversely proportonal to J p 1,p, but ths s not true, f J p 1,p s large but the overall entropy s much larger, δ wll not decrease. Snce δ 1, the consstency of ˆx CC,P CC remans guaranteed, that means P CC P CC. Ths convex combnaton algorthm s equvalent to the CI algorthm wth all MSE matrces nflated by a factor of 1/δ. Snce the proposed mnmzaton crteron s more robust, as dscussed n Secton, combnng t wth the adaptve parameter δ, a fast and fault-tolerant convex combnaton FFCC fuson algorthm can be obtaned: ˆx FFCC = P FFCC ω1 P1 ˆx 1 + ω P ] ˆx 8 P FFCC = ω 1 P1 + ω P ] 9 ω 1 + ω = δ 3 ω 1 = δd p 1 p D p 1 p +D p p 1 31 Fgures 3 and compare the fused estmates of each par of estmates calculated by, FCI and CU, respectvely. In Fgure 3, the overall belef δ =.6, and 7

5 FFCC CU Fgure 3: Overall belef δ =.6 FFCC CU 1 3 Fgure : Overall belef δ =.1 the fuson results are: ˆx 1 = ˆx =.3 ˆx =.1.7 ˆx FFCC = 3.9 ˆx CU = ] 1 P 1 = ] 8 1 P = 1 ] P = ] P FFCC = ] P CU = In Fgure, the overall belef δ =.1 and the fuson results are: ˆx 1 = ˆx = ˆx = ˆx FFCC = ˆx CU = ] 1 P 1 = 1 ] 1 P = 1 ] 1 P = 1 ] 6.6 P FFCC = 6.6 ] P CU = Fgures 3 and llustrate that the MSE matrx of the FFCC estmate can adapt to the overall belef δ. When δ = 1, FFCC s just the FCI algorthm. Wth a decrease n δ,themsematrxsscaledupby1/δ tmes to resolve the nconsstency among estmates. Consderng a smple decentralzed network wth only two nodes and synchronous nformaton exchange, n contrast to our adaptve fault-tolerant mechansm, gven a pre-defned threshold the CI/CU mechansm needs to make a hard decson to determne whether the estmates are nconsstent wth each other. If the userdefned threshold s not exceeded, then CI wll be appled to obtan the fused estmate, whch wll be mantaned for the next update. When an nconsstency s detected, gven that nether node can determne whch of the estmates s spurous, both apply CU to produce a consstent estmate, and the CU estmate wll be mantaned tll the next communcaton. Obvously, however, the FFCC algorthm s easy to be mplemented wthout any pre-defned threshold. Most mportantly, the computatonal complexty of the FFCC algorthm s far less than CI/CU. FFCC can also be generalzed to an arbtrary number of N updates. Let δ j = Hp +Hp j, j 3 H p +Hp j +J p,p j Then the overall belef of the N estmates can be obtaned by N δ N = δ j /m 33,j N where m = = N!/N!. For N, the FFCC algorthm s gven by ˆx FFCC = P FFCC N P FFCC = N ω P ω P ˆx 3 3 N ω = δ N 36 7

6 where ω = j=1 α N j k= β k N =1 j=1 α j Smulaton Examples N k= β k δ N 37 To evaluate the performance of the FCI and FFCC algorthms, we consder a constant-velocty movng target n one dmenson. A target dynamc model s assumed: where F = x k+1 = Fx k + Gw k 38 1 T 1 ] T, G= / T wth samplng tme T = 1 and zero-mean whte Gaussan process nose w k wth varance q. A smple decentralzed fuson system consstng of two nodes s used for trackng ths target, but there s no fuson center n the system. The measurements of the two nodes are modeled as where z k = H x k + v k, =1, 39 H 1 = 1 ], H = 1 ] Each node measures dfferent quanttes about the target: node 1 observes poston wth varance rk 1,node measures velocty wth varance rk.thetwomeasure- ment noses are assumed to be ndependent. Assume a full communcaton rate s employed and the nformaton exchange between nodes s synchronous. Each node apples a Kalman flter to estmate the state. At tme k n the th node, when each node obtans ts own measurement, the local estmates ˇx k k, ˇP k k can be obtaned by usng a standard Kalman flter gven the prevous stored estmate ˆx k k,p k k. Then each node propagates ts own estmates to another node and fuses ts own estmate wth the recevedone to obtan a fnal estmates ˆx k k,p k k accordng to a specfc fuson mechansm. If both nodes apply the same fuson method, then the fnal estmates at tme k are dentcal..1 FCI and It s assumed that both nodes apply the same fuson mechansm: FCI or. Wth the model descrbed above, the total tme span s 1 seconds, gven q =, E v k] = r 1 k =, r k = and the flter s ntalzed at the true state, whch s generated Gaussan dstrbuted wth mean and covarance: ] ] 1 1 x =, P = ] RMS Poston Error RMS Velocty Error 1 1 Poston Error FCI Tme k Velocty Error Tme k Fgure : FCI and FCI Fgure compares the RMS poston errors and RMS velocty errors of FCI and, respectvely, computed from 1 Monte Carlo runs. As can be seen, the two algorthms have smlar performance, and FCI s somewhat better than n estmatng poston, but s slghtly better than FCI n estmatng velocty. The average computatonal tmes of and FCI for Matlab 7..1 over 1 seconds are: t =.87 s t FCI =.13 s respectvely. The fuson estmate was computed by usng the routne presented n 11]. The tmes lsted above were obtaned on a computer wth Core Duo CPU at.33 GHz. Obvously, the computaton of FCI s far less than that of the.. FFCC and CI/CU To compare the performance of the FFCC and CI/CU algorthms, t s assumed that both nodes apply the same fuson mechansm: FFCC or CI/CU. Consderng the same scenaro presented above, t s assumed that node s corrupted by whte Gaussan nose θ k wth mean E θ k ] = 1 and varance varθ k ] = 9 durng the tme from 1s to 6s. The measurement equaton for node s gven by: { z k = H x k + v k + θ k, 1 k 6 z k = H x k + v k, others The model msmatch of the Kalman flter n node makes the estmate nconsstent wth the one obtaned 76

7 by node 1. For CI/CU, to address ths sensor falure problem, a threshold should be defned frst. In order to detect nconsstency as early as possble, we calculated the Mahalanobs dstance between ˇx 1 1, ˇP 1 1 : ˇx 1 1 1, ˇP T Md 1 = ˇx ˇP ˇx ˇP 1 1 ˇx ˇx 1 1 and 1 and the average Mahalanobs dstance over 1 Monte Carlo runs at tme 1s s AMd 1 =8.. Thus the threshold was determned to be 8. Ths s an deal selecton of threshold for CI/CU, not sensble for real systems. Fgure 6 compares the RMS poston errors and RMS velocty errors of FFCC and CI/CU, respectvely, computed from 1 Monte Carlo runs. As can be seen, CI/CU and FFCC have smlar performance n estmatng velocty. However, n estmatng poston, FFCC s better than CI/CU. It also can be shown that the proposed adaptve parameter δ can really reflect the degree of nconsstency between local estmates. The average computatonal tmes of CI/CU and FFCC for Matlab 7..1 over 1 seconds are: t CI/CU =.7 s t FFCC =.1 s respectvely. The CU fuson estmate was computed by usng SeDuM 17] and YALMIP 18]. The tmes lsted above were obtaned on a computer wth Core Duo CPU at.33 GHz. Obvously, FFCC has far less computaton than CI/CU. It s worth notng that numercal problem wll arse n the FFCC fuson f δ. A user-defned lower bound l can be used to sovle ths problem. For example, l =.1, f δ<l,thenletδ = l. Conclusons Ths paper has consdered the estmaton fuson problem underunknowncross-correlatonbetween the errors of estmates to be fused. We have presented a fast CI FCI algorthm based on a proposed mnmzaton crteron, whch leads to a close form soluton. Thus, the computaton of the FCI algorthm s far less than the CI algorthm under the determnant mnmzaton crteron. We have ntroduced an adaptve parameter δ,and a fast and fault-tolerant convex combnaton FFCC fuson algorthm has been worked out wthout a need for any pre-defned threshold. Both the FCI and FFCC algorthms can be generalzed to the case of more than two estmates. The performance of our proposed FCI and FFCC algorthms has been compared wth that of and CI/CU. In llustratve smulaton examples, t RMS Poston Error RMS Velocty Error AMd δ 3 1 Poston Error Tme k Velocty Error Tme k Average Mahalanobs dstance Tme k Average overall "belef" Tme k Fgure 6: FFCC and CI/CU CI/CU FFCC CI/CU FFCC AMd Threshold has been shownthat FCI and have smlar performance when fusng consstent estmates. Wth a slght performance degradaton n estmatng velocty, FFCC s better than another fault-tolerant mechansm CI/CU n estmatng poston. Moreover, FFCC algorthm can save a large amount of computaton. References 1] X. R. L, Y. M. Zhu, J. Wang, and C. Z. Han, Optmal lnear estmaton fuson - part I: Unfed fuson rules, IEEE Transactons on Informaton Theory, vol. 9, no. 9, pp. 19 8, September 3. ] C. Y. Chong, Herarchcal estmaton, n Proceedngs of the MIT/ONR Workshop on C3,Monterey, CA,

8 3] C. Y. Chong, S. Mor, and K. C. Chang, Adaptve dstrbuted estmaton, n Proc. 6th IEEE Conf. Decson and Contr., Los Angeles, CA, Dec. pp , ] Y. Bar-Shalom and L. Campo, The effect of the common process nose on the two-sensor fused-track covarance, IEEE Transactons on Aerospace and Electronc Systems, vol., no. 6, pp. 83 8, November ] K. C. Chang, R. K. Saha, and Y. Bar-Shalom, On optmal trak-to-track fuson, IEEE Transactons on Aerospace and Electronc Systems, vol. 33, no., pp , October ]M.Lggns,C.Y.Chong,I.Kadar,M.G.Alford, V. Vanncola, and S. Thomopoulos, Dstrbuted fuson archtectures and algorthms for target trackng, n Proc. IEEE, vol. 8, pp. 9 17, Jan ] W. Nehsen, Informaton fuson based on fast covarance ntersecton flterng, n Proceedngs of the th Internatonal Conference of Informaton Fuson, Annapols, MD, USA, pp. 91-9,. 1] J. K. Uhlmann. Covarance consstency methods for fault-tolerant dstrbuted data fuson, Informaton. Fuson, pp. 1 1, 3. 16] H. Jeffreys, An nvarant form for the pror probablty n estmaton problems, n Proc. Roy. Soc. Lon., Ser. A, vol. 186, pp. 3-61, ] J. F. Sturm, Usng SeDuM 1., a Matlab toolbox for optmzaton over symmetrc cones, Optmzaton Methods and Software, no. 11-1, pp. 6 63, ] J. Lofberg, YALMIP: A toolbox for modelng and optmzaton n MATLAB, n IEEE Internatonal Symposum on Computer Aded Control Systems Desgn, Tape, pp. 8-89,. 7]S.Mor,W.H.Barker,C.Y.Chong,andK.C. Chang, Track assocaton and track fuson wth non-determnstc target dynamcs, IEEE Transactons on Aerospace and Electronc Systems, vol. 38, no., pp , Aprl. 8] O.E. Drummond, On track and tracklet fuson flterng, n Proceedngs of SPIE, Sgnal and Data Processng of Small Targets, vol.78, pp ,. 9] J. K. Uhlmann. General data fuson for estmates wth unknown cross covarances, n Proceedngs of the SPIE Aerosense Conference, vol.7, pp. 36-7, ] S.J. Juler and J.K. Uhlmann, Non-dvergent estmaton algorthm n the presence of unknown correlatons, n Proc. Amer. Control Conf., vol., pp , ] S.J. Juler and J.K. Uhlmann, General decentralzed data fuson wth Covarance Intersecton CI, n Handbook of Multsensor Data Fuson, D.Hall and J. Llans, Eds. Boca Raton, FL: CRC Press, ch. 1, pp , 1. 1] M. Hurley, An nformaton theoretc justfcaton for covarance ntersecton and ts generalzaton, n Proceedngs of the th Internatonal Conference of Informaton Fuson, Annapols, MD, USA, pp. -11,. 13] T. M. Cover and J. A. Thomas, Elements of Informaton Theory, Wley-Interscence, New York,