Hopfield-Tank
China Institute of Technology
Email: elaine@cc.chit.edu.tw

Abstract
The problem of tracking multiple targets in the presence of clutter is addressed. Data association is the most critical part in multi-target tracking; erroneous data as- sociations can result in lost tracks. The joint probabilistic data association (JPDA) and multiple hypotheses tracking (MHT) algorithm have been previously reported to be suitable for this problem. However, the complexity of this algorithm increases rapidly with the number of targets and returns. The computation for probabilities of enormous feasible events becomes very heavy burden. For real-time processing and tracking performance, it seems that parallel structure is a suitable approach. A Hop- field-Tank Network, which consists of many connected processing elements, is ca- pable of parallel computation, and it is suitable for a solution to the data association problem.
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Tracking (MHT)Hopfield-Tank Neural Network (HTN)

(Nearest Neighbor Standard Filter; NNSF)[3,4]
"1" "0"(Innovation)

(Maximum Likelihood Function)
Y. Bar-Shalom Tse 1975 (Probabilistic Data Association; PDA)[7](All Neighbor)
(Validation Region; Gating)

Y. Bar-Shalom Fortmann 1983
(Joint Probabilistic Data Association; JPDA)[9]
144
Roecker JPDA
"AD HOC JPDA"[10]Emre and Seo [11]

Y. Bar-Shalom Chang [12] JPDA

JPDA
(Neural Network)
JPDA SOFM
-(Hopfield-Tank; HTN) JPDA HTN
”
JPDA : km
[] (Validation Matrix) Gate
[ ] , 1, 2, , , 0,1, 2, ,jt kj m tω = =L TL (1)
jtw t ’1’
’0’
j
m

j jt
145
ˆ ( ) 0,
jt jt
0
jt k t
jt j
≤ =∑ θ L

[] (1) (Binary M
146

(5) 1
ˆ( ) ( ) T
j t
τ ω =
θ t
[] ( )kθ
1 11{ ( ) } { ( ) ( ), } [ ( ) ( ), ] { ( ) }k k kP k Z P k Z k Z p Z k k Z P k Z c
− −= = ⋅θ θ θ θ 1k− (8)
1 1 1 2
j jt j
p Z k k Z p z k z k z k k Z
p z k k Zθ
− −
−
=
=
=∏
1 1
j
N z k if k p z k k Z
V if
τ θ
θ (10)
ˆ[ ( )] [ ( ); ( 1), ( )]j jt t tj j j j jN z k N z k z k k S k−
)1(ˆ −kkz jt j : jt
)(kS jt j : jt
( )1 ( )
1
[ ( ) ( ), ] [ [ ( )]] k
j
=
(Prior Probability) ( )kθ
{ ( )} { ( ), ( ), ( )} { ( ) ( ), ( )} { ( ), ( )}P k P k P k Pδ φ δ φ δ φ= = ⋅θ θ θ θ θ θ θ θ θ (12)
{ ( ) ( ), ( )}P k δ φθ θ θ :
km : k
( )φ θ : θ

−= ⋅ − =θ θθ θ θ θ (13)
{ ( ), ( )} { ( ) ( )} { ( )}P P Pδ φ δ φ φ= ⋅θ θ θ θ θ (14)
{ ( ) ( )}P δ φθ θ : ( )φ θ
∏ =
DP−
tj j D D F j tk
P k Z V N z k P P c m
τ δ δφφ t µ φ−−
= =
= −∏ ∏θ (16)
PMF )(φµF { ( )} ( )FP φ µ φ=θ Possion
! )()(
P k Z N z k P P c
φ τ ttδ δλ −
[] t j
ˆ{ } { } (t k k j jt jtP Z P Z wβ θ
= =∑ θ
i i j j i
j i
148
) iE A X W X xθ ≠
= − ⋅ = − −∑ (20)
1 1( ) ( )2 2i i i ij j i i j i i
E A X X W X iXθ ≠
= − ⋅ = − +∑∑ ∑ (21)
A W x )iθ ≠
= ∑ − (Firing Function)HTN
(1)(2) (3) N


51 52 53 55
0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0
1 2 3 4 5
=
xiX 1xiX =

X X ≠
X X ≠
X N N−∑∑ (25)
: , 1 , 1)(xy xi y i y i x i y x
dist X X X+ ≠
2 2
[( ) ] ( ) 2 2
xi xj xi yi x i j i x i y x
xi xy xi y i y i x i x i y x
A BE X X X X
C DX N dist X X X
≠ ≠
+ − ≠
= ⋅ + ⋅
+ − + ⋅ ⋅ +
∑∑∑ ∑∑∑
xy
W A x y i j B i j x y
C D dist j i j i
δ δ δ δ
TSP
})({)( kZkiPki θβ
−= − k
1 22 ( ) (1 ) /D G Db S k P P Pλ π= − (32)
1 0
τ ≠
i t j i X X
≠
⋅∑∑∑
t i X )−∑ ∑
i t X β−∑∑
[] JPDA
2 2( 1) ( 2 2 2
t t t t DAP i i i j i i i
i t t i t j i t i i t
A B DE X X X X X Xτ
τ
[JPDA ]
: O O 1 2 3(3.3, 2.1) , (1.4, 2.5) , (2.4, 2.225)T T
: TT PP )45.2,0.2(,)0.2,8.2( 21 ==
: 99.0=GP
2χ : Prob 2( ) 0.01 , 9.χ γ γ> = = 21
(1) 2 jtd 2χ
151
0 16.667 0.1 O P d
− ⇒ = = <

1 0 0 θ
1 0 0 θ
1 0 0 θ
0 1 0 θ
0 0 1 θ
1 0 0 θ
0 0 1 θ
0 1 0 θ
1 1exp[ ] exp[ ] 2 2(2 ) (2 )
t t t j j j jt
M Mt t j j
v S v d
1}{1}{0}{1}{ 4321 0
1 ×+×+×+×= kkkk ZPZPZPZP θθθθβ 1}{0}{0}{1}{ 8765 ×+×+×+× kkkk ZPZPZPZP θθθθ
55566.0,3967.0,0,0
t
t
+ = +
= +
k (38)
( )tx k t ( ) ( )t Tx k x x y y= & &
( )tF k t t ( )tG k
153
2
2
01 0 0 2 0 1 0 0 0 1 0 0 0
( ) , ( ) , ( ) 0 0 1 0 0 1 00 20 0 0 1
0
= = =
k σ
x y kmk k sσ σ= =
PDA : 0.95GP = 2χ
9.2γ = 0.95DP =
: 20.2km−
( )x km ( )y km ( )x km s& ( )y km s&
1 1.5 3.5 0.4 0.56 2 1.0 4.0 0.6 0.44 ()()() PDAJPDAHTNPDA
PDA



HTN-JPDA
(Cluster)

1. Bar-Shalom, Y., Fortmann, T. E., Tracking and Data Association. Academic Press, Or-
lando, FL, 1988. 2. Blackman, S. S., Multiple Taget Tracking with Radar Appications, Artech House,
Dedham, Ma., (1986). 3. A. Farina and S. Pardini, "Track-While-Scan Algorithm in a Clutter Environment",
IEEE Trans. Aerospace & Electronic Systems, Vol. AES-14, September 1978, pp.768-779.
4. X. Rong Li and Bar-Shalom, Y., " Tracking in Clutter with Nearest Neighbor Filters: Analysis and Performance", IEEE Trans. Aerospace & Electronic Systems, Vol. 32, No.3 1996, pp.995-1099.
5. Smith, P., and Buechler, G., “A branching algorithm for discriminating and tracking multiple objects,” IEEE Trans. Automatic Control, Vol. AC-20, Feb.1975, pp. 101-104.
6. C.L. Morefield, "Application of 0-1 Integer Programming to Multi-Target Tracking Problems", IEEE Trans. Automatic Control, Vol. AC-22, June.1977, pp. 302-312.
7. Y. Bar-Shalom and E. Tse, " Tracking in a Cluttered Environment with Probabilistic Data Association", Automatica, Vol. 11, September 1975, pp. 451-460.
8. Reid, D.b, " An Algorithm for Tracking Multiple Targets", IEEE Trans. Automatic Control, Vol. AC-24, Dec. 1979, pp. 843-845.
9. Fortmann, T. E., Bar-Shalom, Y., and Scheffe, M., “Sonar tracking of multiple targets using joint probabilistic data association,” Journal of Oceanic Engineering, OE-8, July 1983. pp. 173-184.
10. J.A. Roecker, and G.L. Phillis, "Supoptimal Joint Probabilistic Data Association", IEEE Trans. on Aerospace & Electronic Systems, Vol. 29, No.2, Apr. 1993, pp.510-517.
11. Emre, E. and Seo, J., “A Unifying Approach to Mult-Target Tracking”, IEEE Trans. on Aerospace & Electronic Systems, Vol. AES-25, July. 1989, pp.520-528.
12. K.C. Chang, C.Y. Chong, and Y. Bar-Shalom, “Joint Probabilistic Data Association in Distributed Sensor Networks”, IEEE Trans. Automatic Control, Vol. AC-31, October 1986, pp. 889-897.
13. C.Y. Chong, K.C. Chang, and S.Mori, “Distributed Tracking in Distributed Sensor
155
HTN-JPDA
Networks”, Proc. American Control Conf., Seattle, MA, June 1985. 14. J. L. Speyer, “Computation and transmission requirements for a decentralized LQG
control problem”, IEEE Trans. Automatic Control, Vol. AC-24, no. 2, Apr.1979, pp. 266-269.
15. R.M. Yannone, “Expert System in the Fighter of 1990’s”, IEEE Aerospace & Elec- tronic Systems Magazine, Vol. 1, February. 1986, pp.12-16.
16. Ho. P.F. and Tsai W.H., “Target Detection and Targeting in Infrared Image Sequences by Neural Networks Using Self-Organizing Feature Maps”, 1992 IPPR Conference on Computer Vision, Graphics, and Image Processing, Taiwan, pp. 306-313, Aug. 1992.
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