Fast JPDA multitarget tracking algorithm

James L. Fisher; David P. Casasent

doi:10.1364/AO.28.000371

Applied Optics
Vol. 28,
Issue 2,
pp. 371-376
(1989)
•https://doi.org/10.1364/AO.28.000371

Fast JPDA multitarget tracking algorithm

James L. Fisher and David P. Casasent

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James L. Fisher and David P. Casasent, "Fast JPDA multitarget tracking algorithm," Appl. Opt. 28, 371-376 (1989)

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Abstract

Multitarget tracking requires assigning probabilities that measurements are associated with particular targets. A new and efficient algorithm to achieve this is detailed. Its implementation on an analog optical processor using a new and cost-effective frequency-multiplexing technique is discussed.

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Number of Measurements				Number of Targets (T)
Number of Measurements	1	2	3	4	5	6	7	8	9	10

1	2	3	4	5	6	7	8	9	10	11
2	3	7	13	21	31	43	57	73	91	111
3	4	13	34	73	136	229	358	529	748	1021
4	5	21	73	209	501	1045	1961	3393	5509	8501
5	6	31	136	501	1546	4051	9276	19081	36046	63591
6	7	43	229	1045	4051	13327	37633	93289	207775	424051
7	8	57	358	1961	9276	37633	130922	394353	1047376	2501801
8	9	73	529	3393	19081	93289	394353	1441729	4596553	12975561
9	10	91	748	5509	36046	207775	1047376	4596553	17572114	58941091
10	11	111	1021	8501	63591	424051	2501801	12975561	58941091	234662231

4	f0+f1+f2+f3+f4	f₁	1111	e0(f₁)+e4(f₂)+e3(f₃)+e2(f₄)+e1(f₅)
3	f0+f1+f2+f3	f₂	0111	e0(f₂)+e3(f₆)+e2(f₇)+e1(f₈)
	f0+f1+f2+f4	f₃	1011	e0(f₃)+e4(f₆)+e2(f₉)+e1(f₁₀)
	f0+f1+f3+f4	f₄	1101	e0(f₄)+e4(f₇)+e3(f₉)+e1(f₁₁)
	f0+f2+f3+f4	f₅	1110	e0(f₅)+e4(f₈)+e3(f₁₀)+e2(f₁₁)
2	f0+f1+f2	f₆	0011	e0(f₆)+e2(f₁₂)+e1(f₁₃)
	f0+f1+f3	f₇	0101	e0(f₇)+e3(f₁₂)+e1(f₁₄)
	f0+f2+f3	f₈	0110	e0(f₈)+e3(f₁₃)+e2(f₁₄)
	f0+f1+f4	f₉	1001	e0(f₉)+e4(f₁₂)+e1(f₁₅)
	f0+f2+f4	f₁₀	1010	e0(f₁₀)+e4(f₁₃)+e2(f₁₅)
1	f0+f3+f4	f₁₁	1100	e0(f₁₁)+e4(f₁₄)+e3(f₁₅)
	f0+f1	f₁₂	0001	e0(f₁₂)+e1(f₁₆)
	f0+f2	f₁₃	0010	e0(f₁₃)+e2(f₁₆)
	f0+f3	f₁₄	0100	e0(f₁₄)+e3(f₁₆)
	f0+f4	f₁₅	1000	e0(f₁₅)+e4(f₁₆)
0	f0	f₁₆	0000	e0(f₁₆)

Number of Measurements	Number of Targets
Number of Measurements	3	4	5	6	7
3	7×1	8×1	8×1	8×1	8×1
4	11×1	15×1	16×1	16×1	16×1
5	16×1	26×1	31×1	32×1	32×1
6	22×1	42×1	57×1	63×1	64×1
7	29×1	64×1	99×1	120×1	127×1

Number of Measurements	Number of Targets
Number of Measurements	3	4	5	6	7
3	54/68	144/219	300/544	540/1145	882/2148
4	144/146	528/627	1440/2004	3240/5225	6384/11766
5	300/272	1440/1503	5200/6148	15000/20255	36540/55656
6	540/458	3240/3135	15000/16204	55620/66635	170100/225798
7	882/716	6384/5883	36540/37104	170100/188165	652974/785532

Number of Measurements	Number of Targets
Number of Measurements	3	4	5	6	7
3	63/51	132/144	225/280	342/456	483/672
4	132/78	304/260	540/560	840/960	1204/1456
5	240/111	620/432	1175/1045	1890/1920	2765/3024
6	396/150	1152/672	2370/1830	3996/3654	6006/6048
7	609/195	1988/992	4480/3030	8022/6624	12495/11655

Number of Measurements	Number of Targets
Number of Measurements	3	4	5	6	7
3	0.9/1.3	1.1/1.5	1.3/1.9	1.6/2.5	1.8/3.2
4	1.1/1.9	1.7/2.4	2.7/3.6	3.9/5.4	5.3/8.1
5	1.3/2.5	2.3/3.5	4.4/5.9	7.9/10.5	13.2/18.4
6	1.4/3.1	2.8/4.7	6.3/8.9	13.9/18.2	28.3/37.3
7	1.4/3.7	3.2/5.9	8.2/12.2	21.2/28.4	52.3/67.4

Symbol	Description
d	scalar validation threshold constant
∫_t(z_j)	probability density of the single measurement z_j from target t at time k
H_t(k)	coordinate transformation matrix relating the state space of ${\underline{\hat{x}}}_{t}$ (·) and the measurement space of z_t(·) for target t at time k
j	index of validated measurements; j=1, …,m in Section II, and j=0,1, …,m in Section III
k	discrete time index
m	total number of validated measurements at time k
P_D_,_t	probability of detecting target t
P_t(k\|k−1)	a priori state estimate error covariance matrix for target t at time k
P_t(k\|k)	a posteriori state estimate error covariance matrix for target t at time k
P{χ_i\|Z^k} = P{χ_I}	probability that feasible event χ occurred, conditioned on Z^k
S_t(k)	error covariance matrix associated with ${\underline{\hat{z}}}_{t}$ (k\|k−1)
T	total number of established target tracks at time k
t	Index of established targets; t=0,1, …,T in Section II and t=1, …,T in Section III
${\underline{\hat{x}}}_{t}$ (k\|k−1)	a priori state vector estimate for target t at time k
Z^k	set of all validated measurements z_j(k) from time 0 to time k
z_j(k)	measurement vector j (validated or unvalidated) at time k
${\underline{\hat{z}}}_{t}$ (k\|k−1)	a priori predicted measurement location of target t at time k
β_jt(k)	measurement innovations weighting coefficient at time k; the
	probability that measurement j originated from target t
ν_jt(k)	measurement innovations vector for measurement/target pair (j,t) at time k
χ_jt	an event, i.e., a single measurement/target pair
χ_i	a feasible event, a collection of non-competing events χ_jt such that all measurements j have been considered
Ω_i	(binary) feasibility matrix
ω_jt(χ)	event occurrence indicator

Abstract

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Applied Optics