2012 Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF)
Bonn, 4-6 September 2012
Multisensor Traffic Mapping Filters Roy Streit Metron, Inc. Reston, VA (USA)
[email protected] and
[email protected]
Section II reviews the background and history of the PGFL, and its application to multitarget tracking filters. Section III reviews the PGFL form of the Bayes posterior point process. The Bayesian approach is standard, but the random variables involved are finite point processes. Section IV derives the “governing” PGFL of the traffic mapping filter. Sections V and VI derive the traffic filter under the clutter and scattering models, respectively. Section VII gives the probability generating function (PGF) of the cardinal number of sensor detection opportunities of the Bayes posterior traffic process before making the Poisson point process (PPP) approximation. Conclusions are given in Section VIII.
Abstract—A traffic intensity filter is derived using a probability generating functional approach. Traffic filters estimate, or map, the mean rate at which different regions of state space generate target detection opportunities in a field of distributed sensors. They are Bayesian filters that incorporate sensor measurement likelihood functions and target detection capabilities. Traffic maps contribute to situational awareness for heterogeneous sensor fields. They are practical for applications with large numbers of sensors because their computational complexity is linear in the numbers of sensors and measurements. Keywords-Traffic filter; Intensity filter; PHD filter; Probability generating functionals; Finite point processes; Sensor fields; Situational Awareness
I.
II.
INTRODUCTION
The PGFL approach to finite point processes was introduced in 1962 by Moyal [1] and further developed by Daley and Vere-Jones in [2]. Moyal defined the functional derivatives of the PGFL and used them to show that the PGFL characterizes the pdf of the point process. He defined the factorial moments, of which the first moment is the intensity function. Moyal applied his methods to cluster processes and also to time-dependent Markov population processes. The use of finite point processes for modeling multitarget state and the application of PGFLs to derive Bayesian multitarget tracking filters are due to Mahler; see [3, 4] and the numerous references cited therein. He refers to finite point processes as random finite sets and functional derivatives of the PGFL as the FISST calculus. The PGF of the cardinal number density of the general Bayes posterior process—before the PPP approximation—is straightforward but apparently new nonetheless. The cardinal density for one sensor is of special interest. In that case, the traffic number density is identical to the target number density of the PHD filter and the iFilter [6, 11] for, respectively, the clutter and scattering measurement models. The traffic filter assumptions are matched to the physics of spatial radioisotope distributions such as SPECT (single photon emission computed tomography) [9], a widely used medical imaging technology. The traffic filter (26) is identical to the first step of the SPECT algorithm derived in [9] by the EM method. The relationship between medical imaging algorithms (Shepp-Vardi, 1982) and the PHD filter and iFilter is pointed out by Streit [7, 8]. The connection to the Richardson-Lucy (1972/1974) algorithm for image restoration is discussed in [8].
A traffic process is a finite point process that models target detection opportunities aggregated over the entire sensor field, where a detection opportunity comprises both detections and missed detections of targets. Traffic processes are sensorcentric, not target-centric. The output of a traffic filter is a Bayesian estimate of an intensity function over the target state space—its integral is the expected total number of sensor detection opportunities. Intuitively, the intensity function is a kind of situational awareness map that identifies regions with a high rate (i.e., intensity) of detection opportunities. These regions are correlated with regions where targets are present. Traffic filters are practical for large heterogeneous sensor fields because their computational complexity is linear in the numbers of sensors and measurements. In contrast, previously proposed multisensor multitarget tracking filters based on point process target models, e.g., the PHD filter, are impractical due to high computationally complexity. The need to consider alternative models is thus evident. The traffic filter is due to Streit [5], although it was misidentified there as a multisensor target filter. It was derived by a constructive Bayesian method that did not use probability generating functionals (PGFLs). The PGFL derivation here is new. Two summary statistics are obtained, namely, the intensity function and the cardinal number density. Two sensor measurement models are treated. One is the traditional model in which clutter points are superimposed with target measurements. The other is a scattering model in which no distinction is made between clutter- and targetoriginated measurements. Despite their apparent differences, the clutter and scattering models are closely related and lead to only slightly different traffic filters.
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GENERAL BACKGROUND
43
III.
∞ n ∂G Ξ [h ] = ¦ pNΞ (n )¦ ∂w n =1 k =1
PGFLS AND THE BAYES POSTERIOR POINT PROCESS
The PGFL and Bayes Theorem for finite point processes is reviewed. A didactic exposition can be found in [6]. The following properties make PGFLs useful in practice: i. PGFLs fully characterize a finite point process; ii. The PGFL of the superposition of mutually independent finite point processes is the product of their PGFLs. The first property says, in effect, that the PGFL is fundamental—it is the “governing equation” of the process.
∞
n
n =1
k =1
= ¦ pNΞ ( n )¦
h ( si ) .
∏ h( s ) p i
Ξ X |N
( x, s2 ,..., sn | n ) ds2 " dsn ,
i=2
μ = ³ f Ξ ( s )ds < ∞ , where f Ξ ( s ) ≥ 0 is the intensity function S
of the PPP, and points are i.i.d. in S with pdf f Ξ ( s) μ . Direct calculation using (1) gives its PGFL:
∂G Ξ G Ξ [h + ε w] − G Ξ [h ] . (2) [h ] = εlim + →0 ε ∂w Here, w is a specified real-valued function on S . From (1),
(
)
G Ξ [h ] = exp − ³ f Ξ ( s )ds + ³ h( s) f Ξ ( s)ds .
∞
S
(5)
n
ξ = (n,{x1 ,..., xn }) ∈ ( ( S ) . Evaluating it for h ( s ) = 1 gives the factorial moments. Further discussion can be found in [6]. The process Ξ is a PPP on S if the cardinal number N of points in S is Poisson distributed with mean number
B. Functional Derivatives of the PGFL The “finite set statistics (FISST) calculus” concerns functional differentiation of PGFLs, where functional differentiation has exactly the same meaning as in the classical Calculus of Variations. The functional derivative of G Ξ [h] with respect to the variation w is defined by
n
h( si ) p ΞX |N ( s1 ,..., sn | n) ds1 " dsn .
i =1, i ≠ k
C. Probability Density and Intensity Functions Evaluating (5) for h ( s ) = 0 gives the pdf of the event (1,{x}) ∈ ( ( S ) . Evaluating (5) for h ( s ) = 1 gives what is called the first moment of the process. The first moment is a function of x . The functional derivative of G Ξ [h ] with respect to distinct impulses at x1 ,..., xn is easily derived. Evaluating it at h ( s ) = 0 gives the pdf of the event
h ( si ) is defined
n =0
∏
x (⋅)
specified variation w is the Dirac delta function δ x .
to be one. The PGFL is evaluated only for functions h such that the sum and the integrals in (1) are absolutely convergent. Simply put, the set integral (1) is the expectation of the
׳
δ x ( sk )
n
but its derivative ∂G Ξ [h ] ∂x does. This is because the
s ∈ S , p ( n ) is the discrete pdf (i.e., the probability mass function) of the random cardinal number N of points, and pΞX |N ( s1 ,..., sn | n) is the conditional pdf of the n-tuple
G Ξ [ h + ε w] = ¦ p NΞ ( n )
( s1 ,..., sn | n ) ds1 " dsn .
where the product in (5) is equal to one for n = 1 . The integrals are over S n −1 , not S n . The PGFL G Ξ [h ] does not itself depend on the point x ,
Ξ N
N
Sn
S n −1
(1) § n · Ξ × ³ n ¨ ∏ h ( si ) ¸ p X |N ( s1 ,..., sn | n ) ds1 " dsn , S © i =1 ¹ where h is a real-valued function such that h( s ) ≤ 1 for
i =1
³
׳
n =0
∏
h( si ) p
Simplifying gives ∞ ∂G Ξ [ h ] = ¦ pNΞ (n ) n ∂x n =1
G Ξ [h ] = ¦ pNΞ ( n )
random product
∏
∂G Ξ ∂G Ξ ∂G Ξ [h] ≡ [h] = [h] ∂x ∂δ x ∂w w(⋅)=δ
∞
n
(4) Ξ X |N
Note that the sum starts at n = 1 . Specifying the variation w( s ) = δ x ( s) ≡ δ ( s − x ) to be the Dirac delta function with an impulse (point mass) at s = x ∈ S defines the “set derivative”:
either \ nx , nx ≥ 1 , or some specified subset thereof. The PGFL of the finite point process Ξ is defined by
i =1
w( sk )
i =1, i ≠ k
pairs of the form (n,{s1 ,..., sn }), si ∈ S , and S is the state space. For n = 0 , the event is (0, ∅ ) . The state space S is
∏
Sn
n
×
A. Event Space and Definition of the PGFL The finite point process Ξ is a random variable defined on the event space ( ( S ) , where ( ( S ) is the set of all ordered
( s1 ,..., sn ) . For n = 0 , p ΞX |N (⋅ | n ) = 1 and
³
S
S
It is straightforward to verify for PPPs that ∂G Ξ f Ξ ( x) = [h ] . ∂x h (⋅)≡1
(3)
[ h( si ) + ε w( si ) ] p XΞ |N ( s1 ,..., sn | n ) ds1 " dsn . n ∏ i =1
(6)
(7)
For this reason, the first moment of general finite point processes is often called the intensity function.
Moyal [1, Section 4] proves that (3) is an analytic function of ε in some open region of the complex plane containing the origin. Using equation (2) gives, since integrals and sums are absolutely convergent,
D. Joint Point Processes and Their PFGLs Let ϒ be a finite point process with events υ = (m,{ y1 ,..., ym }) ∈ ( (Y ) , where the space Y is in general unrelated to the space S . The events of the joint point process
44
( ϒ, Ξ) are elements of the Cartesian product ( (Y ) × ( ( S ) . Extending the definition (1) to the joint process ( ϒ, Ξ) gives ∞
1 ∂ m +1G ϒΞ [0,1], x ∈ S . (14) p (υ ) ∂z1 " ∂zm ∂x Expressions (13) - (14) hold for general finite point processes. The cardinal number of points in the Bayes posterior process Ξ | ϒ is N Ξ|ϒ . The discrete probability mass function f Ξ |ϒ ( x ) =
· § m ·§ n g ( y ) ∏ i ¸ ¨ ∏ h( s j ) ¸ S n ³Y m ¨ © i =1 ¹ © j =1 ¹ (8)
∞
ϒΞ G ϒΞ [ g , h ] = ¦¦ pMN ( m, n ) ³ m=0 n =0
ϒΞ YX |MN
×p
ϒΞ MN
( y1 ,..., ym , s1 ,..., sn | m, n ) dy1 " dym ds1 " dsn ,
N Ξ|ϒ is called the cardinal number density. The PGF of N Ξ|ϒ is the GPFL (13) evaluated for the constant functions h( s ) = x ; that is, the PGF of N Ξ|ϒ is
ϒΞ YX | MN
(⋅) and p (⋅) are the discrete and continuous pdfs of the joint process ( ϒ, Ξ) . The functions g and h are defined on Y and S , respectively. The products in (8) are one for m = 0 and n = 0 . The PGFL reduces to those of the single processes: G Ξ [ h ] = G ϒΞ [1, h ] and G ϒ [ g ] = G ϒΞ [ g ,1] . The joint PGFL can sometimes be written as a composition of generating functions. Such compositions are a hallmark of the classical theory of branching processes (see, e.g., [10]). The branching process form is apparent in the point processes defined below by governing equations (19) and (21). where p
F Ξ |ϒ ( x ) =
p ϒΞ (υ , ξ ) , p ϒ (υ )
IV.
(9)
Taking functional derivatives of G ϒΞ [ g , h ] with respect to
(10)
where υ = (m,{z1 ,..., zm }) and ξ = (n,{x1 ,..., xn }) . The pdf of the unconditional process ϒ is, since G ϒ [ g ] = G ϒΞ [ g ,1] ,
∂ mG ϒΞ [0,1] . ∂z1 " ∂zm The pdf (9) of the Bayes posterior point process Ξ | ϒ is p ϒ (υ ) =
1 ∂ m + n G ϒΞ [0,0] . p ϒ (υ ) ∂z1 " ∂zm ∂x1 " ∂xn This expression holds for general finite point processes. p Ξ|ϒ (ξ | z1 ,..., zm ) =
(16)
TRAFFIC PROCESSES
A. Target and Traffic State Processes To reduce the notational burden, the target and traffic processes are the point processes predicted to the current time from the previous time step. For the filters considered here, Ξ and Σ are PPPs because the Bayes posterior processes at the previous time step are approximated by PPPs to close the Bayes recursions. Prediction involves thinning, superposition, and Markovian target motion with transition function Ψ (⋅ | ⋅) . For clutter models, Ψ (⋅ | ⋅) is defined on S × S . For scattering
impulses at z1 ,..., zm and x1 ,..., xn gives
∂ m + n G ϒΞ [0,0], ∂z1 " ∂zm ∂x1 " ∂xn
(15)
A traffic process Σ is a finite point process that models the total number and locations of sensor detection opportunities. The word “traffic” relates to the notion that sensor detections generate message traffic on a communications network. Missed detections correspond to missed messages. The traffic count includes both detections and missed detections, i.e., traffic comprises all target detection opportunities.
where p ϒΞ (υ , ξ ) is the pdf of the joint process ( ϒ, Ξ) .
p ϒΞ (υ , ξ ) =
1 ∂ mG ϒΞ [0, h ] . p (υ ) ∂z1 " ∂zm h (⋅) ≡ x ϒ
The cardinal number density is a discrete pdf, hence 1 d n Ξ |ϒ p NΞ|ϒ ( n ) = F (0) . n ! dx n
E. PGFL Form of Bayes Theorem By Bayes Theorem, the pdf of the process Ξ | ϒ is
p Ξ|ϒ (ξ | υ ) ≡
ϒ
(11)
models it is defined on S + × S + , where S + = S ∪ φ and φ ∉ S . The point φ plays an important role, as will be seen in the next subsection. Prediction does not alter the character of the process models—if the process is a PPP at the previous time step, the predicted process is a PPP at the current time. This elementary fact can be proved in several ways [8]. A critique of the point process approximation for targets is given in [6]. An important distinction is that the integral of the intensity of the target process is the expected number of targets, while the integral of the intensity of the traffic process is the expected total number of sensor detection opportunities.
(12)
F. Summary Statistics of the Bayes Posterior Process Since the event space ( ( S ) is very large, it is useful to provide summary statistics of the posterior process Ξ | ϒ . Two statistics are of interest here. The process Ξ | ϒ is a point process on S . Its intensity is the first functional derivative of its PGFL evaluated at h = 1 . Using (12) in (5), and modifying the notation to accommodate conditioning on ϒ = υ , gives 1 ∂ mG ϒΞ G Ξ|ϒ [h | z1 ,..., zm ] = ϒ [0, h ]. (13) p (υ ) ∂z1 " ∂zm
B. Sensor Measurement Processes Let L ≥ 1 denote the number of sensors. The measurement process ϒ A , A = 1,..., L , is assumed to be a finite point process on the measurement space Y A , that is, it is a random variable on the event space ( (Y A ) . The sensors are independent when conditioned on the state of the target process Ξ . The sensor
The intensity, f Ξ|ϒ ( x ) , of Ξ | ϒ is the functional derivative of (13) with respect to an impulse at x ∈ S and evaluated at h =1:
45
measurement pdf of sensor A is p A ( y | s ) , and the probability
L sensors. An important, but surprising, property of PPPs is that the sensor processes Σ A are mutually independent PPPs when the assignments are independent. This is called the Coloring Theorem, where in this case the colors (i.e., labels) are the sensors. Further discussion can be found in, e.g., [8]. The probability of detection P D A ( s ) is the probability that a target at s is detected by sensor A , given that it is present. The probability β A ( s ) is very different—it represents the fraction of the total number of detection opportunities that are detection opportunities for sensor A . Thus, β A ( s ) is a fieldlevel probability that depends on geometrical considerations concerning the point s and the entire sensor field, i.e., the constellation. Consequently, it is unrelated to the sensor-level probability P D A ( s ) .
of target detection is P DA ( s ) . Two different measurement models are used. The traditional model is an exogenous model, in which clutter points are superposed with target measurements to produce the sensor measurement set. The other model is an endogenous scattering model—it makes no distinction between scatterers that are targets and those that are clutter. It assumes all sensor measurements are generated by scatterers with states in the augmented space S + = S ∪ φ , where φ corresponds to scatterers that are not assigned a definite state in S . The pdfs p A ( y | s ) and P DA ( s ) are also defined for s ∈ S + . Thus,
p A ( y | φ ) is the pdf of measurements due to a scatterer whose state in S is unknown, i.e, whose state is φ , and P DA (φ ) is the probability that a scatterer in φ generates a measurement. To compare the scattering model with the traditional clutter model, interpret a scatterer in state s ∈ S as a target in the same state, and a scatterer in state φ as a clutter generator that, like a target, produces at most one measurement. Scatterer transitions Ψ ( s | φ ) and Ψ (φ | s ) are interpreted as target birth and death, respectively, while Ψ (φ | φ ) is the probability that a scatterer in φ remains in φ . Because φ is a discrete state, realizations of the PPP multitarget state process on S + can have multiple points in φ . Thus, the scattering model can accommodate variable numbers of what would be considered clutter points in the traditional model.
D. PGFL of the Traffic Process The PGFL of Σ A is (cf. (6)) G Σ A [h ] = exp ª − ³ f Σ A ( s ) ds + ³ h( s ) f Σ A ( s ) ds º , (18) S ¬ S ¼ ΣA where f ( s ) is given by (17). The joint sensor-level processes ( ϒ A , Σ A ) are parametrically tied to Σ through the intensity function f Σ , but they are independent processes because Σ A are mutually independent and ϒ A is conditionally independent of all processes except Σ A . The sensor PGFL is the same as that of a PHD filter: A A A A A G ϒ Σ [ g A , h ] = G Clutter [ g A ] G Σ A ª h( s )G ϒ |Σ [ g A | s ]º . (19) ¬ ¼ The sensor processes are mutually independent, so the PGFL of the joint process ( ϒ1 ," , ϒ L , Σ) is the product:
C. Superposition of Sensor Traffic Processes The traffic process Σ is a PPP on the space S or on the space S + , depending on whether on the measurement model is the clutter model or the scattering model, respectively. Either way, the intensity function is denoted by f Σ (⋅) . The traffic process Σ models the totality of target detection opportunities across the entire sensor field. Thus, it is the superposition of mutually independent sensor PPPs, denoted by Σ A , whose intensity functions are f ΣA (s) = β A (s) f Σ (s) , (17) where, for each s ∈ S , β A ( s ) ≥ 0 and
L
G ϒ "ϒ Σ [ g 1 ,..., g L , h ] = ∏ G ϒ Σ [ g A , h ] . 1
L
A A
(20)
A =1
The PGFLs use the same function h (not different functions h1 ,..., h L ) because Σ is the superposition of the processes Σ A . The joint PGFL (20) fully characterizes the Bayes posterior traffic process, i.e., it is the governing equation of the process. In sharp contrast, the PGFL of the multisensor target process for the clutter model is very different from (20):
G ϒ "ϒ Ξ [ g 1 ,..., g L , h ] 1
L ¦ A=1 β A ( s ) = 1 . To see
L
L A ª º = G Clutter [ g 1 ,..., g L ] G Ξ « h ( s )∏ G ϒ [ g A | s ]» , A =1 ¬ ¼ where G Ξ [⋅] is the PGFL of the target process and
this, note that f Σ ( s ) Δs is the expected total number of sensor detection opportunities of targets in a region Δs of state space located at s and of volume Δs . For Δs sufficiently small,
L
G Clutter [ g 1 ,..., g L ] = ∏ G Clutter [ g A ] ,
realizations of Σ have at most one point in Δs . The sensor to which this point (i.e., detection opportunity) corresponds is unknown. Let β A ( s ) denote the probability that it corresponds
A
(21)
(22)
A =1
A
where G Clutter [ g A ] is the PGFL for the clutter on sensor A . The PGFL (21) uses only one function h because there is one target process. The number of terms in the functional derivatives of (21) is too large for practical application. The functional derivatives of the multisensor target PGFL for the scattering model are also high complexity. For further discussion, see [6].
to sensor A . Thus, β A ( s ) f Σ ( s ) Δs is the mean number of sensor detection opportunities of targets in Δs by sensor A . Dividing by Δs gives (17). The probability functions β 1 ( s ),..., β L ( s ) are assumed to independently assign a point s of the process Σ to one of the
46
V.
computational complexity of the summand depends on the number of particles employed. The integral 1 L Nˆ traffic ( S ) = ³ f Σ|ϒ "ϒ ( s ) ds (27)
TRAFFIC MAPPING FILTER WITH CLUTTER MODEL
In the clutter model, the measurement set for sensor A is the superposition of exogenous PPP clutter process with intensity λ A ( y ) and a number of target processes. The conditional PGFL for sensor A is exactly the same as for a PHD filter: A
(
A
G ϒ |Σ ª¬ g A | s1 ,..., sn º¼ = exp − ³ A λ A ( y )dy + ³ A g A ( y )λ A ( y )dy Y Y n
(
S
is the expected traffic, i.e., the number of sensor detection opportunities of targets in S , not the number of targets in S .
)
VI.
)
× ∏ 1 − P D A ( si ) + P D A ( si ) ³ A g A ( y ) p A ( y | si ) dy , i =1
Y
In this model, every measurement originates from a scatterer (source), and no distinction is made between scatterers that are targets and those that are clutter. As discussed in Section IV.B, to accommodate measurements that correspond to previously unknown scatterers, the state space S is augmented by the state φ to denote a scatterer whose precise location in S is unknown. Let S + = S ∪ φ . Integrals over S + are defined for real-valued
(23) where g ( y ) is a function defined on Y . Using (23) in the A
A
joint PGFL (19) and the PGFL (18) of Σ A gives the joint PGFL of ( ϒ A , Σ A ) . Substituting this expression (omitted due to lack of space) into (20) and substituting (17) gives the PGFL of the joint process ( ϒ1 ," , ϒ L , Σ) as
functions h( s ) , s ∈ S + , by
G ϒ "ϒ Σ [ g 1 ,..., g L , h ] 1
L
³
{
= exp ª ¦ A =1 − ³ A λ A ( y ) dy + ³ A g A ( y )λ A ( y ) dy Y Y ¬ L
S+
− ³ β ( s ) f ( s ) ds + ³ h( s ) β ( s ) f ( s ) ds A
Σ
A
S
Σ
S
³
A
DA
A
}
Σ
S YA
The interpretation of f Σ ( s) is unchanged for s ∈ S ; however, the intensity f Σ (φ ) is the expected number of scatterers with state φ , i.e., it is dimensionless. This is consistent with (28). The measurement set for sensor A is the superposition of a number of target processes. These processes are conditionally independent, so the PGFL for sensor A is A A G ϒ |Σ ª¬ g A | s1 ,..., sn º¼ n (29) = ∏ 1 − P D A ( si ) + P D A ( si ) ³ A g A ( y ) p A ( y | si ) dy .
Z A = {z1A ,..., zmA A }, ziA ∈ Y A ,
is L
1
1 1 L L ∂ m +"+ m G ϒ "ϒ Σ [ g 1 ,..., g L , h ] = G ϒ "ϒ Σ [ g 1 ,..., g L , h ] ∂Z 1 " ∂Z L
L
mA
{
}
(25)
× ∏∏ λ ( z ) + ³ h( s ) p ( z | s ) P ( s ) β ( s ) f ( s )ds . A =1 i =1
A
A i
A
S
A i
DA
A
Σ
(28)
S
The intensity function f Σ ( s ) is defined on S + = S ∪ φ .
g ( y )h( s ) p ( y | s ) P ( s ) β ( s ) f ( s ) dyds º . ¼ The functional derivative of (24) with respect to impulses at the measurement set Z ≡ ( Z 1 ,..., Z L ) , where for sensor A +³
A
h( s ) ds ≡ h(φ ) + ³ h( s ) ds .
This definition is used in the PGFL. Functional derivatives extend to S + by defining the Dirac delta function so that δφ (φ ) = 1 and δ x (φ ) = δφ ( s ) = 0 for s, x ∈ S .
(24)
S
− ³ h( s ) P D A ( s ) β A ( s ) f Σ ( s ) ds
TRAFFIC MAPPING FILTER WITH SCATTERING MODEL
i =1
(
)
Y
The clutter term in (23) is noticeably absent from (29). The joint PGFL of ( ϒ A , Σ A ) is the same as (19), but without the
A final derivative is needed with respect to an impulse at x ∈ S . This derivative is straightforward to evaluate, but tedious. Details are omitted. The traffic filter for clutter is found by substituting (25) and the (omitted) derivative with respect to x into expressions (11) and (14). The result is L ª 1 L f Σ|ϒ "ϒ ( x ) = f Σ ( x )¦ β A ( x ) «1 − P D A ( x ) A =1 ¬ (26) º mA P D A ( x ) p A ( ziA | x ) ». + ¦ A A A A DA ( s ) β A ( s ) f Σ ( s ) ds » i =1 λ ( zi ) + ³ p ( zi | s ) P S ¼ The intensity function (26) has units of number per unit state space. For L = 1 the filter (26) reduces to the usual (single sensor) PHD filter. The computational complexity of the filter (26) is of linear computational complexity in the number of sensors L and the number of measurements M = m1 + " m L . In sequential Monte Carlo (SMC), or particle filter, implementations, the
clutter term. The PGFL of ( ϒ1 ," , ϒ L , Σ) is, from (20), 1
L
G ϒ "ϒ Σ [ g 1 ,..., g L , h ] = exp ª ¦ A =1 ¬ L
{− ³
S+
β A ( s ) f Σ ( s ) ds + ³ h( s ) β A ( s ) f Σ ( s ) ds S+
− ³ + h ( s ) P D A ( s ) β A ( s ) f Σ ( s ) ds S
}
g A ( y )h ( s ) p A ( y | s ) P D A ( s ) β A ( s ) f Σ ( s ) dyds º . ¼ The functional derivatives with respect to the data Z are +³
S+
1
³
YA
L
1 L 1 L ∂ m +"+ m G ϒ "ϒ Σ [ g 1 ,..., g L , h ] = G ϒ "ϒ Σ [ g 1 ,..., g L , h ] 1 L ∂Z " ∂Z
L
mA
(30)
× ∏∏ ³ + h( s ) p ( z | s ) P ( s ) β ( s ) f ( s )ds. A =1 i =1
A
S
A i
DA
A
Σ
The derivative of this expression with respect to an impulse at x ∈ S + is straightforward. Details are omitted.
47
The traffic filter for the scattering model is found by substituting (30) and the (omitted) derivative with respect to x into expressions (11) and (14). The result is
f
1
Σ| ϒ " ϒ
L
L
A =1
traffic
A
A i A
S
μφ = (1 − P DA (φ ) ) β A (φ ) f Σ (φ ) .
The third PGF is that of a process that always produces m A scatterers. The total traffic count is at least the total number of measurements, M . The traffic count includes scatterers in state φ , and these can occur with multiplicity greater than one. The PGF of the traffic for targets just in S is not available. VIII. CONCLUSIONS A traffic mapping filter is a sensor-centric filter whose output is a map of the rate at which different regions of target state space generate traffic, i.e., target detection opportunities. High activity regions in the traffic map are correlated with regions containing targets, so they are potentially useful for subsequent search and resource allocation decisions. Traffic filters are of linear computational complexity in the numbers of sensors and measurements. This makes them suitable for applications with large sensor fields.
³
S+
REFERENCES [1]
J.E. Moyal, “The General Theory of Stochastic Population Processes,” Acta Mathematica, vol. 108, 1-31, 1962. [2] D.J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure, Springer, 1988 (Second Edition, 2008). [3] R.P.S. Mahler, “Multitarget Bayes Filtering via First-Order Multitarget Moments,” IEEE Trans. AES, vol. AES-39, 1152-1178, 2003. [4] R.P.S. Mahler, Statistical Multisource-Multitarget Information Fusion, Artech House, Boston, 2007. [5] R. Streit, “Multisensor Multitarget Intensity Filter,” 11th International Conference on Information Fusion, Cologne, Germany, June 2008. [6] R. Streit, “The Probability Generating Functional for Finite Point Processes, and Its Application to the Comparison of PHD and Intensity Filters,” Journal of Advances in Information Fusion, submitted 9 January 2012. [7] R.L. Streit, “PHD Intensity Filtering Is One Step of an EM Algorithm for Positron Emission Tomography,” 12th International Conference on Information Fusion, Seattle, July 2009. [8] R.L. Streit, Poisson Point Processes—Imaging, Tracking, and Sensing, Springer, New York, 2010. [9] M.I. Miller, D.L. Snyder, and T.R. Miller, “Maximum-Likelihood Reconstruction for Single-Photon Emission Computed Tomography,” IEEE Trans. on Nuclear Science, NS-32(1), 769-778, 1985. [10] K.B. Athreya and P.E. Ney, Branching Processes, Springer-Verlag, 1972. (Reprinted by Dover, 2004) [11] R.L. Streit and L.D. Stone, “Bayes Derivation of Multitarget intensity Filters,” 11th International Conference on Information Fusion, Cologne, Germany, June 2008.
VII. CARDINAL NUMBER DENSITY OF THE BAYES POSTERIOR TRAFFIC PROCESS BEFORE PPP APPROXIMATION A. Clutter Model The PGF of the total traffic count of the Bayes posterior traffic process before PPP approximation is given by (15). In this case, using the derivative (25) gives L ° Σ| ϒ F ( x ) = ∏ ®exp ª( x − 1) ³ (1 − P DA ( s ) ) β A ( s ) f Σ ( s )ds º S ¬ ¼ A =1 ° ¯
p A ( ziA | s ) P DA ( s ) β A ( s ) f Σ ( s )ds ½° ×∏ A A ¾. DA A A ( s ) β A ( s ) f Σ ( s )ds ° i =1 λ ( zi ) + ³ p ( zi | s ) P S ¿ This PGF is the product of L sensor-level PGFs, where each sensor PGF is the product of two PGFs. This means that the total traffic count, before PPP approximation, is the convolution of the traffic counts in each sensor.
λ A ( ziA ) + x
}
A
μS = ³ (1 − P DA ( s ) ) β A ( s ) f Σ ( s ) ds
The traffic in S is same as (27) but uses the intensity (33).
mA
{
The PGF is the product of L sensor-level PGFs, each of which is itself the product of three PGFs. Two of the PGFs are for Poisson distributed scatterers with means
(31) º m P ( x ) p ( z | x) ». +¦ DA A A ( s ) β ( s ) f Σ ( s ) ds » i =1 ³ + p ( zi | s ) P S ¼ Expanding the integral in (31) using (28) and substituting λˆ A ( z ) = p A ( z | φ ) P DA (φ ) β A (φ ) f Σ (φ ) (32) gives the equivalent form L ª 1 L f Σ|ϒ "ϒ ( x ) = f Σ ( x )¦ β A ( x ) «1 − P D A ( x ) A =1 ¬ (33) º mA P D A ( x ) p A ( ziA | x ) ». +¦ A A DA A A ˆ ( s ) β A ( s ) f Σ ( s ) ds » i =1 λ ( zi ) + ³ p ( zi | s ) P S ¼ The traffic filter (33) reduces to (26) by restricting the filter to S and assuming a priori known sensor clutter models. The computational complexity of the filter (33) is linear in the number of sensors L and the total number of measurements M . The complexity of the summand is governed by the number of SMC particles employed. The expected total traffic count in S + is 1 L (S + ) = Nˆ f Σ|ϒ "ϒ ( s ) ds . (34) DA
}
A
≡ ∏ x m exp [( x − 1) μS ] exp ¬ª ( x − 1) μφ ¼º .
ª ( x ) = f Σ ( x )¦ β A ( x ) «1 − P D A ( x ) A =1 ¬ L
A
{
L
F Σ|ϒ ( x ) = ∏ x m exp ª( x − 1) ³ + (1 − P DA ( s ) ) β A ( s ) f Σ ( s )ds º ¬ ¼ S A =1
³
S
B. Scattering Model In this case, using the derivative (30) gives the PGF of the total traffic count due to scatterers in S + :
48
