8.4    GENERAL MULTITARGET DISTRIBUTED FUSION

In this section, I show how to directly generalize the single-target T²F theory of Section 8.2 to the multitarget situation. The section is organized as follows:

1.  Section 8.4.1: Multitarget T²F when the track sources are independent. Approach: multitarget generalization of Equation 8.10.

2.  Section 8.4.2: Multitarget T²F when the track sources are dependent because of known double-counting. Approach: multitarget generalization of Equation 8.16.

3.  Section 8.4.3: Multitarget T²F when the track sources are arbitrary and their correlations are completely unknown. Approach: multitarget generalization of XM fusion, Equations 8.34, 8.35, 8.36, 8.37, 8.38.

8.4.1    MULTITARGET T²F OF INDEPENDENT SOURCES

Suppose that multiple targets are being tracked, and that s independent sources, relying on their own dedicated local sensors, provide track data about these targets to a T²F site. The jth sensor suite collects a time-sequence $\stackrel{j}{Z^k}:\stackrel{j}{Z_1},\dots \stackrel{j}{Z_k}$ where $\stackrel{j}{Z_l}$ denotes the set of measurements supplied by the jth source’s sensors at time t_l. The source does not pass this information directly to the fusion site. Rather, it passes the following information:

•  Measurement-updated multitarget track data, in the form of multitarget probability distributions ${\stackrel{j}{f}}_{k|k}(X)\stackrel{\text{abbr}}{=}{\stackrel{j}{f}}_{k|k}(X|\stackrel{j}{Z^{(k)}})$

•  Time-updated multitarget track data, in the form of multitarget distributions ${\stackrel{j}{f}}_{k+1|k}(X)\stackrel{\text{abbr}}{=}{\stackrel{j}{f}}_{k+1|k}(X|\stackrel{j}{Z^{(k)}})$

Let $f_{k|k}(X)\stackrel{\text{abbr}}{=}{f}_{k|k}(X|Z^{(k)})$ be the fusion node’s determination of the multitarget state, given all the accumulated track data supplied by the sensor sources. Then the exact multitarget generalization of Equation 8.10 is the following multitarget track-merging formula, first introduced in Ref. [3]:

$f_{k+1|k+1}(X)\propto \frac{{\stackrel{1}{f}}_{k+1|k+1}(X)}{{\stackrel{1}{f}}_{k+1|k}(X)}\cdots \frac{{\stackrel{s}{f}}_{k+1|k+1}(X)}{{\stackrel{s}{f}}_{k+1|k}(X)}\cdot f_{k+1|k}(X).$

(8.89)

This is the fundamental formula for multitarget T²F with independent sources. As in Section 8.2.2, it is being assumed here that the sources provide their data in lockstep, simultaneously at every time-step. Once again, however, it also applies to the asynchronous case. If each source has its own data rate, then the measurement-collection times t₁,…,t_k can be taken to refer to the arrival times of data from all of the track sources, taken collectively. If at time t_l only s_l of the sources provide data, then Equation 8.10 is replaced by the corresponding formula for only those sources.

The approximations described in Section 8.2.2 apply equally well here. Suppose that the sources do not pass on their time-update track data but, rather, only their measurement-update track data. Presume that all of the sources employ identical target motion models. Then (in principle) the fusion site can construct time-update track data for the sources, using the multitarget prediction integral

${\stackrel{j}{f}}_{k+1|k}(X)={\displaystyle \int f_{k+1|k}(X|X^{\prime })\cdot {\stackrel{j}{f}}_{k|k}(X)\text{δ}X,}$

(8.90)

and then apply Equation 8.89.

Alternatively, assume that the sources’ time-updated track data is identical to the fusion site’s: ${\stackrel{j}{f}}_{k+1|k}(X)=f_{k+1|k}(X)$ for all j = 1, …, s. Then Equation 8.89 reduces to

$f_{k+1|k+1}(X)\propto {\stackrel{1}{f}}_{k+1|k+1}(X)\cdots {\stackrel{s}{f}}_{k+1|k+1}(X)\cdot f_{k+1|k}{(X)}^{1-s},$

(8.91)

which is the multitarget version of Bayes parallel combination, Equation 8.14.

Equations 8.89 and 8.91 are computationally intractable in general. The task of devising more tractable approximations of them will be taken up in Sections 8.5.1 and 8.5.2.

8.4.2    MULTITARGET T²F WITH KNOWN DOUBLE-COUNTING

Suppose now that the data sources share sensors, but that it is known which sensors are being shared by which sources. As in Section 8.2.3, define $Z_{k+1}={\stackrel{1}{Z}}_{k+1}\cup \dots \cup {\stackrel{s}{Z}}_{k+1}$. Let ${\stackrel{12}{Z}}_{k+1}$ be the measurements supplied to the second source that are not in ${\stackrel{1}{Z}}_{k+1},{\stackrel{13}{Z}}_{k+1}$ the measurements supplied to the third source that are not in ${\stackrel{1}{Z}}_{k+1}\cup {\stackrel{12}{Z}}_{k+1},{\stackrel{14}{Z}}_{k+1}$ measurements supplied to the fourth source that are not in ${\stackrel{1}{Z}}_{k+1}\cup {\stackrel{12}{Z}}_{k+1},{\stackrel{13}{Z}}_{k+1}$, and so on. Let ${\stackrel{(j)}{Z}}_{k+1}={\stackrel{j}{Z}}_{k+1}-{\stackrel{1j}{Z}}_{k+1}$. Then the multitarget version of Equation 8.16 is

$f_{k+1|k+1}(X)\propto \frac{{\stackrel{1}{f}}_{k+1|k+1}(X)}{{\stackrel{1}{f}}_{k+1|k}(X)}.\frac{{\stackrel{2}{f}}_{k+1|k+1}(X)}{{\stackrel{2}{f}}_{k+1|k}(\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{f}}_{k+1|k+1}(X)}{{\stackrel{s}{f}}_{k+1|k}(\stackrel{(s)}{Z})}\cdot f_{k+1|k}(X).$

(8.92)

This is the fundamental formula for multitarget T²F with known double-counting. As in Section 8.2.3, the jth source must know which sensors it shares with each of sources 1, …, j − 1, and must pass on ${\stackrel{j}{f}}_{k+1|k}=(x|\stackrel{(j)}{Z})$ in addition to ${\stackrel{j}{f}}_{k+1|k}=(x)$.

Equation 8.92 is computationally intractable in general. More tractable approximations of it will be taken up in Sections 8.5.3 and 8.5.4.

8.4.3    MULTITARGET XM FUSION

Suppose that multiple targets are being observed by two track sources. At time-step k, the first source provides a multitarget distribution ${\stackrel{1}{f}}_{k+1|k+1}(X)$ and the second source provides a multitarget distribution ${\stackrel{2}{f}}_{k+1|k+1}(X)$. Then the multitarget version of the single-target XM fusion formula, Equation 8.34, is [3]

${\stackrel{\text{ω}}{f}}_{k+1|k+1}(X)=\frac{{\stackrel{1}{f}}_{k+1|k+1}{(X)}^{1-\text{ω}}\cdot {\stackrel{2}{f}}_{k+1|k+1}{(X)}^{\text{ω}}}{{\displaystyle \int {\stackrel{1}{f}}_{k+1|k+1}{(Y)}^{1-\text{ω}}\cdot {\stackrel{2}{f}}_{k+1|k+1}{(Y)}^{\text{ω}}\text{δ}Y}}.$

(8.93)

This is the general formula for the XM fusion of multitarget track sources with completely unknown correlations. As I did in Ref. [3] and at the end of Section 8.2.5, I argue that the most theoretically reasonable optimal XM fusion procedure is as follows:

${\stackrel{\text{XM}}{f}}_{k+1|k+1}(X)={\stackrel{\widehat{\text{ω}}}{f}}_{k+1|k+1}(X)$

(8.94)

where

$\widehat{\omega }=\arg \underset{\text{ω}}{\sup }\underset{X}{\sup }{\stackrel{\text{ω}}{f}}_{k+1|k+1}(X)\cdot \frac{c^{|X|}}{|X|!}$

(8.95)

where c is as defined in Equation 8.51: a fixed constant which has the same units of measurement as the single-target state x.

However this may be, Equations 8.93, 8.94, 8.95 are computationally intractable in general. More tractable approximations of these equations will be taken up in Sections 8.5.5 and 8.5.6.

8.5    CPHD/PHD FILTER DISTRIBUTED FUSION

In this section, I derive CPHD filter-based approximations of the multitarget T²F approaches described in Section 8.4. The section is organized as follows:

1.  Sections 8.5.1 and 8.5.2: Multitarget T²F when the track sources are independent. Approach: CPHD and PHD filter approximations of Equation 8.89.

2.  Section 8.5.3: Multitarget T²F when the track sources are dependent because of known double-counting. Approach: CPHD and PHD filter approximations of Equation 8.92.

3.  Sections 8.5.5 and 8.5.6: Multitarget T²F when the track sources are arbitrary and their correlations are completely unknown. Approach: CPHD and PHD filter approximations of Equation 8.93, as proposed by Clark et al.

8.5.1    CPHD FILTER T²F OF INDEPENDENT SOURCES

Suppose as in Section 8.4.1 that multiple targets are being tracked, and that s independent sources, relying on their own dedicated local sensors, provide track data about these targets to a T²F site. The jth sensor suite collects its measurements, processes them using a CPHD filter, and then passes on the following to a central T²F site:

•  Measurement-update multitarget track data, in the form of spatial distributions ${\stackrel{j}{s}}_{k|k}(x)\stackrel{\text{abbr}}{=}{\stackrel{j}{s}}_{k|k}(x|\stackrel{j}{Z^{(k)}})$ and cardinality distributions ${\stackrel{j}{p}}_{k|k}(n)\stackrel{\text{abbr}}{=}{\stackrel{j}{p}}_{k|k}(n|\stackrel{j}{Z^{(k)}})$

•  Time-update multitarget track data, in the form of spatial distributions ${\stackrel{j}{s}}_{k|k+1}(x)\stackrel{\text{abbr}}{=}{\stackrel{j}{s}}_{k|k+1}(x|\stackrel{j}{Z^{(k)}})$ and cardinality distributions ${\stackrel{j}{p}}_{k|k+1}(n)\stackrel{\text{abbr}}{=}{\stackrel{j}{p}}_{k|k+1}(n|\stackrel{j}{Z^{(k)}})$

Then the multitarget track merging formula—i.e., a CPHD filter approximation of Equation 8.89—is as follows (see Section 8.7.1):

$p_{k+1|k+1}(n)=\frac{1}{{\text{μ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n)}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)$

(8.96)

$s_{k+1|k+1}(x)=\frac{1}{{\text{σ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x)}\cdot s_{k+1|k}(x)$

(8.97)

where

${\text{μ}}_{k+1|k+1}{\displaystyle \sum _{n>0}\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n)}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)}$

(8.98)

${\text{σ}}_{k+1|k+1}{\displaystyle \int \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x)}\cdot s_{k+1|k}(x)d}x.$

(8.99)

Suppose that we use the approximations $p_{k+1|k}(n)={\stackrel{1}{p}}_{k+1|k}(n)=\cdots ={\stackrel{s}{p}}_{k+1|k}(n)$ and $s_{k+1|k}(x)={\stackrel{1}{s}}_{k+1|k}(x)=\cdots ={\stackrel{s}{s}}_{k+1|k}(x)$. Then we get the CPHD filter analog of the Bayes parallel combination formula, Equation 8.14:

$p_{k+1|k+1}(n)=\frac{1}{{\text{μ}}_{k+1|k+1}}\cdot {\stackrel{1}{p}}_{k+1|k+1}(n)\cdots {\stackrel{s}{p}}_{k+1|k+1}(n)\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}{(n)}^{1-s}$

(8.100)

$s_{k+1|k+1}(x)=\frac{1}{{\text{σ}}_{k+1|k+1}}\cdot {\stackrel{1}{s}}_{k+1|k+1}(x)\cdots {\stackrel{s}{s}}_{k+1|k+1}(x)\cdot s_{k+1|k}{(x)}^{1-s}$

(8.101)

where

${\text{μ}}_{k+1|k+1}={\displaystyle \sum _{n\ge 0}{\stackrel{1}{p}}_{k+1|k+1}(n)\cdots {\stackrel{s}{p}}_{k+1|k+1}(n)\cdot {\text{σ}}_{k+1|k+1}^n}$

(8.102)

${\text{σ}}_{k+1|k+1}={\displaystyle \int {\stackrel{1}{s}}_{k+1|k+1}(x)\cdots {\stackrel{s}{s}}_{k+1|k+1}(x)\cdot s_{k+1|k}{(x)}^{1-s}dx.}$

(8.103)

Remark 3: Equations 8.100 and 8.101 have been employed as the basis for the principled approximate multisensor CPHD and PHD filters [39] mentioned in Section 8.3.5.

8.5.2    PHD FILTER T²F OF INDEPENDENT SOURCES

What is the analog of Equation 8.97 for PHD filters? That is, suppose that the track sources use PHD filters rather than CPHD filters, and thus pass on PHDs rather than spatial distributions and cardinality distributions. Then what is the formula for the merged PHD? This turns out to be (see Section 8.7.2)

$D_{k+1|k+1}(x)=\frac{{\stackrel{1}{D}}_{k+1|k+1}(x)}{{\stackrel{1}{D}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{D}}_{k+1|k+1}(x)}{{\stackrel{s}{D}}_{k+1|k}(x)}\cdot D_{k+1|k}(x).$

(8.104)

This merging formula is potentially problematic because, in the single-target case, it does not reduce to the correct single-target formula. For example, suppose that $D_{k+1|k}(x)={\stackrel{1}{D}}_{k+1|k}(x)=\cdots ={\stackrel{s}{D}}_{k+1|k}(x)$, in which case

$D_{k+1|k+1}(x)={\stackrel{1}{D}}_{k+1|k+1}(x)\cdots {\stackrel{s}{D}}_{k+1|k+1}(x)\cdot D_{k+1|k}{(x)}^{1-s}.$

(8.105)

Equation 8.105 should reduce to Bayes parallel combination, Equation 8.14. However, it does not. To see this, note that with the single-target Bayes recursive filter, there are (1) no missed detections or false alarms; (2) the integrals of D_{k+1 | k}(x) = f_{k+1 | k}(x) and ${\stackrel{i}{D}}_{k+1|k+1}(x)={\stackrel{i}{f}}_{k+1|k+1}(x)$ for all i should equal 1; and (3) the integral of D_{k+1 | k}+1(x) should equal 1.

By way of contrast, Equations 8.96 and 8.97 do reduce to the correct single-target formula in the single-target case. Thus one must conclude:

•  Equation 8.104 is unlikely to provide an accurate approximate track-merging formula when the number of targets in the scenario is small.

Remark 4: Let ${\stackrel{i}{D}}_{k+1|k+1}(x)={\stackrel{i}{L}}_{{\stackrel{i}{Z}}_{k+1}}(x)\cdot {\stackrel{i}{D}}_{k+1|k}(x)$ be the PHD filter measurement-update formula for the ith source, as defined in Equation 8.67. Then Equation 8.105 becomes

$D_{k+1|k+1}(x)={\stackrel{1}{L}}_{{\stackrel{1}{{}_Z}}_{k+1}}(x)\cdots {\stackrel{s}{L}}_{{\stackrel{s}{{}_Z}}_{k+1}}(x)\cdot D_{k+1|k}(x).$

(8.106)

This is the multisensor PHD measurement-update formula as described in Equation 8.106 of reference [43]. It follows that this update formula is likely to be inaccurate when the number of targets is small.

8.5.3    CPHD FILTER T²F WITH KNOWN DOUBLE-COUNTING

Suppose that the data sources share sensors, but that it is known which sensors are being shared by which sources. The sources use CPHD filters to process these measurements: As in Section 8.4.2, define $Z_{k+1}={\stackrel{1}{Z}}_{k+1}\cup \cdots \cup {\stackrel{s}{Z}}_{k+1}$. Let ${\stackrel{12}{Z}}_{k+1}$ be the measurements supplied to the second source that are not in ${\stackrel{1}{Z}}_{k+1},{\stackrel{13}{Z}}_{k+1}$ the measurements supplied to the third source that are not in ${\stackrel{1}{Z}}_{k+1}\cup {\stackrel{12}{Z}}_{k+1}{\stackrel{14}{Z}}_{k+1}$ the measurements supplied to the fourth source that are not in ${\stackrel{1}{Z}}_{k+1}\cup {\stackrel{12}{Z}}_{k+1}\cup {\stackrel{12}{Z}}_{k+1}$, and so on. Let ${\stackrel{(j)}{Z}}_{k+1}={\stackrel{(j)}{Z}}_{k+1}-{\stackrel{1j}{Z}}_{k+1}$.

At time-step k, the jth source provides spatial distributions ${\stackrel{j}{s}}_{k|k}(x)$ and ${\stackrel{j}{s}}_{k+1|k}(x|\stackrel{(j)}{Z})$ and cardinality distributions ${\stackrel{j}{p}}_{k|k}(n)$ and ${\stackrel{j}{p}}_{k+1|k}(n|\stackrel{(j)}{Z})$. Then the CPHD filter version of Equation 8.92 is

$P_{k+1|k+1}(n)=\frac{1}{{\text{μ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}.\frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{2}{p}}_{k+1|k}(n|\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n|\stackrel{(s)}{Z})}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)$

(8.107)

$s_{k+1|k+1}(x)=\frac{1}{{\text{σ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}.\frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{2}{s}}_{k+1|k}(x|\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x|\stackrel{(s)}{Z})}\cdot s_{k+1|k}(x)$

(8.108)

where

${\text{μ}}_{k+1|k+1}={\displaystyle \sum _{n\ge 0}\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}.\frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{2}{p}}_{k+1|k}(n|\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n|\stackrel{(s)}{Z})}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)}$

(8.109)

${\text{σ}}_{k+1|k+1}={\displaystyle \int \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}.\frac{{\stackrel{2}{s}}_{k+1|k+1}(x)}{{\stackrel{2}{s}}_{k+1|k}(x|\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x|\stackrel{(s)}{Z})}\cdot s_{k+1|k}(x)dx.}$

(8.110)

8.5.4    PHD FILTER T²F WITH KNOWN DOUBLE-COUNTING

The PHD filter version of these equations is

$D_{k+1|k+1}(x)={\displaystyle \int \frac{{\stackrel{1}{D}}_{k+1|k+1}(x)}{{\stackrel{1}{D}}_{k+1|k}(x)}.\frac{{\stackrel{2}{D}}_{k+1|k+1}(x)}{{\stackrel{2}{D}}_{k+1|k}(x|\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{D}}_{k+1|k+1}(x)}{{\stackrel{s}{D}}_{k+1|k}(x|\stackrel{(s)}{Z})}\cdot D_{k+1|k}(x).}$

(8.111)

This update formula is unlikely to offer good performance when the number of targets is small.

8.5.5    CPHD FILTER XM FUSION

Clark et al. have considered the special case of Equation 8.93 when the distributions are i.i.d. cluster processes—that is, when track fusion is based on CPHD or PHD filters [4–6]. Suppose that multiple targets are being observed by two track sources equipped with CPHD filters. At time-step k, the first source provides a spatial distribution ${\stackrel{0}{s}}_{k|k}(x)$ and cardinality distribution ${\stackrel{0}{p}}_{k|k}(n)$; and the second source provides a spatial distribution ${\stackrel{1}{s}}_{k|k}(x)$ and cardinality distribution ${\stackrel{1}{p}}_{k|k}(n)$.

Given this, the CPHD filter approximation of the multitarget XM fusion formula, Equation 8.93, is (see Section 8.7.4)

${\stackrel{\text{ω}}{p}}_{k+1|k+1}(n)=\frac{1}{{\stackrel{\text{ω}}{\text{μ}}}_{k+1|k+1}}\cdot {\stackrel{0}{p}}_{k+1|k+1}{(n)}^{1-\text{ω}}\cdot {\stackrel{1}{p}}_{k+1|k+1}{(n)}^{\text{ω}}\cdot \stackrel{\text{ω}}{{\text{σ}}_{k+1|k+1}^n}$

(8.112)

${\stackrel{\text{ω}}{s}}_{k+1|k+1}(x)=\frac{1}{{\stackrel{\text{ω}}{\text{σ}}}_{k+1|k+1}}\cdot {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}$

(8.113)

where

${\stackrel{\text{ω}}{\text{μ}}}_{k+1|k+1}={\displaystyle \sum _{n\ge 0}{\stackrel{0}{p}}_{k+1|k+1}{(n)}^{1-\text{ω}}\cdot {\stackrel{1}{p}}_{k+1|k+1}{(n)}^{\text{ω}}\cdot \stackrel{\text{ω}}{{\text{σ}}_{k+1|k+1}^n}}$

(8.114)

${\stackrel{\text{ω}}{\text{σ}}}_{k+1|k+1}={\displaystyle \int {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}dx.$

(8.115)

From a theoretical point of view, optimization of Equations 8.112 and 8.113 would be obtained via Equation 8.95:

$\widehat{\text{ω}}=\arg \underset{\text{ω}}{\sup }\underset{X}{\sup }{\stackrel{\text{ω}}{f}}_{k+1|k+1}(X)\cdot \frac{c^{|X|}}{|X|!}=\arg \underset{\text{ω}}{\sup }\underset{X}{\sup }{p}_{k|k}(|X|)\cdot {(c{s}_{k+1|k+1})}^X.$

(8.116)

However, this formula will usually be computationally problematic, as will be the multitarget version of the Chernoff information, Equation 8.35. A very approximate approach would be to first apply Chernoff optimization to the spatial distributions:

$\widehat{\text{ω}}=\arg \underset{\text{ω}}{\inf }{\displaystyle \int {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}dx.}$

(8.117)

Then setting

$\stackrel{⌣}{\sigma }={\displaystyle \int {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-{\stackrel{⌣}{\text{ω}}}_1}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{{\stackrel{⌣}{\text{ω}}}_1}dx,}$

(8.118)

one could apply Chernoff optimization once again to the cardinality distributions:

${\stackrel{⌣}{\text{ω}}}_2=\arg \underset{\text{ω}}{\inf }{\displaystyle \sum _{n\ge 0}{\stackrel{0}{p}}_{k+1|k+1}{(n)}^{1-\text{ω}}\cdot {\stackrel{1}{p}}_{k+1|k+1}{(n)}^{\text{ω}}\cdot {\stackrel{⌣}{\text{σ}}}^n}$

(8.119)

The final distributions would then be

$\begin{array}{cc}{\stackrel{\text{XM}}{p}}_{k+1|k+1}(n)={\stackrel{{\stackrel{⌣}{\text{ω}}}_2}{p}}_{k+1|k+1}(n),& {\stackrel{\text{XM}}{s}}_{k+1|k+1}(x)={\stackrel{{\stackrel{⌣}{\text{ω}}}_1}{s}}_{k+1|k+1}(x)\end{array}.$

(8.120)

Clark et al. have implemented Equations 8.112, 113, 114, 8.115 and have considered a broad range of optimization procedures [5,6,44]. They have further demonstrated that these equations lead to good distributed-fusion performance.

8.5.6    PHD FILTER XM FUSION

The PHD filter approximation of the multitarget XM fusion formula, Equation 8.93, can be shown to be (see Section 8.7.5)

${\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)={\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}.$

(8.121)

This formula is an equality, not a proportionality. Thus it does not reduce to the single-target XM fusion formula, Equation 8.34, in the single-target case. Thus it is unlikely to perform well when the number of targets is small.

How might we optimize Equation 8.121? Once again, Equation 8.116 will be computationally problematic. One alternative is as follows. It can be shown (see Section 8.7.6) that Chernoff information can be defined for PHDs and has the form

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=\underset{0\le \text{ω}\le \text{1}}{\sup }(K_{\text{ω}}-{\stackrel{\text{ω}}{N}}_{k+1|k+1})$

(8.122)

where

${\stackrel{\text{ω}}{N}}_{k+1|k+1}={\displaystyle \int {\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)dx={\displaystyle \int {\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}}\cdot {\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}dx}$

(8.123)

is the expected number of targets corresponding to ω, and where

$K_{\text{ω}}=(1-\text{ω}){\stackrel{0}{N}}_{k+1|k+1}+\text{ω}{\stackrel{1}{N}}_{k+1|k+1}$

(8.124)

is the weighted expected number of targets. Equation 8.122 is, at least in principle, potentially computationally tractable. Thus one would chose

${\stackrel{\text{XM}}{D}}_{k+1|k+1}(x)={\stackrel{\stackrel{⌣}{\text{ω}}}{D}}_{k+1|k+1}(x)$

(8.125)

where

$\stackrel{⌣}{\text{ω}}\text{=}\text{arg}\underset{\text{ω}}{\sup }(K_{\text{ω}}-{\stackrel{\text{ω}}{N}}_{k+1|k+1}).$

(8.126)

As an example, suppose that expected target numbers are identical: ${\stackrel{0}{N}}_{k+1|k+1}={\stackrel{1}{N}}_{k+1|k+1}=N_{k+1|k+1}$. Then K_ω = N_{k+1 | k+1} is constant and it is easily shown that

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=N_{k+1|k+1}\cdot \underset{0\le \text{ω}\le 1}{\sup }\left(1-{\displaystyle \int {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}dx\right).$

(8.127)

As another example, let the spatial distributions be identical: ${\stackrel{0}{s}}_{k+1|k+1}(x)={\stackrel{1}{s}}_{k+1|k+1}(x)=s_{k+1|k+1}(x)$. Then it is easily shown that

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=\underset{0\le \text{ω}\le 1}{\sup }\left((1-\text{ω}){\stackrel{0}{N}}_{k+1|k+1}|\text{ω}{\stackrel{1}{N}}_{k+1|k+1}-\stackrel{0}{N_{k+1|k+1}^{1-\text{ω}}}\stackrel{1}{N_{k+1|k+1}^{\text{ω}}}\right).$

(8.128)

8.6    COMPUTATIONAL ISSUES

In this section, I address the practical computability of the T²F CPHD/PHD filter formulas derived in the previous sections. There are two general fusion architectures that can be envisioned. In the first architecture, the track sources use GM-CPHD or GM-PHD filters, and transmit their Gaussian-mixture PHDs to the T²F site. In the second architecture, the track sources use particle-CPHD or particle-PHD filters, and transmit their particle-PHDs to the T²F site. In either case, the most serious obstacle to practical implementation is the following:

•  The exact fusion formulas in Sections 8.5.1 through 8.5.3 involve division by PHDs.

•  The XM fusion formulas in Sections 8.5.5 through 8.5.6 involve fractional powers of PHDs.

I deal with these two situations in the two sections that follow.

8.6.1    IMPLEMENTATION: EXACT T²F FORMULAS

In what follows, I consider implementation of the exact fusion formulas in Sections 8.5.1 through 8.5.3. I consider two cases: the track sources employ GM-CPHD or GM-PHD filters; or the track sources employ particle-CPHD or particle-PHD filters.

8.6.1.1    Case 1: GM-PHD Tracks

Suppose that the track sources send their PHDs to the T²F site in the form of Gaussian mixtures—or, more precisely, as finite sets of the form {(w₁,x₁,P₁),…,(w_n,x_n,P_n)} where each triple (w_i,x_i,P_i) is a Gaussian component. Here I sketch the outlines of a possible implementation approach, using a hybridization of particle and Gaussian-mixture techniques.

For the sake of clarity, consider the simplest CPHD/PHD filter track-merging formula, Equation 8.105:

$D_{k+1|k+1}(x)={\stackrel{1}{D}}_{k+1|k+1}(x)\cdots {\stackrel{s}{D}}_{k+1|k+1}(x)\cdot D_{k+1|k+1}{(x)}^{1-s}.$

(8.129)

If we can devise an implementation solution in this case, it should be possible to devise solutions for the more complex track-merging formulas in Sections 8.5.1 through 8.5.3. Rewrite Equation 8.129 as

$D_{k+1|k+1}(x)={\stackrel{1}{D}}_{k+1|k+1}(x)\cdots {\stackrel{s}{D}}_{k+1|k+1}(x)\cdot D_{k+1|k+1}{(x)}^{-s}.D_{k+1|k}(x).$

(8.130)

where D_{k+1 | k}(x) and each ${\stackrel{i}{D}}_{k+1|k+1}(x)$ is a Gaussian mixture. Then:

1.  Use standard GM-PHD filter merging and pruning techniques to reduce the product ${\stackrel{i}{D}}_{k+1|k+1}(x)\cdots {\stackrel{s}{D}}_{k+1|k+1}(x)$ to a new Gaussian-mixture PHD ${\stackrel{1\cdots s}{D}}_{k+1|k+1}(x)$.

2.  Draw a statistical sample from the normalized predicted PHD:

$x_{k+1|k}^1,\dots ,x_{k+1|k}^v\sim \frac{D_{k+1|k}(x)}{N_{k+1|k}}.$

(8.131)

3.  Approximate D_{k+1 | k}(x) as the Dirac mixture

$D_{k+1|k}(x)\cong \frac{N_{k+1|k}}{\text{v}}{\displaystyle \sum _{i=1}^{\text{v}}{\delta }_{x_{k+1|k}^i}(x).}$

(8.132)

4.  Determine the corresponding particle approximation of D_{k+1 | k+1}(x):

${\tilde{D}}_{k+1|k+1}(x)=\frac{N_{k+1|k}}{\text{v}}{\displaystyle \sum _{i=1}^{\text{v}}{\stackrel{1\dots s}{D}}_{k+1|k+1}(x_{k+1|k}^i)\cdot D_{k+1|k}{(x_{k+1|k}^i)}^{-s}\cdot {\delta }_{x_{k+1|k}^i}(x).}$

(8.133)

For this formula to be effective, there have to be enough particles nearby the means of the Gaussian components of ${\stackrel{1\cdots s}{D}}_{k+1|k+1}(x)$.

5.  Employ some particle-resampling technique to convert Equation 8.133 into distribution-sample form:

${\stackrel{⌣}{D}}_{k+1|k+1}(x)=\frac{{\tilde{N}}_{k+1|k+1}}{\text{v}}{\displaystyle \sum _{i=1}^{\text{v}}{\text{δ}}_{y_{k+1|k}^i}(x)}$

(8.134)

where $y_{k+1|k}^1,\dots ,y_{k+1|k}^{\text{v}}$ are the resampled particles and where

${\tilde{N}}_{k+1|k+1}=N_{k+1|k}{\displaystyle \sum _{i=1}^{\text{v}}{\stackrel{1\dots s}{D}}_{k+1|k+1}(x_{k+1|k}^i)}\cdot D_{k+1|k}{(x_{k+1|k}^i)}^{-s}.$

(8.135)

6.  Use the EM algorithm, or some other particle-regularization procedure, to approximate ${\stackrel{⌣}{D}}_{k+1|k+1}(x)$ as a Gaussian mixture.

7.  Iterate.

8.6.1.2    Case 2: Particle-PHD Tracks

This approach can be modified to address the case in which the track sources send their PHDs to the T²F site in the form of Dirac mixtures:

1.  The ${\stackrel{i}{D}}_{k+1|k+1}(x)$ are Dirac mixtures. Use the EM algorithm, or some other particle-regularization procedure, to approximate each of them as a Gaussian mixture.

2.  Apply steps 1, 4, and 5 from the previous implementation approach.

8.6.2    IMPLEMENTATION: XM T²F FORMULAS

In what follows, I consider implementation of the XM fusion formulas in Sections 8.5.5 through 8.5.6. I consider two cases: the track sources employ GM-CPHD or GM-PHD filters; or the track sources employ particle-CPHD or particle-PHD filters.

8.6.2.1    Case 1: GM-PHD Tracks

Suppose that the track sources send their PHDs to the T²F site in the form of Gaussian mixtures. For the sake of conceptual clarity, consider the simplest of the XM fusion formulas, Equation 8.121:

${\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)={\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}.$

(8.136)

If we can devise an implementation solution in this case, it should be possible to devise solutions for the more complex track-merging formulas in Section 8.5.5.

One approach is to adapt the approximation suggested by Julier, one that appears to be surprisingly effective [15]:

${\left({\displaystyle \sum _{i=1}^n{x}_i}\right)}^{\text{ω}}\cong {\displaystyle \sum _{i=1}^n{x}_i^{\text{ω}},}{\left({\displaystyle \sum _{i=1}^n{x}_i}\right)}^{1-\text{ω}}\cong {\displaystyle \sum _{i=1}^n{x}_i^{\text{1}-\text{ω}}}$

(8.137)

Thus, if

${\stackrel{0}{D}}_{k+1|k+1}(x)={\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}\stackrel{0}{w_i}\cdot N_{\stackrel{0}{P_i}}}(x-{\stackrel{0}{x}}_i)$

(8.138)

${\stackrel{1}{D}}_{k+1|k+1}(x)={\displaystyle \sum _{i=1}^{\stackrel{1}{\text{v}}}\stackrel{1}{w_j}\cdot N_{\stackrel{1}{P_j}}}(x-{\stackrel{1}{x}}_j),$

(8.139)

it can be shown (see Section 8.7.7) that the corresponding XM fusion formula is

${\left\{({\stackrel{0}{w}}_i,{\stackrel{0}{x}}_i,{\stackrel{0}{P}}_i)\right\}}_i,{\left\{({\stackrel{1}{w}}_j,{\stackrel{1}{x}}_j,{\stackrel{1}{P}}_j)\right\}}_j\to {\left\{({\stackrel{\text{ω}}{w}}_{i,j},{\stackrel{\text{ω}}{x}}_{i,j},{\stackrel{\text{ω}}{P}}_{i,j})\right\}}_{i,j}$

(8.140)

where for $i=1,\dots ,\stackrel{0}{v}$ and $j=1,\dots ,\stackrel{1}{v}$,

$\stackrel{\text{ω}}{P_{i,j}^{-1}}=(1-\text{ω})\stackrel{0}{P_i^{-1}}+\text{ω}\stackrel{1}{P_j^{-1}}$

(8.141)

$\stackrel{\text{ω}}{P_{i,j}^{-1}}{\stackrel{\text{ω}}{x}}_{i,j}=(1-\text{ω})\stackrel{0}{P_i^{-1}}{\stackrel{0}{x}}_i+\text{ω}\stackrel{1}{P_j^{-1}}{\stackrel{1}{x}}_j$

(8.142)

${\stackrel{\text{ω}}{w}}_{i,j}=\frac{\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{1}{w_i^{\text{ω}}}}{\sqrt{{\text{ω}}^N{(1-\text{ω})}^N}}\cdot N_{{\stackrel{0}{P}}_i/(1-\text{ω})+{\stackrel{1}{P}}_j/\text{ω}}({\stackrel{1}{x}}_j-{\stackrel{0}{x}}_i)$

(8.143)

and where N is the dimension of the underlying Euclidean space.

As for optimization of ω, it follows that

${\stackrel{\text{ω}}{N}}_{k+1|k+1}={\displaystyle \int {\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)dx}$

(8.144)

$={\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}{\displaystyle \sum _{j=1}^{\stackrel{1}{\text{v}}}\frac{\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{0}{w_j^{\text{ω}}}}{\sqrt{{\text{ω}}^N{(1-\text{ω})}^N}}\cdot N_{{\stackrel{0}{P}}_i/(1-\text{ω})+{\stackrel{1}{P}}_j/\text{ω}}({\stackrel{1}{x}}_j-{\stackrel{0}{x}}_i)}}$

(8.145)

and thus from Equation 8.122 that the Chernoff information is, approximately, the supremum with respect to ω of the quantity

$K_{\text{ω}}-{\stackrel{\text{ω}}{N}}_{k+1|k+1}$

(8.146)

$={\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}{\displaystyle \sum _{j=1}^{\stackrel{1}{\text{v}}}\left(\frac{(1-\text{ω})\cdot {\stackrel{0}{w}}_i}{\stackrel{1}{\text{v}}}+\frac{\text{ω}\cdot {\stackrel{1}{w}}_j}{\stackrel{0}{\text{v}}}-\frac{\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{0}{w_j^{\text{ω}}}}{\sqrt{{\text{ω}}^N{(1-\text{ω})}^N}}\cdot N_{{\stackrel{0}{P}}_i/(1-\text{ω})+{\stackrel{1}{P}}_j/\text{ω}}({\stackrel{1}{x}}_j-{\stackrel{0}{x}}_i)\right)}.}$

(8.147)

8.6.2.2    Case 2: Particle-PHD Tracks

This approach can be modified to address the case in which the track sources send their PHDs to the T²F site in the form of Dirac mixtures. Use the EM algorithm, or another particle-regularization procedure, to approximate the particle-PHDs ${\stackrel{0}{D}}_{k+1|k+1}(x)$ and ${\stackrel{1}{D}}_{k+1|k+1}(x)$ as Gaussian mixtures. Then proceed as before.

8.7    MATHEMATICAL DERIVATIONS

8.7.1    PROOF: CPHD T²F FUSION—INDEPENDENT SOURCES

Suppose that the track distributions of the sources are i.i.d.c. processes, i.e.,

$f_{k+1|k}(X)=|X|!\cdot p_{k+1|k}(|X|) s_{k+1|k}^X$

(8.148)

${\stackrel{j}{f}}_{k+1|k+1}(X)=|X|!\cdot {\stackrel{j}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{j}{s_{k+1|k+1}^X}$

(8.149)

${\stackrel{j}{f}}_{k+1|k}(X)=|X|!\cdot {\stackrel{j}{p}}_{k+1|k}(|X|)\cdot \stackrel{j}{s_{k+1|k}^X}$

(8.150)

Then I show that the multitarget track-merging distribution of Equation 8.89 is also that of an i.i.d.c. process:

$f_{k+1|k+1}(X)=|X|!\cdot p_{k+1|k+1}(|X|)\cdot s_{k+1|k+1}^X$

(8.151)

where

$p_{k+1|k+1}(n)=\frac{1}{{\text{μ}}_{k+1|k+1}}.\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n)}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)$

(8.152)

$s_{k+1|k+1}(x)=\frac{1}{{\text{σ}}_{k+1|k+1}}.\frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x)}\cdot s_{k+1|k}(x)$

(8.153)

and where

${\text{μ}}_{k+1|k+1}={\displaystyle \sum _{n\ge 0}\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n)}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)}$

(8.154)

${\text{σ}}_{k+1|k+1}={\displaystyle \int \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x)}\cdot s_{k+1|k}(x)dx}$

(8.155)

For, from Equation 8.89,

$f_{k+1|k+1}(X)\propto =\frac{{\stackrel{1}{f}}_{k+1|k+1}(X)}{{\stackrel{1}{f}}_{k+1|k}(X)}\cdots \frac{{\stackrel{s}{f}}_{k+1|k+1}(X)}{{\stackrel{s}{f}}_{k+1|k}(X)}\cdot f_{k+1|k}(X)$

(8.156)

$=\frac{|X|!\cdot {\stackrel{1}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{1}{s_{k+1|k+1}^X}}{\begin{array}{r}\hfill |X|!\cdot {\stackrel{1}{p}}_{k+1|k}(|X|)\cdot \stackrel{1}{s_{k+1|k}^X}\\ \hfill \cdot |X|!\cdot p_{k+1|k}(|X|)\cdot s_{k+1|k}^X\end{array}}\cdots \frac{|X|!\cdot {\stackrel{s}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{s}{s_{k+1|k+1}^X}}{|X|!\cdot {\stackrel{s}{p}}_{k+1|k}(|X|)\cdot \stackrel{s}{s_{k+1|k}^X}}$

(8.157)

$=|X|!\cdot \frac{{\stackrel{1}{p}}_{k+1|k+1}(|X|)}{{\stackrel{1}{p}}_{k+1|k}(|X|)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(|X|)}{{\stackrel{s}{p}}_{k+1|k}(|X|)}\cdot p_{k+1|k}(|X|)·{\left(\frac{{\stackrel{1}{s}}_{k+1|k+1}}{{\stackrel{1}{s}}_{k+1|k}}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}}{{\stackrel{s}{s}}_{k+1|k}}\cdot s_{k+1|k}\right)}^X$

(8.158)

$\propto |X|!\cdot p_{k+1|k+1}(|X|)\cdot s_{k+1|k+1}^X$

(8.159)

where the probability distributions p_{k+1 | k+1}(n) and s_{k+1 | k+1}(x) are defined by

$p_{k+1|k+1}(n)\propto =\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n)}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)$

(8.160)

$s_{k+1|k+1}(x)\propto \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x)}\cdot s_{k+1|k}(x).$

(8.161)

Thus

$f_{k+1|k+1}(X)=K\cdot |X|!\cdot p_{k+1|k+1}(|X|)\cdot s_{k+1|k+1}^X$

(8.162)

for some K which is independent of X. Integrating both sides, and making note of Equation 8.63, we get

$1={\displaystyle \int f_{k+1|k+1}(X)}\text{δ}X=K\cdot {\displaystyle \int |X|!\cdot p_{k+1|k+1}(|X|)}\cdot s_{k+1|k+1}^X\text{δ}X=K\cdot 1=K$

(8.163)

which completes the derivation.

8.7.2    PROOF: PHD T²F FUSION—INDEPENDENT SOURCES

Suppose that the track distributions of the sources are Poisson processes, i.e.,

$f_{k+1|k}(X)=e^{-N_{k+1|k}}\cdot D_{k+1|k}^X$

(8.164)

${\stackrel{j}{f}}_{k+1|k+1}(X)=e^{-{\stackrel{j}{N}}_{k+1|k+1}}\cdot \stackrel{j}{D_{k+1|k+1}^X}$

(8.165)

${\stackrel{j}{f}}_{k+1|k}(X)=e^{-{\stackrel{j}{N}}_{k+1|k}}\cdot \stackrel{j}{D_{k+1|k}^X.}$

(8.166)

Then I show that the XM fusion of the sources is also an i.i.d.c. process, where

$N_{k+1|k+1}={\displaystyle \int \frac{{\stackrel{1}{D}}_{k+1|k+1}(x)}{{\stackrel{1}{D}}_{k+1|k}(x)}}\cdots \frac{{\stackrel{s}{D}}_{k+1|k+1}(x)}{{\stackrel{s}{D}}_{k+1|k}(x)}\cdot D_{k+1|k}(x)dx$

(8.167)

$D_{k+1|k+1}(x)=\frac{{\stackrel{1}{D}}_{k+1|k+1}(x)}{{\stackrel{1}{D}}_{k+1|k}(x)}\cdots \frac{{\stackrel{s}{D}}_{k+1|k+1}(x)}{{\stackrel{s}{D}}_{k+1|k}(x)}\cdot D_{k+1|k}(x).$

(8.168)

For, in this case

$f_{k+1|k+1}(X)\propto \frac{{\stackrel{1}{f}}_{k+1|k+1}(X)}{{\stackrel{1}{f}}_{k+1|k}(X)}\cdots \frac{{\stackrel{s}{f}}_{k+1|k+1}(X)}{{\stackrel{s}{f}}_{k+1|k}(X)}\cdot f_{k+1|k}(X)$

(8.169)

$=\frac{e^{-{\stackrel{1}{N}}_{k+1|k+1}}\cdot \stackrel{1}{D_{k+1|k+1}^X}}{e^{-{\stackrel{1}{N}}_{k+1|k}}\cdot \stackrel{1}{D_{k+1|k}^X}}\cdots \frac{e^{-{\stackrel{s}{N}}_{k+1|k+1}}\cdot \stackrel{s}{D_{k+1|k+1}^X}}{e^{-{\stackrel{s}{N}}_{k+1|k}}\cdot \stackrel{s}{D_{k+1|k}^X}}\cdot e^{-N_{k+1k}}\cdot D_{k+1|k}^X$

(8.170)

$\propto {\left(\frac{{\stackrel{1}{D}}_{k+1|k+1}}{{\stackrel{1}{D}}_{k+1|k}}\cdots \frac{{\stackrel{s}{D}}_{k+1|k+1}}{{\stackrel{s}{D}}_{k+1|k}}\cdot D_{k+1|k}\right)}^X$

(8.171)

$=e^{-N_{k+1|k+1}}\cdot D_{k+1|k+1}^X$

(8.172)

as claimed.

8.7.3    PROOF: CPHD FILTER WITH DOUBLE-COUNTING

Suppose that the track distributions of the s track sources are i.i.d.c. processes, i.e.,

$f_{k+1|k}(X)=|X|!\cdot p_{k+1|k}(|X|)\cdot s_{k+1|k}^X$

(8.173)

${\stackrel{j}{f}}_{k+1|k+1}(X)=|X|!\cdot {\stackrel{j}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{j}{s_{k+1|k+1}^X}$

(8.174)

${\stackrel{j}{f}}_{k+1|k}(X|\stackrel{(j)}{Z})=|X|!\cdot {\stackrel{j}{p}}_{k+1|k}(|X|)\cdot \stackrel{j}{s_{k+1|k}^X}$

(8.175)

where

${\stackrel{j}{p}}_{k+1|k}(n)\stackrel{\text{abbr}\text{.}}{=}{\stackrel{j}{p}}_{k+1|k}(n|\stackrel{(j)}{Z})$

(8.176)

${\stackrel{j}{s}}_{k+1|k}(x)\stackrel{\text{abbr}\text{.}}{=}{\stackrel{j}{s}}_{k+1|k}(x|\stackrel{(j)}{Z})$

(8.177)

Then I show that the multitarget track-merging distribution of Equation 8.92 is also that of an i.i.d.c. process:

$f_{k+1|k+1}(X)=|X|!\cdot p_{k+1|k+1}(|X|)\cdot s_{k+1|k+1}^X$

(8.178)

where

$p_{k+1|k+1}(n)=\frac{1}{{\text{μ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdot \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{2}{p}}_{k+1|k}(n+\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n+\stackrel{(s)}{Z})}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)$

(8.179)

$s_{k+1|k+1}(x)=\frac{1}{{\text{σ}}_{k+1|k+1}}\cdot \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdot \frac{{\stackrel{2}{s}}_{k+1|k+1}(x)}{{\stackrel{2}{s}}_{k+1|k}(x+\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x+\stackrel{(s)}{Z})}\cdot s_{k+1|k}(x)$

(8.180)

and where

${\text{μ}}_{k+1|k+1}={\displaystyle \sum _{n\ge 0}\frac{{\stackrel{1}{p}}_{k+1|k+1}(n)}{{\stackrel{1}{p}}_{k+1|k}(n)}\cdot \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{2}{p}}_{k+1|k}(n+\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{p}}_{k+1|k+1}(n)}{{\stackrel{s}{p}}_{k+1|k}(n+\stackrel{(s)}{Z})}\cdot {\text{σ}}_{k+1|k+1}^n\cdot p_{k+1|k}(n)}$

(8.181)

${\text{σ}}_{k+1|k+1}={\displaystyle \int \frac{{\stackrel{1}{s}}_{k+1|k+1}(x)}{{\stackrel{1}{s}}_{k+1|k}(x)}\cdot \frac{{\stackrel{2}{s}}_{k+1|k+1}(x)}{{\stackrel{2}{s}}_{k+1|k}(x+\stackrel{(2)}{Z})}\cdots \frac{{\stackrel{s}{s}}_{k+1|k+1}(x)}{{\stackrel{s}{s}}_{k+1|k}(x+\stackrel{(s)}{Z})}\cdot s_{k+1|k}(x)dx}$

(8.182)

The derivation is essentially identical to that in Section 8.7.1.

8.7.4    PROOF: CPHD FILTER XM FUSION

We are to prove the CPHD filter version of XM fusion as defined in Equations 8.112 and 8.113. Thus assume that the original multitarget distributions redistributions of i.i.d.c. processes:

${\stackrel{0}{f}}_{k+1|k+1}(X)=|X|!\cdot {\stackrel{0}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{0}{s_{k+1|k+1}^X}$

(8.183)

${\stackrel{0}{f}}_{k+1|k}(X)=|X|!\cdot {\stackrel{0}{p}}_{k+1|k}(|X|)\cdot \stackrel{0}{s_{k+1|k}^X}$

(8.184)

${\stackrel{1}{f}}_{k+1|k+1}(X)=|X|!\cdot {\stackrel{1}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{1}{s_{k+1|k+1}^X}$

(8.185)

${\stackrel{1}{f}}_{k+1|k+1}(X)=|X|!\cdot {\stackrel{1}{p}}_{k+1|k}(|X|)\cdot \stackrel{1}{s_{k+1|k}^X}$

(8.186)

$f_{k+1|k}(X)=|X|!\cdot p_{k+1|k+1}(|X|)\cdot s_{k+1|k}^X$

(8.187)

Then Equation 8.93 becomes

$\begin{array}{ll}{\stackrel{\text{ω}}{f}}_{k+1|k+1}(X)\hfill & \propto {\stackrel{0}{f}}_{k+1|k+1}{(X)}^{1-\text{ω}}\cdot {\stackrel{1}{f}}_{k+1|k+1}{(X)}^{\text{ω}}\hfill \\ \hfill & \propto {\left(|X|!\cdot {\stackrel{0}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{0}{s_{k+1|k+1}^X}\right)}^{1-\text{ω}}{\left(|X|!\cdot {\stackrel{1}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{1}{s_{k+1|k+1}^X}\right)}^{\text{ω}}\hfill \end{array}$

(8.188)

$\begin{array}{l}=|X|!\cdot {\stackrel{0}{p}}_{k+1|k+1}{(|X|)}^{1-\text{ω}}\cdot {\stackrel{1}{p}}_{k+1|k+1}{(|X|)}^{\text{ω}}\cdot {\left(\stackrel{0}{s_{k+1|k+1}^{1-\text{ω}}}\cdot \stackrel{1}{s_{k+1|k+1}^{1-\text{ω}}}\right)}^X\hfill \\ \propto |X|!\cdot {\stackrel{\text{ω}}{p}}_{k+1|k+1}(|X|)\cdot \stackrel{\text{ω}}{s_{k+1|k+1}^X}\hfill \end{array}$

(8.189)

where

${\stackrel{\text{ω}}{p}}_{k+1|k+1}(n)\propto {\stackrel{0}{p}}_{k+1|k+1}{(n)}^{1-\text{ω}}\cdot {\stackrel{1}{p}}_{k+1|k+1}{(n)}^{\text{ω}}\cdot \stackrel{\text{ω}}{{\text{σ}}_{k+1}^n}$

(8.190)

${\stackrel{\text{ω}}{s}}_{k+1|k+1}(x)\propto {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}$

(8.191)

and where

${\stackrel{\text{ω}}{\text{σ}}}_{k+1}={\displaystyle \int {\stackrel{0}{s}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{s}}_{k+1|k+1}{(x)}^{\text{ω}}dx.}$

(8.192)

This concludes the proof.

8.7.5    PROOF: PHD FILTER XM FUSION

We are to prove the PHD filter version of XM fusion as defined in Equation 8.121. Thus assume that the original multitarget distributions are distributions of Poisson processes:

${\stackrel{0}{f}}_{k+1|k+1}(X)=e^{-{\stackrel{0}{N}}_{k+1|k+1}}\cdot \stackrel{0}{D_{k+1|k+1}^X}$

(8.193)

${\stackrel{0}{f}}_{k+1|k}(X)=e^{-{\stackrel{0}{N}}_{k+1|k}}\cdot \stackrel{0}{D_{k+1|k}^X}$

(8.194)

${\stackrel{1}{f}}_{k+1|k+1}(X)=e^{-{\stackrel{1}{N}}_{k+1|k+1}}\cdot \stackrel{1}{D_{k+1|k+1}^X}$

(8.195)

${\stackrel{1}{f}}_{k+1|k+1}(X)=e^{-{\stackrel{1}{N}}_{k+1|k}}\cdot \stackrel{1}{D_{k+1|k}^X}$

(8.196)

$f_{k+1|k}(X)=e^{-N_{k+1|k}}\cdot D_{k+1|k}^X$

(8.197)

Then from Equation 8.93, we get

${\stackrel{\text{ω}}{f}}_{k+1|k+1}(X)\propto {\left(e^{-{\stackrel{0}{N}}_{k+1|k+1}}\cdot \stackrel{0}{D_{k+1|k+1}^X}\right)}^{1-\text{ω}}{\left(e^{-{\stackrel{1}{N}}_{k+1|k+1}}\cdot \stackrel{1}{D_{k+1|k+1}^X}\right)}^{\text{ω}}$

(8.198)

$=e^{-(1-\text{ω}){\stackrel{0}{N}}_{k+1|k+1}-\text{ω}{\stackrel{1}{N}}_{k+1|k+1}}\cdot {\left(\stackrel{0}{D_{k+1|k+1}^{1-\text{ω}}}\cdot \stackrel{1}{D_{k+1|k+1}^{\text{ω}}}\right)}^X$

(8.199)

$\propto {\left(\stackrel{0}{D_{k+1|k+1}^{1-\text{ω}}}\cdot \stackrel{1}{D_{k+1|k+1}^{\text{ω}}}\right)}^X$

(8.200)

$\propto e^{-{\stackrel{\text{ω}}{N}}_{k+1|k+1}}\cdot \stackrel{\text{ω}}{D_{k+1|k+1}^X}$

(8.201)

where

${\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)={\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}$

(8.202)

${\stackrel{\text{ω}}{N}}_{k+1|k+1}={\displaystyle \int {\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot {\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}dx.}$

(8.203)

8.7.6    PROOF: PHD FILTER CHERNOFF INFORMATION

We are to show that Chernoff information directly generalizes as follows:

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=\underset{0\le \text{ω}\le \text{1}}{\sup }(K_{\text{ω}}-{\stackrel{\text{ω}}{N}}_{k+1|k+1})$

(8.204)

where

$K_{\text{ω}}=(1-\text{ω}){\stackrel{0}{N}}_{k+1|k+1}+\text{ω}{\stackrel{1}{N}}_{k+1|k+1}.$

(8.205)

Applying Equation 8.35 to the multitarget distributions ${\stackrel{0}{f}}_{k+1|k+1}(X)$ and ${\stackrel{1}{f}}_{k+1|k+1}{(X)}^{\text{ω}}$, we get

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=\underset{0\le \text{ω}\le \text{1}}{\sup }\left(-\log {\displaystyle \int {\stackrel{0}{f}}_{k+1|k+1}{(X)}^{1-\text{ω}}.}{\stackrel{1}{f}}_{k+1|k+1}{(X)}^{\text{ω}}\text{δ}X\right)$

(8.206)

$=\underset{0\le \text{ω}\le \text{1}}{\sup }\left(-\log {{\displaystyle \int \left(e^{-{\stackrel{0}{N}}_{k+1|k+1}}\stackrel{0}{D_{k+1|k+1}^X}\right)}}^{\text{1}-\text{ω}}{\left(e^{-{\stackrel{1}{N}}_{k+1|k+1}}\stackrel{1}{D_{k+1|k+1}^X}\right)}^{\text{ω}}\text{δ}X\right)$

(8.207)

$=\underset{0\le \text{ω}\le \text{1}}{\sup }\left(K_{\text{ω}}-\log {{\displaystyle \int \left(\stackrel{0}{D_{k+1|k+1}^{\text{1}-\text{ω}}}\cdot \stackrel{1}{D_{k+1|k+1}^{\text{ω}}}\right)}}^X\text{δ}X\right).$

(8.208)

According to Equation 8.61,

${{\displaystyle \int \left(\stackrel{0}{D_{k+1|k+1}^{\text{1}-\text{ω}}}\cdot \stackrel{1}{D_{k+1|k+1}^{\text{ω}}}\right)}}^X\text{δ}X=e^{{\stackrel{\text{ω}}{N}}_{k+1|k+1}}$

(8.209)

where

${\stackrel{\text{ω}}{N}}_{k+1|k+1}={\displaystyle \int {\stackrel{0}{D}}_{k+1|k+1}{(x)}^{1-\text{ω}}\cdot }{\stackrel{1}{D}}_{k+1|k+1}{(x)}^{\text{ω}}dx.$

(8.210)

Thus

$C({\stackrel{0}{D}}_{k+1|k+1},{\stackrel{1}{D}}_{k+1|k+1})=\underset{0\le \text{ω}\le 1}{\sup }\left(K_{\text{ω}}-\log e^{{\stackrel{\text{ω}}{N}}_{k+1|k+1}}\right)$

(8.211)

$=\underset{0\le \text{ω}\le 1}{\sup }(K_{\text{ω}}-{\stackrel{\text{ω}}{N}}_{k+1|k+1}).$

(8.212)

Thus we can define Equation 8.212 to be the “Chernoff information” of the PHDs ${\stackrel{0}{D}}_{k+1|k+1}(x)$ and ${\stackrel{1}{D}}_{k+1|k+1}(x)$.

8.7.7    PROOF: XM IMPLEMENTATION

We are to establish Equations 8.141, 8.142, 8.143. Substituting Equations 8.138 and 8.139 into Equation 8.136 we get

${\stackrel{\text{ω}}{D}}_{k+1|k+1}(x)\cong \left({\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}\stackrel{0}{w_i^{1-\text{ω}}}\cdot N_{\stackrel{0}{P_i}}{(x-{\stackrel{0}{x}}_i)}^{1-\text{ω}}}\right)\left({\displaystyle \sum _{j=1}^{\stackrel{1}{\text{v}}}\stackrel{0}{w_j^{\text{ω}}}\cdot N_{\stackrel{1}{P_j}}{(x-{\stackrel{1}{x}}_j)}^{\text{ω}}}\right)$

(8.213)

$={\displaystyle \sum }_{i=1}^{\stackrel{0}{\text{v}}}{\displaystyle \sum }_{j=1}^{\stackrel{1}{\text{v}}}\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{1}{w_j^{\text{ω}}}\cdot N_{\stackrel{0}{P_i}}{(x-{\stackrel{0}{x}}_i)}^{1-\text{ω}}\cdot N_{\stackrel{1}{P_j}}{(x-{\stackrel{1}{x}}_j)}^{\text{ω}}$

(8.214)

$\begin{array}{l}={\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}{\displaystyle \sum _{j=1}^{\stackrel{1}{\text{v}}}\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{1}{w_j^{\text{ω}}}\cdot \sqrt{\frac{\det 2\text{π}{\stackrel{0}{P}}_i/(1-\text{ω})}{\det 2\text{π}{\stackrel{0}{P}}_i}}\cdot \sqrt{\frac{\det 2\text{π}{\stackrel{1}{P}}_j/\text{ω}}{\det 2\text{π}{\stackrel{1}{P}}_j}}}}\\ \cdot N_{{\stackrel{0}{P}}_i/(1-\text{ω})}(x-{\stackrel{0}{x}}_i)\cdot N_{{\stackrel{1}{P}}_j/\text{ω}}(x-{\stackrel{1}{x}}_j)\end{array}$

(8.215)

$={\displaystyle \sum _{i=1}^{\stackrel{0}{\text{v}}}{\displaystyle \sum _{j=1}^{\stackrel{1}{\text{v}}}\frac{\stackrel{0}{w_i^{1-\text{ω}}}\cdot \stackrel{1}{w_j^{\text{ω}}}}{\sqrt{{\text{ω}}^N{(1-\text{ω})}^N}}\cdot N_{{\stackrel{0}{P}}_i/(1-\text{ω})+{\stackrel{1}{P}}_j/\text{ω}}({\stackrel{1}{x}}_j-{\stackrel{0}{x}}_i)\cdot N_{{\stackrel{\text{ω}}{P}}_i{,}_j}(x-{\stackrel{\text{ω}}{x}}_{i,j})}}$

(8.216)

where

$\stackrel{\text{ω}}{P_{i,j}^{-1}}=(1-\text{ω})\stackrel{0}{P_i^{-1}}+\stackrel{1}{\text{ω}{P}_j^{-1}}$

(8.217)

$\stackrel{\text{ω}}{P_{i,j}^{-1}}{\stackrel{\text{ω}}{x}}_{i,j}=(1-\text{ω})\stackrel{0}{P_i^{-1}}{\stackrel{0}{x}}_i+\stackrel{1}{\text{ω}{P}_j^{-1}}{\stackrel{1}{x}}_j$

(8.218)

This establishes the result.

8.8    CONCLUSIONS

In this chapter, I have proposed the elements of a general theoretical foundation for multisource-multitarget track-to-track fusion (T²F). After summarizing three major single-target T²F situations and approaches—exact T²F without double-counting, exact T²F with double-counting, and approximate T²F in the manner of Clark et al.—I showed how to directly generalize them to the multisource-multitarget case. Since the resulting algorithms are computationally intractable in general, I showed how to derive approximate versions of them using CPHD and PHD filter-based approaches. I also suggested notional implementation techniques for these approaches.

The ideas proposed in this chapter are, of course, just a beginning. Further research into practical implementation is necessary. Also, since the XM fusion approach is a generalization of the CI approach, it is necessarily also an overly conservative approach. More work needs to be conducted on modifications of single-target XM fusion that model different assumptions about the degree of correlations between the original distributions. If such modifications can be devised, then it should be possible to generalize them to the multitarget case using the methodology advocated in this chapter.

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