**4. Anti-Clutter ET-GIW-PHD**

The difference between ET-GIW-PHD and anti-clutter ET-GIW-PHD is how to determine the source of measurements in the cell, specifically, the difference is how to obtain *dW* in Equation (6). Using only the number of measurements in ET-GIW-PHD to determine whether the measurements in the cell is the target or not may be erroneous. In contrast, our anti-clutter ET-GIW-PHD uses hypothesis testing to deal with this problem. The number of measurements, the kinematic state and extent state of target and clutter spatial distribution are taken into account to obtain the likelihood ratio test statistic.

There are two hypotheses:

$$H\_0: \mathbf{Z}\_W \subset \mathbb{C},\tag{13}$$

$$H\_1: \mathbf{Z}\_W \subset \mathbf{T},\tag{14}$$

where **<sup>Z</sup>***<sup>W</sup>* <sup>=</sup> {*z*1, *<sup>z</sup>*2, ..., *<sup>z</sup>nW* } is the measurements of Wth cell, *nW* is the number of the measurements, C and T represent clutter and target respectively.

The likelihood ratio test statistic for hypothese is given by

$$\eta = \frac{L(\mathbf{Z}\_W|\mathbf{T})}{L(\mathbf{Z}\_W|\mathbf{C})},\tag{15}$$

where *L*(**Z***W*|T) and *L*(**Z***W*|C) are the likelihood to measure the set **Z***<sup>W</sup>* given **Z***<sup>W</sup>* ⊂ T and **Z***<sup>W</sup>* ⊂ C respectively and *L*(**Z***W*|T) and *L*(**Z***W*|C) will be presented later.

If log(·) is applied to Equation (15), log(*η*) can be obtained.

$$\log \eta = \log L(\mathbf{Z}\_W|\mathbf{T}) - \log L(\mathbf{Z}\_W|\mathbf{C}).\tag{16}$$

Because log(·) is monotony increase, log(*η*) is also the test statistic for hypothesis *H*<sup>0</sup> versus *H*1. If these measurements are generated by the target, *L*(**Z***W*|T) will achieve a large value and *L*(**Z***W*|C) will be small. Consequently, the statistics log(*η*) will grow to a large value. Using a threshold, we can distinguish between targets and clutter. Specifically, if log(*η*) is greater than the threshold, the measurements in the cell is considered to be generated by targets. Otherwise, these measurements are considered to be clutter. The expression of *L*(**Z***W*|T) and *L*(**Z***W*|C) are given below, the setting of the threshold is discussed.

If the measurements are generated by a target, different extent models lead to a different expression of *L*(**Z***W*|T). In this paper, *L*(**Z***W*|T) is deduced based on the model in [27],

$$\begin{split} L(\mathbf{Z}\_{W}|\mathbf{T}) &= \sum\_{j=n\_{W}}^{\infty} p\_{d}^{n\_{W}} (1-p\_{d})^{j-n\_{W}} \mathsf{C}\_{j}^{n\_{W}} \frac{\gamma^{j}}{j!} e^{-\gamma} \cdot \prod\_{i=1}^{n\_{W}} \mathcal{N}(z\_{i}; (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k}, \mathbf{X}\_{k}) \\ &= \sum\_{j=n\_{W}}^{\infty} p\_{d}^{n\_{W}} (1-p\_{d})^{j-n\_{W}} \frac{j!}{n\_{W}!(j-n\_{W})!} \frac{\gamma^{j}}{j!} e^{-\gamma} \cdot \prod\_{i=1}^{n\_{W}} \mathcal{N}(z\_{i}; (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k}, \mathbf{X}\_{k}) \\ &= \frac{p\_{d}^{n\_{W}} \gamma^{n\_{W}}}{n\_{W}!} e^{-p\_{d} \gamma} \sum\_{j=n\_{W}}^{\infty} \frac{((1-p\_{d})\gamma)^{j-n\_{W}}}{(j-n\_{W})!} e^{-(1-p\_{d})\gamma} \cdot \prod\_{i=1}^{n\_{W}} \mathcal{N}(z\_{i}; (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k}, \mathbf{X}\_{k}) \\ &= \frac{p\_{d}^{n\_{W}} \gamma^{n\_{W}}}{n\_{W}!} e^{-p\_{d} \gamma} \prod\_{i=1}^{n\_{W}} \mathcal{N}(z\_{i}; (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k}, \mathbf{X}\_{k}), \end{split} \tag{17}$$

where **I***<sup>d</sup>* is an unit matrix with *d* dimension, **X***<sup>k</sup>* is the extension of target at time *k*, **H***<sup>k</sup>* is the 1D observation matrix.

The clutters are assumed to be uniformly distributed over the surveillance area [27], then

$$\begin{split} L(\mathbf{Z}\_{W}|\mathbf{C}) &= \beta\_{FA}^{n\_{W}} \sum\_{j=n\_{W}}^{\infty} \frac{\lambda\_{j}^{j}}{j!} e^{-\lambda\_{c}} C\_{j}^{n\_{W}} p\_{d}^{n\_{W}} (1 - p\_{d})^{j - n\_{W}} \\ &= \beta\_{FA}^{n\_{W}} \sum\_{j=n\_{W}}^{\infty} \frac{\lambda\_{c}^{j}}{j!} e^{-\lambda\_{c}} \frac{j!}{n\_{W}!(j - n\_{W})!} p\_{d}^{n\_{W}} (1 - p\_{d})^{j - n\_{W}} \\ &= \beta\_{FA}^{n\_{W}} p\_{d}^{n\_{W}} \frac{\lambda^{n\_{W}}}{n\_{W}!} e^{-p\_{d} \lambda\_{c}} \sum\_{j=n\_{W}}^{\infty} \frac{((1 - p\_{d})\lambda\_{c})^{j - n\_{W}}}{(j - n\_{W})!} e^{-(1 - p\_{d})\lambda\_{c}} \\ &= \beta\_{FA}^{n\_{W}} P\_{d}^{n\_{W}} \frac{\lambda\_{c}^{n\_{W}}}{n\_{W}!} e^{-p\_{d} \lambda\_{c}} \sum\_{i=0}^{\infty} \frac{((1 - p\_{d})\lambda\_{c})^{i}}{i!} e^{-(1 - p\_{d})\lambda\_{c}} \\ &= \beta\_{FA}^{n\_{W}} \frac{p\_{d}^{n\_{W}} \lambda\_{c}^{n\_{W}}}{n\_{W}!} e^{-p\_{d} \lambda\_{c}}. \end{split} \tag{18}$$

where *βFA* = *λcck*, *λ<sup>c</sup>* is the mean number of clutter measurements, *ck* is the spatial distribution of the clutter over the surveillance volume.

Substitute Equations (17) and (18) into Equation (16), we have

log(*η*) = log *L*(**Z***W*|T) − log *L*(**Z***W*|C) = *nW* ∑ *j*=1 log(<sup>N</sup> (*zj*;(**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*, **<sup>X</sup>***k*))+ log( *pnW <sup>d</sup> <sup>γ</sup>nW nW*! *<sup>e</sup>* <sup>−</sup>*pdγ*) <sup>−</sup> log(*βnW FA pnW <sup>d</sup> <sup>λ</sup>nW c nW*! *<sup>e</sup>* <sup>−</sup>*pdλ<sup>c</sup>* ) = *nW* ∑ *j*=1 {−0.5 log 2*π* − 0.5 log |**X***k*|} − *nW* ∑ *j*=1 {0.5(*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*)**X**−<sup>1</sup> *<sup>k</sup>* (*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*) T} + (−*pdγ* + *nW* log *pdγ* − *nW* ∑ *i*=1 log *i*) − (−*pdλ<sup>c</sup>* + *nW* log *pdλ<sup>c</sup>* − *nW* ∑ *i*=1 log *i*) − *nW* log *βFA* = −0.5 *nW* ∑ *j*=1 {(*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*)**X**−<sup>1</sup> *<sup>k</sup>* (*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*) <sup>T</sup>} − 0.5*nW* log 2*<sup>π</sup>* <sup>−</sup> 0.5*nW* log <sup>|</sup>**X***k*<sup>|</sup> <sup>−</sup> *pd*(*<sup>γ</sup>* <sup>−</sup> *<sup>λ</sup>c*) + *nW* log( *<sup>γ</sup> λc* ) − *nW* log *βFA* = −0.5*G* + *D*, (19)

where

$$\mathbf{G} = \sum\_{j=1}^{n\_{\text{IV}}} \left( \mathbf{z}\_{j} - (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k} \right) \mathbf{X}\_{k}^{-1} \left( \mathbf{z}\_{j} - (\mathbf{H}\_{k} \otimes \mathbf{I}\_{d}) \mathbf{x}\_{k} \right)^{\mathrm{T}} \tag{20}$$

$$\begin{split} D &= -0.5n\_{W} \log 2\pi - 0.5n\_{W} \log |\mathbf{X}\_{k}| - p\_{d}(\gamma - \lambda\_{c}) \\ &+ n\_{W} \log(\frac{\gamma}{\lambda\_{c}}) - n\_{W} \log \beta\_{\text{FA}}. \end{split} \tag{21}$$

Because *<sup>z</sup><sup>j</sup>* is subject to Gaussian distribution with mean (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*x<sup>k</sup>* and covariance **<sup>X</sup>***k*, *<sup>z</sup><sup>j</sup>* <sup>∝</sup> <sup>N</sup> ((**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*xk*, **<sup>X</sup>***k*), thus

$$\mathbf{G} = \sum\_{j=1}^{m\_W} \left( \mathbf{z}\_j - (\mathbf{H}\_k \otimes \mathbf{I}\_d) \mathbf{x}\_k \right) \mathbf{X}\_k^{-1} \left( \mathbf{z}\_j - (\mathbf{H}\_k \otimes \mathbf{I}\_d) \mathbf{x}\_k \right)^\mathrm{T} \approx \mathcal{X}^2 (n\_W), \tag{22}$$

where *G* is subject to chi-square distribution with degree of freedom *nW*. In Equation (21), *γ*, *λ<sup>c</sup>* and *βFA* are priori known, the volume of the target extension is proportional to |**X***k*|, the size of the target could be assumed to be unchanged, then *D* could be considered as a constant.

The confidence coefficient is set to *α* and a threshold is introduced (denoted by *g*), suppose hypothesi *H*<sup>1</sup> is true, then

$$P(\log \eta > \mathbf{g}) = P(-0.5G + D > \mathbf{g}) = P\{G < 2(D - \mathbf{g})\} = \mathbf{a},\tag{23}$$

then

$$\mathcal{Z}(D-\mathcal{g}) = \mathcal{X}\_{1-a}^2 \tag{24}$$

$$\mathcal{g} = D - 0.5 \mathcal{X}\_{1-a\prime}^2 \tag{25}$$

where

$$\mathfrak{a} = \int\_{\mathcal{X}\_a^2(n\_W)}^{\infty} \mathcal{X}^2(n\_W) d\mathbf{x}.\tag{26}$$

From Equation (23), the probability of *log*(*η*) < *g* is 1 − *α*. Generally, *α* is set to be a value close to 1 and *log*(*η*) < *g* is a small probability event. If *log*(*η*) < *g* is satisfied, hypothesi *H*<sup>1</sup> should be rejected. Finally, we have

If *log*(*η*) < *g*, the measurements are generated by clutter, then

$$d\_{W} = 1 + \sum\_{l=1}^{l\_{k|k-1}} e^{-\gamma(l)} \left(\frac{\gamma^{(l)}}{\beta\_{FA}}\right)^{|W|} p\_D^{(l)} \Lambda\_k^{(l,W)} w\_{k|k-1}^{(l)}.\tag{27}$$

If log *η* ≥ *g*,the measurements are generated by targets, then

$$d\_{W} = \sum\_{l=1}^{J\_{k|k-1}} e^{-\gamma(l)} \left(\frac{\gamma^{(l)}}{\beta\_{FA}}\right)^{|W|} p\_{D}^{(l)} \Lambda\_{k}^{(l,W)} w\_{k|k-1}^{(l)}.\tag{28}$$

The pseudo-code for anti-clutter ET-GIW-PHD is illustrated in Table 2.

**Table 2.** Pseudo-code for anti-clutter extended target Gaussian inverse Wishart probability hypothesis density (ET-GIW-PHD) filter.

> 1: **Input:** Sequence of measurement sets 2: **Initialize:** parameter initialization 3: **for** *k* = 1 : *K* (*K* is totally time steps) 4: Measurements partition 5: Prediction 6: Correction, see Table 3. 7: Prune and merge 8: Extract target state 9: **end for** 10: **Output:** Sequence of estimated targets.

The difference between ET-GIW-PHD and anti-clutter ET-GIW-PHD lies in correction step. Pseudo-code for anti-clutter ET-GIW-PHD filter correction is shown in Table 3, pseudo-code for other steps (prediction, prune and merge etc) can be found in [39].

**Table 3.** Pseudo-code for anti-clutter ET-GIW-PHD filter correction.

1: **Input:** GIW components {*w<sup>j</sup> <sup>k</sup>*|*k*−1, *<sup>ξ</sup> j <sup>k</sup>*|*k*−1} *Jk*|*k*−<sup>1</sup> *<sup>j</sup>*=<sup>1</sup> , measurements partitions {*pρ*}*<sup>n</sup> ρ*=1 2: **Undetected target case:** 3: **for** *<sup>j</sup>* <sup>=</sup> 1 : *Jk*|*k*−<sup>1</sup> 4: *w<sup>j</sup> <sup>k</sup>*|*<sup>k</sup>* <sup>←</sup> ' <sup>1</sup> <sup>−</sup> (<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*γ*)*pD* ( *wj <sup>k</sup>*|*k*−<sup>1</sup> *<sup>ξ</sup> j <sup>k</sup>*|*<sup>k</sup>* <sup>←</sup> *<sup>ξ</sup> j k*|*k*−1 5: **end for** 6: **Detected target case:** 7:*l* = 0 8: **for** *ρ* = 1 : *n* 9: **for** *W* = 1 : |*pρ*| 10: *l* = *l* + 1 11: **for** *<sup>j</sup>* <sup>=</sup> 1 : *Jk*|*k*−<sup>1</sup> 12: update *ξ j <sup>k</sup>*|*k*−<sup>1</sup> using Kalman filter, see details in [39], *<sup>ξ</sup> j*+*l*·*Jk*|*k*−<sup>1</sup> *k*|*k update* ←−−− *ξ j k*|*k*−1 13: *<sup>w</sup>*(*j*+*l*·*Jk*|*k*−1,*W*) *<sup>k</sup>*|*<sup>k</sup>* <sup>←</sup> *<sup>e</sup>*−*γ*(*j*) *γ*(*j*) *λ<sup>c</sup> ck* |*W*| *p* (*j*) *<sup>D</sup>* <sup>Λ</sup>(*j*,*W*) *<sup>k</sup> <sup>w</sup>*(*j*) *k*|*k*−1 14: *<sup>G</sup><sup>j</sup>* <sup>=</sup> *nw* ∑ *j*=1 (*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*<sup>x</sup> j <sup>k</sup>*|*k*−1)(**X***<sup>j</sup> <sup>k</sup>*|*k*−1) −1 (*z<sup>j</sup>* <sup>−</sup> (**H***<sup>k</sup>* <sup>⊗</sup> **<sup>I</sup>***d*)*<sup>x</sup> j <sup>k</sup>*|*k*−1) T 15: **end for** 16: *<sup>D</sup>* <sup>=</sup> <sup>−</sup>0.5*nW* log 2*<sup>π</sup>* <sup>−</sup> 0.5*nW* log <sup>|</sup>**X***k*|*k*−1| − *pd*(*<sup>γ</sup>* <sup>−</sup> *<sup>λ</sup>c*) + *nW* log( *<sup>γ</sup> λc* ) − *nW* log *βFA*. 17: log(*η*) = −0.5 arg min *j G<sup>j</sup>* + *D* 18: *<sup>g</sup>* <sup>=</sup> *<sup>D</sup>* <sup>−</sup> 0.5<sup>X</sup> <sup>2</sup> <sup>1</sup>−*α*, 19: *d* (*ρ*,*W*) *<sup>W</sup>* = ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ 1 + *Jk*|*k*−<sup>1</sup> ∑ *l*=1 *e*−*γ*(*l*) *γ*(*l*) *βFA*,*<sup>k</sup>* |*W*| *p* (*l*) *<sup>D</sup>* <sup>Λ</sup>(*l*,*W*) *<sup>k</sup> <sup>w</sup>*(*l*) *<sup>k</sup>*|*k*−<sup>1</sup> log *<sup>η</sup>* <sup>&</sup>lt; *<sup>g</sup> Jk*|*k*−<sup>1</sup> ∑ *l*=1 *e*−*γ*(*l*) *γ*(*l*) *βFA*,*<sup>k</sup>* |*W*| *p* (*l*) *<sup>D</sup>* <sup>Λ</sup>(*l*,*W*) *<sup>k</sup> <sup>w</sup>*(*l*) *<sup>k</sup>*|*k*−<sup>1</sup> log *<sup>η</sup>* <sup>≥</sup> *<sup>g</sup>* 20: *<sup>w</sup>*(*j*+*l*·*Jk*|*k*−1,*W*) *<sup>k</sup>*|*<sup>k</sup>* <sup>←</sup> *<sup>w</sup>* (*j*+*l*·*J <sup>k</sup>*|*k*−1,*W*) *k*|*k dW* 21: **end for** 22: *<sup>ω</sup>p<sup>ρ</sup>* <sup>←</sup> <sup>Π</sup>|*p<sup>ρ</sup>* <sup>|</sup> *<sup>W</sup>*=1*d* (*ρ*,*W*) *W* 23: **end for** 24: *<sup>ω</sup>p<sup>ρ</sup>* <sup>←</sup> *<sup>ω</sup>p<sup>ρ</sup> n* ∑ *ρ*=1 *ωp<sup>ρ</sup>* for *ρ* = 1 : *n* 25:*Jk*|*<sup>k</sup>* <sup>←</sup> *Jk*|*k*−1(*<sup>l</sup>* <sup>+</sup> <sup>1</sup>) *Jtmp* <sup>=</sup> *Jk*|*k*−<sup>1</sup> 26: **for** *ρ* = 1 : *n* 27: **for** *<sup>j</sup>* <sup>=</sup> 1 : *Jk*|*k*−1|*pρ*<sup>|</sup> 28: *<sup>w</sup>*(*j*+*Jtmp* ) *<sup>k</sup>*|*<sup>k</sup>* <sup>←</sup> *<sup>w</sup>*(*j*+*Jtmp* ) *<sup>k</sup>*|*<sup>k</sup> <sup>ω</sup>p<sup>ρ</sup>* 29: **end for** 30: *Jtmp* <sup>←</sup> *Jtmp* <sup>+</sup> *Jk*|*k*−1|*pρ*<sup>|</sup> 31: **end for** 32: **Output:** GIW components {*w<sup>j</sup> k*|*k* , *ξ j <sup>k</sup>*|*k*} *Jk*|*k j*=1
