*3.4. Target Confirmation Based on Sequential Probability Ratio Test*

Revisions of posterior weights will bring a new problem: once Poisson-distributed false alarms are captured by the probability density of the spontaneously appearing targets, the proposed algorithm will regard them as targets and maintain corresponding particles and weights by prediction step although no measurement available afterwards. In order to distinguish real targets from false alarms captured by the probability density of newborn targets, the measurement of each target extracted from posterior PHD should be recorded. Suppose *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* , *<sup>n</sup>* <sup>=</sup> 1, ··· , *<sup>N</sup>*<sup>ˆ</sup> *<sup>k</sup>*|*<sup>k</sup>* is the target with label *<sup>l</sup>* at time *k* extracted from the posterior PHD, and the measurement set at time *k* collected by the sensor is *Zk* = {*z*1, ··· , *zm*}, then the likelihood function of *Zk* with respect to *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is defined as

$$L\_{l,k} = \max\_{z \in Z\_k} L\_z(\mathfrak{x}\_{l,k}^{(n)}). \tag{11}$$

Obviously, the parameter *Ll*,*k*, *k* = 1, 2, ··· can tell us whether the target with label *l* is a real target. For simplification, the proposed algorithm binarizes the above likelihood function as

$$L'\_{l,k} = \begin{cases} 1, & L\_{l,k} \ge L^{th} \\ 0, & L\_{l,k} < L^{th} \end{cases} \tag{12}$$

where *Lth* is the threshold judging if there is a measurement of one target. The probability that there exists corresponding measurement of target *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is

$$\begin{aligned} &\Pr\{L'\_{l,k} = 1\} \\ &= \Pr\left(\max\_{z \in Z\_k} L\_z(\mathbf{x}\_{l,k}^{(n)}) \ge L^{th}\right) \\ &= 1 - \Pr\{L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{th}, \forall z \in Z\_k\} \end{aligned} \tag{13}$$

Furthermore, the bigger the cumulative sum *sum k L <sup>l</sup>*,*k*, the more we can confirm that the target with label *l* is a real target.

In order to confirm targets in real time, this paper proposes the method based on sequential probability ratio test. Without loss of generality, suppose the single-target likelihood function is Gaussian

$$L\_z(\mathbf{x}) = \sqrt{\frac{1}{\left(2\pi\right)^2 \det(\mathbf{C})}} \exp\left(-\frac{1}{2} (z - H(\mathbf{x}))^T \mathbf{C}^{-1} (z - H(\mathbf{x}))\right) \tag{14}$$

where *z* = [*z*1, *z*2] *<sup>T</sup>*, *<sup>C</sup>* = *diag* σ2 <sup>1</sup>, <sup>σ</sup><sup>2</sup> 2 is the covariance matrix of observation noises, *H*(*x*) is the deterministic state-to-measurement transform model, and target *x* is located at the coordinate origin, the probability that the likelihood function of single measurement is above the threshold is

*Sensors* **2019**, *19*, 2842

$$\Pr\{L\_z(\mathbf{x}) > L^{th}\} = \Pr\left(\frac{z\_1^2}{\sigma\_1^2} + \frac{z\_2^2}{\sigma\_2^2} < -2\ln\left(2\pi L^{th}\sqrt{\sigma\_1^2 \sigma\_2^2}\right)\right) = \begin{array}{c} \iint\limits\_{\frac{z\_1^2}{\sigma\_1^2} + \frac{z\_2^2}{\sigma\_2^2} < -2\ln\left(2\pi L^{th}\sqrt{\sigma\_1^2 \sigma\_2^2}\right)} \frac{\sqrt{\pi}}{\sqrt{\sigma\_1^2 \sigma\_2^2}} \end{array} \tag{15}$$

which indicates the measurement *z* lies inside the ellipse *z*<sup>2</sup> 1/σ<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>z</sup>*<sup>2</sup> 2/σ<sup>2</sup> <sup>2</sup> <sup>=</sup> <sup>−</sup><sup>2</sup> ln% <sup>2</sup>π*Lth* σ2 1σ2 2 & and where *f*(*z*1, *z*2) is the spatial distribution of *z*. Suppose false alarms obey uniform distribution spatially, then the probability that the likelihood function of single clutter is above the threshold is *p*<sup>0</sup> = *Se*/*SFOV*, where *Se* is the area of the above ellipse and *SFOV* is the area of the whole FoV. If a real target is detected, the probability that the likelihood function of target measurement is above the threshold is *<sup>q</sup>*<sup>0</sup> <sup>=</sup> *z*2 1 σ2 1 + *z*2 2 σ2 2 <sup>&</sup>lt;−2 ln (2π*Lth* σ2 1σ2 2) *Lz*(*x*)*dz*1*dz*2. Suppose σ<sup>2</sup> = σ<sup>2</sup> <sup>1</sup> <sup>=</sup> <sup>σ</sup><sup>2</sup> <sup>2</sup>, then the probability *q*<sup>0</sup> with respect to

observation noise variance σ<sup>2</sup> under different thresholds is depicted in Figure 1, which indicates that *q*<sup>0</sup> is close to 1 under a suitable threshold.

**Figure 1.** Probability that the likelihood function of target measurement is above the threshold.

Next, we consider the situation of multiple measurements. When the target *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is a false alarm, the measurement set *Zk* can be organized as the union of measurements from targets and clutter. The likelihood function of *Zk* with respect to *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is

$$L\_{l,k} = \max\left\{ \max\_{z \in Z\_k \backslash K\_k} L\_z(\mathbf{x}\_{l,k}^{(n)}), \max\_{z \in K\_k} L\_z(\mathbf{x}\_{l,k}^{(n)}) \right\},\tag{16}$$

Therefore, Equation (13) is

$$\begin{split} & \Pr\{L'\_{l,k} = 1\} \\ &= 1 - \Pr\{L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, \forall z \in Z\_k \} \text{Pr}\{L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, \forall z \in K\_k\} \\ & \approx 1 - \Pr\{L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, \forall z \in K\_k\} \\ &= 1 - \left(\Pr\{L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, z \in K\_k\}\right)^D \\ &= 1 - (1 - p\_0)^D \end{split} \tag{17}$$

where *<sup>D</sup>* is the number of clutters, following Pr(*<sup>D</sup>* = *<sup>d</sup>*) = <sup>λ</sup>*de*−λ/*d*!, *<sup>d</sup>* = 0, 1, ··· . The approximation is reasonable, because the false alarm always appears before or after corresponding real target, which results that it can neither be associated with the measurement of its corresponding real target nor that of other real targets. Considering *D* is a random variable, the expectation of Equation (17) is

$$E\left[1 - (1 - p\_0)^D\right] = 1 - \sum\_{d=0}^{\infty} (1 - p\_0)^d \frac{\lambda^d e^{-\lambda}}{d!} = 1 - e^{-p\_0 \lambda}.\tag{18}$$

On the other hand, when the target *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is a real target, the measurement set *Zk* can be divided into three parts: the measurement generated from target *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* , measurements generated from other targets and clutters. The likelihood function of *Zk* with respect to *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is

$$L\_{l,k} = \max\left\{ L\_{\Theta\_k(\mathbf{x}\_{l,k}^{(n)})} (\mathbf{x}\_{l,k}^{(n)}), \max\_{z \in K\_k} L\_z(\mathbf{x}\_{l,k}^{(n)}), \max\_{z \in Z\_k(\mathcal{K}\_k) \Theta\_k(\mathbf{x}\_{l,k}^{(n)})} L\_z(\mathbf{x}\_{l,k}^{(n)}) \right\}. \tag{19}$$

Therefore, Equation (13) is

$$\begin{split} &\Pr\left(L'\_{l,k}=1\right) \\ &= p\_D(\mathbf{x}\_{l,k}^{(n)}) \left[1 - (1-q\_0)(1-p\_0)^D \Pr\left(L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, \forall z \in Z\_k | \mathcal{K}\_k\right) \Theta\_k(\mathbf{x}\_{l,k}^{(n)})\right] \\ &+ \left(1-p\_D(\mathbf{x}\_{l,k}^{(n)})\right) \left[1 - (1-p\_0)^D \Pr\left(L\_z(\mathbf{x}\_{l,k}^{(n)}) < L^{\text{th}}, \forall z \in Z\_k | \mathcal{K}\_k\right)\right] \\ &\approx p\_D(\mathbf{x}\_{l,k}^{(n)}) \left[1 - (1-q\_0)(1-p\_0)^D\right] + \left(1-p\_D(\mathbf{x}\_{l,k}^{(n)})\right) \left[1 - (1-p\_0)^D\right] \\ &\approx p\_D(\mathbf{x}\_{l,k}^{(n)}) + \left(1-p\_D(\mathbf{x}\_{l,k}^{(n)})\right) \left[1 - (1-p\_0)^D\right] \\ &\approx p\_D(\mathbf{x}\_{l,k}^{(n)}) \end{split} \tag{20}$$

in which we consider if the real target *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* is detected. Three approximations are reasonable when all real targets are far from each other, *q*<sup>0</sup> is close to 1, and *pD*(*x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* ) >> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*p*0λ, respectively.

In summary, random variable *L <sup>l</sup>*,*<sup>k</sup>* obeys two-point distribution

$$\Pr\left(L'\_{\,\,l,k} = 1\right) = p\_\prime \Pr\left(L'\_{\,\,l,k} = 0\right) = 1 - p\_\prime \tag{21}$$

where success probability *<sup>p</sup>* = *<sup>p</sup>*<sup>1</sup> = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*p*0<sup>λ</sup> when the target with label *<sup>l</sup>* is a false alarm, and *<sup>p</sup>* = *<sup>p</sup>*<sup>2</sup> = *pD*(*x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* ) when the target with label *<sup>l</sup>* is a real target. Then, target confirmation can be represented as a hypothesis test problem

$$H: p = p\_1 \leftrightarrow K: p = p\_2. \tag{22}$$

SPRT tells us: when *sum k L <sup>l</sup>*,*<sup>k</sup>* ≤ *An* is true, to accept *H*, mark *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* as a false alarm and eliminate corresponding particles; when *sum k L <sup>l</sup>*,*<sup>k</sup>* ≥ *Bn* is true, to reject *H* and mark *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* as a real target; otherwise, to make no decision and maintain particles of *x* (*n*) *<sup>l</sup>*,*<sup>k</sup>* . The parameters *An* and *Bn* are

$$\begin{array}{l} A\_{\text{II}} = \left( \frac{\beta}{1-\alpha} - n \ln \frac{1-p\_2}{1-p\_1} \right) / \ln \frac{p\_2(1-p\_1)}{p\_1(1-p\_2)}\\ B\_{\text{II}} = \left( \frac{1-\beta}{\alpha} - n \ln \frac{1-p\_2}{1-p\_1} \right) / \ln \frac{p\_2(1-p\_1)}{p\_1(1-p\_2)} \end{array} \tag{23}$$

where α, β are Type I error rate and Type II error rate, respectively, and *n* is the cumulative time of the target with label *l* from emerging to current step.

It should be mentioned that confirmation of a real target always lags behind its emerging. Fortunately, we can make up point-valued estimates of the target at previous times.
