Cascaded GLRT Radar/Infrared Lidar Information Fusion Algorithm for Weak Target Detection

Abstract

To deal with the problem of weak target detection, a cascaded generalized likelihood ratio test (GLRT) radar/infrared lidar heterogeneous information fusion algorithm is proposed in this paper. The algorithm makes full use of the target characteristics in microwave/infrared spectrum and the scanning efficiency of different sensors. According to the correlation of target position in the multi-sensor view field, the GLRT statistic derived from the radar measurements is compared with a lower threshold so as to generate initial candidate targets with high detection probability. Subsequently, the lidar is guided to scan the candidate regions and the final decision is made by GLRT detector to discriminate the false alarm. To get the best detection performance, the optimal detection parameters are obtained by nonlinear optimization for the cascaded GLRT Radar/Infrared lidar heterogeneous information fusion detection algorithm. Simulation results show that the cascaded GLRT heterogeneous information fusion detector comprehensively utilizes the advantages of radar and infrared lidar sensors in detection efficiency and performance, which effectively improves the detection distance upon radar weak targets within the allowable time.

1 Introduction

Some important targets with collaborate design of shape and material are capable of backscattering the incidence electromagnetic wave weakly and the radar detection performance degrades a lot. Single-mode sensors are no longer satisfy the detection requirements, and multi-sensor fusion detection has become a development trend [1, 2], such as multi-radar sensors fusion [3], multi-infrared sensors fusion [4], radar/infrared fusion [5], radar/optical fusion [6], lidar point cloud/optical fusion [7], etc.

Since it is hard to control the targets characteristics in microwave and infrared frequency bands simultaneously, radar and infrared sensors have become an important combination mode for fusion detection. In [8], the infrared imaging/active radar fusion detection of weak target is realized through spatiotemporal registration and radar virtual detection image generation from infrared image. In [9], the relevance of radar/infrared characteristics is used for multi-target association. However, the maximum detection range of passive infrared sensors usually mismatches to that of radar. With the development of laser phased array technology, the combination of radar and infrared lidar will exhibits potential in aerial target detection.

Although it’s easy to realize the spatiotemporal registration for co-platform radar/infrared lidar, the target characteristics in microwave/infrared spectrum has great difference which increased the difficulty of fusion detection. Besides, the mechanisms of radar and infrared lidar are different from each other. The wide beam of the radar can lead to quicker scanning but the detection angle resolution is low; the narrow beam of lidar can lead to higher detection angle resolution but the scanning and detection speed is low. Therefore, this paper proposed a cascaded GLRT radar/infrared lidar heterogeneous information fusion algorithm to solve the fusion detection problem of that radar/infrared lidar cross-spectrum sensors have difference on target characteristics and detection mechanisms.

Aiming at the problem of long distance and weak targets detection, the radar/infrared lidar heterogeneous fusion detection method is studied in this paper. Based on the target location prior constraint relationship of multi-sensor, a low detection threshold is set for radar detection firstly, and then the infrared lidar is guided by radar detection results for further detection and false alarm elimination. The organization of the paper is as follows: Sect. 2 describes the radar/infrared lidar measurement model, Sect. 3 provides the method of heterogeneous fusion detection, and the simulation experiments of typical scenarios are demonstrated in Sect. 4, and Sect. 5 concludes the paper.

2 Measurement Model of Radar and Infrared Lidar

2.1 Radar Echo Model

When the radar transmits a series of pulses with carrier frequency $f_c$, the echo of a target at distance $R$ can be expressed as follows

$$ S_r (t) = \sum_{k = 0}^{CPI - 1} {\sqrt {P_{t_R } \cdot \sigma_R \cdot K} \cdot {\text{rect}}\left[ {\frac{{t - \tau_{Ra} - kT_{r_R } }}{{T_{p_R } }}} \right] \cdot \exp \{ {\text{j}}2{\uppi }f_c (t - \tau_{Ra} )\} + e_R (t)} $$

(1)

where $P_{t_R }$ is emitted peak power, $\sigma_R$ is the radar cross section of target, $CPI$ is the number of pulses, $T_{r_R }$ is pulse repetition period and $T_{p_R }$ is the pulse width. $\tau_{Ra} = 2R/c$ is the echo delay time of the target, $K = \frac{G^2 \lambda_R^2 }{{(4\pi )^3 R^4 }}$ is the propagation decay factor, $e_R (t)$ is complex white Gaussian noise due to the receiver [11] with variance $P_{n_R }$, and

$$ P_{n_R } = kT_0 BN_F $$

(2)

$k = 1.38 \times 10^{{ - }23} \,J/K$ is the Boltzmann constant, $T_0 { = }290\,K$, $B$ is the bandwidth of receiver and $N_F$ is the noise coefficient of receiver.

2.2 Lidar Echo Model

The lidar echo of a target at distance $R$ can be expressed as follows [13].

$$ S_r (t) = \sum_{k = 0}^{CPI - 1} {\sqrt {P_{t_R } \cdot \sigma_R \cdot K} \cdot {\text{rect}}\left[ {\frac{{t - \tau_{Ra} - kT_{r_R } }}{{T_{p_R } }}} \right] \cdot \exp \{ {\text{j}}2{\uppi }f_c (t - \tau_{Ra} )\} } + e_R (t). $$

(3)

$Tau = T_{half} /\sqrt {8\ln 2}$, $P_{t_L }$ is emitted peak power, $\sigma_L$ is the lidar cross section of target, $T_{p_L }$ is the pulse width, $\tau_{Li} { = }\frac{2R}{c}$ is target echo delay time, $K = \frac{G_T }{{(4\pi R^2 )^2 }} \cdot \frac{\pi D_r^2 }{4}$ is the propagation decay, $e_L (t)$ is the background light noise including the sunlight reflected by the target and scattered by the atmosphere and the direct sunlight [14]. The noise variance

$$ P_b = \frac{{\uppi }}{{{16}}}\eta_{r_L } \Delta \lambda \theta_{t_L }^2 D_r^2 [\rho T_a H_\lambda \cos \theta \cos \varphi + \frac{\beta }{4\alpha }(1 - T_a )H_\lambda + {\uppi }L_\lambda ] $$

(4)

In case of air-to-air lidar detection, by reviewing the paper [15], the angle between the sun ray and the target surface is taken $\theta = 0$, the angle between the normal line of the target surface and the receiving axis is taken $\varphi = 0$. In addition, the transmittance of receiving optical system is $\eta_t = 1$, receiving field angle $\theta_{t_L } = 1\,{\text{mrad}}$, target reflection coefficient $\rho = 0.8$, the narrowband filter bandwidth $\Delta \lambda = 50\,{\text{nm}}$, atmospheric transmittance $T_a = 0.87$. Atmospheric attenuation coefficient and scattering coefficient are $\alpha = 1$ and $\beta = 1$ combined with the detection requirements of more than 100 km [16]. The spectral radiance of atmospheric scattering and the spectral irradiance on the ground of sunlight are $L_\lambda { = 3}{\text{.04}} \times {10}^{ - 6} {\text{W}} /({\text{cm}}^2 \cdot {\text{sr}} \cdot {\text{nm}})$ and $H_\lambda { = 6}{\text{.5}} \times {10}^{ - 5} {\text{W}} /({\text{cm}}^2 \cdot {\text{nm}})$ are simulated by MOTRAN4.0 software. When $\lambda = 1064\,{\text{nm}}$, the $P_b \approx 1.1 \times 10^{ - 7} \,{\text{W}}$.

3 Radar/Infrared Lidar Fusion Detection Algorithm

The Radar/Infrared lidar fusion detection algorithm proposed in this paper is asynchronously cascaded, the radar target detection is finished firstly, then based on the radar detection results and position correlation, the infrared lidar is used for further detection and false alarm discrimination. The algorithm flow is shown in Fig. 1

Fig. 1.

Algorithm flow chart of cascade detection algorithm, the $P_{FA1}$ and $P_{FA2}$ are false alarm probability for radar detection and lidar detection

The radar/lidar heterogeneous fusion detection method includes two cascade target detection: the radar detection and the lidar false alarm discrimination. The received radar and lidar echo signals are converted into a discrete signal by the digital analogue digital converter (ADC), so the echo used for target detection is discrete sequence signal and the detection model [12] can be described as Eq. (5) uniformly.

$$ \left\{ {\begin{array}{*{20}c} {H_0 {\bf{:X}} = {\bf{w}}} \\ {H_1 {\bf{:X}} = A{\bf{S}} + {\bf{w}}} \\ \end{array} } \right. $$

(5)

For radar detection, ${\bf{X}}$ and ${\bf{S}}$ are observation signal and signal wave with length $N$. ${\bf{w}} \sim CN(0,\,\sigma^2 {\bf{I}}_N )$, and the probability density function (PDF) is ${\bf{X}} \sim \mathcal{C}\mathcal{N}(0,\,\sigma^2 {\bf{I}}_N )$ for $H_0$ and ${\bf{X}} \sim \mathcal{C}\mathcal{N}(A{\bf{S}},\,\sigma^2 {\bf{I}})$ for $H_1$, A, $\sigma^2$ are both unknown parameters. The test statistics variable $T$ can be constructed by GLRT [12].

$$ T = \frac{{({\bf{S}}({\bf{S}}^H {\bf{S}})^{ - 1} {\bf{S}}^H {\bf{X}})^H ({\bf{S}}({\bf{S}}^H {\bf{S}})^{ - 1} {\bf{S}}^H {\bf{X}})/m}}{{(({\bf{I}} - {\bf{S}}({\bf{S}}^H {\bf{S}})^{ - 1} {\bf{S}}^H ){\bf{X}})^H (({\bf{I}} - {\bf{S}}({\bf{S}}^H {\bf{S}})^{ - 1} {\bf{S}}^H ){\bf{X}})/n}}{ = }\frac{{({\bf{P}}_{\bf{S}} {\bf{X}})^H ({\bf{P}}_{\bf{S}} {\bf{X}})/m}}{{(({\bf{I}} - {\bf{P}}_{\bf{S}} {\bf{X}})^H (({\bf{I}} - {\bf{P}}_{\bf{S}} {\bf{X}})/n}} $$

(6)

The PDF of test statistics variable $T$ is $T \sim F_{m,n}$ for $H_0$ and $T \sim F_{m,n} (\lambda )$ for $H_1$, $m = 2{\text{rank}} ({\bf{P}}_{\bf{S}} )$, $n = 2N - m$, $\lambda { = }\frac{{2(A{\bf{S}})^H (A{\bf{S}})}}{\sigma^2 }$.

For lidar false alarm discrimination, ${\bf{X}}$ and ${\bf{S}}$ are two-dimensional observation signal and signal wave with size $M \times N$, $M$ is the number of beams, $N$ is the number of distance bins. ${\bf{S}} = [s_{mn} ]_{M \times N}$ has an unknown parameter $m_0$( $m_0$ is the index of a beam containing targets), ${\bf{w}} = [w_{mn} ]_{M \times N}$, and $w_{mn} \rm{ \sim }\mathcal{N}(0,\,\sigma^2 )$. Assuming that ${\bf{\vec{X}}}$ and ${\bf{\vec{S}}}$ are both one-dimensional vectors stretched from the two-dimensional matrix ${\bf{X}}$ and ${\bf{S}}$, the PDF is ${\bf{\vec{X}}} \sim \mathcal{N}(0,\,\sigma^2 {\bf{I}}_{MN} )$ for $H_0$ and ${\bf{\vec{X}}} \sim \mathcal{N}(A{\bf{\vec{S}}},\,\sigma^2 {\bf{I}}_{MN} )$ for $H_1$, A, $\sigma^2$ are both unknown parameters. Then $T$ constructed by GLRT [12] is as Eq. (7) shows.

$$ T = \mathop {\arg \max }\limits_{m_0 \in M} \frac{{({\bf{P}}_{\bf{S}} {\bf{\vec{X}}})^T ({\bf{P}}_{\bf{S}} {\bf{\vec{X}}})/p}}{{((I - {\bf{P}}_{\bf{S}} ){\bf{\vec{X}}})^T ((I - {\bf{P}}_{\bf{S}} ){\bf{\vec{X}}})/(MN - p)}} $$

(7)

$p = rank({\bf{P}}_{{\bf{ S}}} ) = rank({\bf{\vec{S}}}({\bf{\vec{S}}}^T {\bf{\vec{S}}})^{ - 1} {\bf{\vec{S}}}^T )$. Because the correlation of the $M$ random variables is hard to analysis, it’s difficult to calculate the PDF of $T$. Considering that the lidar echo of a point target for the different beam is independent, if we use ${\bf{\vec{x}}}_m$ (the beam echo that beam index is $m$) to substitute ${\bf{\vec{X}}}$, use ${\bf{\vec{s}}}_m$ (the beam wave with index $m$) to substitute ${\bf{\vec{S}}}$, the $T$ is changed to

$$ T = \mathop {\arg \max }\limits_{m_0 \in M} \frac{{({\bf{P}}_{{\bf{\vec{s}}}_{m_0 } } {\bf{\vec{x}}}_{m_0 } )^T ({\bf{P}}_{{\bf{\vec{s}}}_{m_0 } } {\bf{\vec{x}}}_{m_0 } )/p_{m_0 } }}{{(({\bf{I}}_N - {\bf{P}}_{{\bf{\vec{s}}}_{m_0 } } ){\bf{\vec{x}}}_{m_0 } )^T (({\bf{I}}_N - {\bf{P}}_{{\bf{\vec{s}}}_{m_0 } } ){\bf{\vec{x}}}_{m_0 } )/(N - p_{m_0 } )}} $$

(8)

when $m_0$ is given, the ${\bf{\vec{s}}}_{m_0 }$ will be definite. And for different values of $m_0 {\bf{,}}\,{\bf{\vec{s}}}_{m_0 }$ are same, so the values of $p_{m_0 } = rank({\bf{P}}_{{\bf{\vec{s}}}_{m_0 } } ) = rank({\bf{\vec{S}}}_{m_0 } ({\bf{\vec{S}}}_{m_0 }^T {\bf{\vec{S}}}_{m_0 } )^{ - 1} {\bf{\vec{S}}}_{m_0 }^T )$ are same, and the observation echo in different beams are independent. Thus, the PDF of the test statistics variable is easy to analysis [17], the false alarm probability and detection probability are as Eq. (9) shows. $F_t$ is the distribution function of $t$ whose PDF is $F_{p,N - p} ,\,F_{t_2 }$ is the distribution function of $t_2$ whose pdf is $F^{\prime}_{p,N - p} (\lambda ),\,\lambda { = }\frac{{(A{\bf{\vec{s}}}_{m_0 } )^T (A{\bf{\vec{s}}}_{m_0 } )}}{\sigma^2 }$.

$$ \left\{ {\begin{array}{*{20}c} {P_{FA} = \Pr \{ T > \gamma |H_0 \} = 1 - (F_t (\gamma ))^M } \\ {{\text{P}}_D = \Pr \{ T > \gamma |H_1 \} = \Pr \{ t_1 > \gamma ,\,t_2 > \gamma |H_1 \} = 1 - F_t (\gamma )^{M - 1} F_{t_2 } (\gamma )} \\ \end{array} } \right. $$

(9)

In summary, suppose the Radar test statistics variable is $T_1$, the infrared lidar test statistics variable is $T_1$, the total $P_{FA}$ and $P_D$ of the detection system can be calculated

$$ \left\{ \begin{gathered} P_{FA} = \Pr \{ T_1 > \gamma_1 ,\,T_2 > \gamma_2 |H_0 \} = \Pr \{ T_1 > \gamma_1 |H_0 \} \cdot \Pr \{ T_2 > \gamma_2 |H_0 \} = P_{FA1} \cdot P_{FA2} \hfill \\ P_D = \Pr \{ T_1 > \gamma_1 ,\,T_2 > \gamma_2 |H_1 \} = \Pr \{ T_1 > \gamma_1 |H_1 \} \cdot \Pr \{ T_2 > \gamma_2 |H_1 \} = P_{D1} \cdot P_{D2} \hfill \\ \end{gathered} \right. $$

(10)

For a given $P_{FA}$, to get the best $P_D$ and satisfy the engineering application requirements for algorithm complexity at the same time, the following nonlinear optimization strategy are given to get the optimized false alarm probability parameters for cascade detection

$$ \left\{ {\begin{array}{*{20}c} {P_D = \mathop {\arg \max }\limits_{P_{FA1} ,\,P_{FA2} } P_{D1} \cdot P_{D2} } \\ {0 < P_D \le 1,0 < P_{FA1} < a,0 < P_{FA2} < 1} \\ {P_{FA1} \cdot P_{FA2} = P_{FA} } \\ \end{array} } \right. $$

(11)

The value of $a$ is related to the signal processing speed of the detection system. In actual engineering applications, it’s necessary to minimize the time required for signal processing to achieve real-time updates of detection results. Assuming that the system need finish the lidar false alarm elimination within a given time $T_{\lim }$, the expected detection time can approximately satisfy the inequality that

$$ E(T_D ) \approx N_{B_{Ra} } \cdot N_{R_{Ra} } \cdot P_{FA1} /n_{L^{\prime}} \le T_{\lim } \Rightarrow P_{FA1} \le \frac{{T_{\lim } \cdot n_{L^{\prime}} }}{{N_{B_{Ra} } \cdot N_{R_{Ra} } }} = a $$

(12)

$N_{B_{Ra} }$ is the number of Radar echo beams, $N_{R_{Ra} }$ is the number of Radar distance bins, $P_{FA1}$ is the false alarm probability for radar detection, $n_{L^{\prime}}$ is the number of false alarm eliminations completed by the signal processing system per unit time. $a = 10^{ - 1}$ in the paper, and the value of $a$ can be changed for different detection situations.

In addition, the single radar detection model is the same as radar detection. Besides, for single lidar detection, the $T$ can be constructed as $T = \frac{{({\bf{P}}_{\bf{S}} {\bf{X}})^T ({\bf{P}}_{\bf{S}} {\bf{X}})/p}}{{(({\bf{I}} - {\bf{P}}_{\bf{S}} {\bf{X}})^T (({\bf{I}} - {\bf{P}}_{\bf{S}} {\bf{X}})/(N - p)}}$, $p = {\text{rank}} ({\bf{\vec{S}}}({\bf{\vec{S}}}^T {\bf{\vec{S}}})^{ - 1} {\bf{\vec{S}}}^T )$, $\lambda { = }\frac{{(A{\bf{S}})^T (A{\bf{S}})}}{\sigma^2 }$.

4 Experiment and Analysis

4.1 Simulation Parameter Set

To verify the effectiveness of the cross-spectrum fusion detection algorithm, a stationary target detection simulation experiment is done in this part. (For moving targets, motion compensation can be used to convert target detection into an equivalent stationary target detection situation). The simulation parameters are as follows Table 1.

Table 1. The experiment simulation parameters

Figure 2 shows the SNR varies with detection distance. It can be seen that the SNR of lidar echo is higher than that of the radar echo for the same detection distance.

Fig. 2.

The signal-to-noise ratio (SNR) of radar echo and lidar echo for different distances

4.2 Simulation Results and Analysis

Three comparative experiments are carried out to verify the effectiveness of fusion detection method, single radar detection, single lidar detection and radar/infrared lidar fusion detection. The number of Monte Carlo simulations are 1000, according to the evaluation method proposed in the paper [18], the multi-sensor information fusion performance is analyzed as follow.

Detection performance curve.

Figure 3 shows the detection performance curve of the three detectors. The variations of $P_D$ with detection distance when $P_{FA}$ are $10^{ - 5}$ and $10^{ - 3}$ are given respectively. It shows that the detection probability of the fusion detection is obviously higher than single radar detection with the same detection distance; when the detection probability is 0.8, the combined detection result has the detection distance increment of 14 km and 12 km respectively compared with single-use radar detection in case of $P_{FA} = 10^{ - 5}$ and $P_{FA} = 10^{ - 3}$. Besides, when $P_{FA}$ is low (corresponding to the case that $P_{FA} = 10^{ - 5}$), the detection result of fusion detection is close to that of single lidar detection.

Fig. 3.

The detection performance curve for single radar detection, single lidar detection and radar/infrared lidar fusion detection when $P_{FA}$ is equal to $10^{ - 5}$ and $10^{ - 3}$

Detection Time.

The simulation scene is a two-dimensional plane with the azimuth angle range of [–5°, 5°], and the number of detection units of the azimuth and distance dimension for radar and lidar detection is shown in Table 2.

Table 2. Detection unit parameters

In addition, the detection time of different detection methods is analyzed in Table 3.

Table 3. The detection time of different detection methods

$n_R ,n_L$ and $n_{L^{\prime}}$ are the number of detection times completed by the signal processing system per unit time for radar detection, lidar detection and radar/lidar fusion detection. $N_{B_{Li} }$ is the number of lidar beams and $N_{R_{Li} }$ represents the number of range units of the lidar echo. According to the simulation experiment, we can obtain that $n_R \approx n_L \approx 20n_{L^{\prime}} ,P_{FA1} \le 10^{ - 1}$, thus

$$ T_{D_{Ra} } < T_{D_C } \rm{ < < }T_{D_{Li} } $$

(13)

Figure 4 shows the variations of detection time with $P_{FA}$ of the three detectors when R = 121 km. The time is calculated by MATLAB 2018b. The computer used in the experiment is a Lenovo Legion R7000 2020 notebook computer with 16G running memory, and the CPU is configured with an 8-core AMD Ryzen 7 4800 H.

Fig. 4.

The detection time of radar detection, lidar detection and radar/infrared lidar fusion detection for R = 121 km. The detection time is the average time for 1000 times simulation

It can be clearly seen that the detection time of the radar/infrared lidar fusion detection algorithm is much shorter than that of using a single lidar for detection.

5 Conclusion

Radar and infrared lidar are both active sensors, and they are complementary in working principle and detection performance. Based on the target characteristics and detection mechanism differences between radar and infrared lidar, this paper proposed a radar/infrared lidar cascade GLRT fusion algorithm for weak target detection and the optimal detection parameters are obtained by nonlinear optimization. The experimental simulation results show that the proposed fusion detection method has certain effectiveness: the heterogeneous information fusion detector comprehensively utilizes the advantages of radar and infrared lidar sensors in detection efficiency and performance, which effectively improves the detection distance upon radar weak targets within the allowable time. For further study, the joint statistics variable of radar/infrared lidar can be considered to constructed to make the best use of the target characteristics’ correlation between microwave and infrared.