# UAV Detection with Transfer Learning from Simulated Data of Laser Active Imaging
## 3 Data Simulation  
To simulate the imaging process, we need to have complete knowledge of the laser active imaging system. The active imaging system uses its light source to irradiate the target area and receives the reflected signal of the target. Hence, it is not limited by the illumination of the scene. At the same time, the laser has the characteristics of high brightness and high collimation, so it is an ideal light source. The commonly used laser active illumination imaging system is shown in Figure 2. The laser is emitted to irradiate the object, and then the camera images at the image plane by receiving the reflected signal of the object.  
Figure 2: Schematic diagram of our laser active imaging system.  
In the process of laser illumination, the laser can be modeled as a Gaussian beam [25], so the intensity of light can be written as:  
\[I(r,z)=I_{0}\bigg{(}\frac{w_{0}}{w(z)}\bigg{)}^{2}\exp\bigg{(}\frac{-2r^{2}}{w^{2} (z)}\bigg{)} \tag{1}\]  
where  
\[w(z)=w_{0}\sqrt{1+\bigg{(}\frac{z}{z_{R}}\bigg{)}^{2}} \tag{2}\]  
is the spot size in the forward propagation direction \(z\). \(r\) is the radial coordinate, taking the optical axis center as a reference. \(I_{0}=I(0,0)\) is the irradiance at the center of beam waist \(w_{0}\). \(z_{R}=\pi w^{2}{}_{0}/\lambda\) is the Rayleigh length, with \(\lambda\) being the wavelength.  
Reflection occurs when the laser reaches the target surface. It is a non-trivial problem to determine the reflectivity of each point on the target plane. For simplicity, the reflectivity was approximated by the gray level of the panchromatic visible image. Ignoring the influence of atmospheric turbulence, the irradiance at the target plane can be approximated as:  
\[I_{0}(x_{0},y_{0})\approx I_{c}\exp\left(\frac{-2\times\Big{(}(x_{0}-x_{c})^{2 }+(y_{0}-y_{c})^{2}\Big{)}}{w^{2}(z)}\right).I_{gray} \tag{3}\]  
where \(I_{gray}\) is the gray level of the image in natural light; \(I_{c}\) denotes the irradiance of the center point (\(x_{c}\), \(y_{c}\)) of the Gaussian beam at the target plane. Then the process from the object surface to the imaging surface can be represented based on standard statistical optics:  
\[\begin{array}{ll}I_{im}(x_{i},y_{i})&=\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}J_{0}^{\prime}(x_{0},y_{0 };x_{0}^{\prime},y_{0}^{\prime})\\ &\times h(x_{i}-x_{0},y_{i}-y_{0})\times h^{*}(x_{i}-x_{0}^{\prime},y_{i}-y_{ 0}^{\prime})dx_{0}dy_{0}dx_{0}^{\prime}dy_{0}^{\prime}\end{array} \tag{4}\]  
where \((x_{i},y_{i})\) denotes the position in the image plane; \(J_{0}^{\prime}(x_{0},y_{0};x_{0}^{\prime},y_{0}^{\prime})\) is the mutual intensity of \((x_{0}\), \(y_{0})\) and \((x_{0}^{\prime},y_{0}^{\prime})\), two points in the target plane; \(h(x_{i}\), \(y_{i})\) is the amplitude spread function of the imaging system. This representation of the imaging process is general, but it is difficult to determine the mutual intensity. For simplicity, the mutual intensity is usually modeled in two extreme cases: fully coherent and completely incoherent. The real laser works between these two extremes.