# Compressive Sensing Imaging Based on Modulation of Atmospheric Scattering Medium
## 3 Numerical Simulation  
When a laser beam propagates in the atmosphere, the modulation effect of the atmosphere can be described by the atmospheric point spread function (PSF). Monte Carlo (MC) is a common method used to simulate the atmospheric PSF [30; 31; 32; 33]. For MC simulation, the atmosphere environment of light propagation is constructed with the moderate spectral resolution atmospheric transmittance (MODTRAN) algorithm and computer model, which divides the atmosphere, from 0 to 100 km high, into 34 layers. Physical parameters for each layer, including air pressure, temperature, absorption, and scatter coefficients are defined. The MODTRAN model is used to calculate the optical thickness of molecules and aerosols at different heights and the MC method is used to simulate photon propagation in the atmosphere. For CS imaging based on modulation of atmospheric scattering medium, a laser beam passes through the atmosphere and illuminates the target. The direction of laser emission relative to the target is represented by the zenith angle \(\theta_{0}\) and azimuth angle \(\varphi_{0}\). During the laser propagation, photons randomly collide with molecules or aerosols in the atmosphere. The molecule collision probability may be expressed as \(\tau_{m}/[\tau_{m}+\tau_{a}]\) where \(\tau_{m}\) and \(\tau_{a}\) are the molecule and aerosol optical thickness, respectively. A random number \(r_{0}\) (\(0\leq r_{0}\leq 1\)) is defined to describe the collision property. If \(0\leq\tau_{0}\leq\tau_{m}/[\tau_{m}+\tau_{a}]\), photons will collide with molecules. If \(\tau_{m}/[\tau_{m}+\tau_{a}]<r_{0}\leq 1\), photons will collide with aerosols. After any collision, the direction of photon propagation will be decided by the scattering phase function. For molecule and aerosol collisions, the phase functions used are the Rayleigh [34] and Henyey-Greenstein [35] phase functions, respectively. The Rayleigh phase function is defined as:  
\[P(\theta)=\frac{3}{16\pi}\Big{(}1+cos^{2}\theta\Big{)} \tag{2}\]  
where \(\theta\) is the scattering angle. The Henyey-Greenstein phase function is defined as:  
\[P(\theta)=\frac{1}{4\pi}\cdot\frac{3}{2}\cdot\frac{\Big{(}1-g^{2}\Big{)}\Big{(} 1+cos^{2}\theta\Big{)}}{\Big{(}2+g^{2})\big{(}1+g^{2}-2 gcos\theta\Big{)}^{3/2}} \tag{3}\]  
where \(g\) is the asymmetry factor. The normalized phase function is defined as:  
\[p(\theta)=\frac{\int_{0}^{\theta}P(\theta^{\prime})d\theta^{\prime}}{\int_{0 }^{2\pi}P(\theta^{\prime})d\theta^{\prime}} \tag{4}\]  
Figure 1: Compressive sensing (CS) imaging model based on modulation of atmospheric scattering medium. where the random numbers \(r_{1}\) and \(r_{2}\) (\(0\leq r_{1,2}\leq 1\)) are used to decide the direction after collision, based on the following equations:  
\[r_{1}=p(\theta) \tag{5}\]  
\[\varphi=2\pi r_{2} \tag{6}\]  
After the direction of the new trajectory is determined, the photon will propagate along a free path \(l\) represented by the random number \(r_{3}\) (\(0\leq r_{3}\leq 1\)) in the direction (\(\theta,\varphi\)) until it collides with another atmosphere particle [35]:  
\[l=-ln(1-r_{3}) \tag{7}\]  
where \(l\) is the optical distance between the two collisions. The collision and propagation processes are repeated until the photon strikes the target surface or is extinguished in the atmosphere. The positions of all photons on the target surface are recorded to obtain the atmospheric PSF. Examples of the atmospheric PSF are shown in Figure 2. The laser wavelength was 532 nm. The 1976 U.S. Standard Atmosphere model was used for the atmosphere and the rural extinction model was used for the aerosols. Since the scattering characteristics of the atmosphere were used for CS imaging, this requires that the atmosphere be turbid. Generally, the degree of turbidity of the atmosphere can be expressed by the surface meteorological range (also called standard visibility). The lower the surface meteorological range, the more turbid the atmosphere.