# Compressive Sensing Imaging Based on Modulation of Atmospheric Scattering Medium
## 3 Numerical Simulation
In the simulation, we set the surface meteorological range to vary randomly over time from 0.5 km to 2 km. The laser passed through the entire atmosphere and the transmission distance was set to 100 km. The emitting directions were random with two propagation paths of zenith and azimuth angles of 160\({}^{\circ}\) and 355\({}^{\circ}\), respectively, and 176\({}^{\circ}\) and 54\({}^{\circ}\), respectively. The pixel number of the simulation was \(32\times 32\) and the side length of every pixel was 30 m. The PSF obtained was a single modulation \(\mathcal{O}_{\text{i}}\) in CS. By changing the emitting direction randomly, 1024 modulation patterns were simulated, which were used to generate the measurement matrix meeting the requirements of full sampling. The measurement matrix is crucial for the CS method. General modulation measurement matrices (such as the Hadamard matrix and the Gaussian random matrix) are designed in advance. In this case, the atmospheric modulation measurement matrix was essentially determined by the real-time changes in the atmosphere. Moreover, the modulation mode of the atmosphere was grayscale (as shown in Figure 2), while the general modulation is binary. This is an innovative application of CS methods. To ensure that  
Figure 2: Atmospheric modulation patterns for two different propagation paths. The propagating direction was determined by the zenith and azimuth angles: (**a**,**b**) are the 3D images, (**c**,**d**) are the corresponding 2D images (modulation patterns). the original signal can be reconstructed from the observations, the measurement matrix needs to meet certain requirements. The restricted isometry property (RIP) [36; 37] is the first important condition to consider. However, it is difficult to judge whether a matrix satisfies the RIP or not, and it is not practical to use the RIP to guide the design of the measurement matrix. Donoho et al. proposed that if the correlation coefficients between the selected sparse basis (\(\Psi\)) and measurement matrix (\(\Phi\)) are small, then the RIP condition will be satisfied with a high probability. It is inferred that if the diagonal elements of the Gram matrix \(G\)[38] (\(G=\widetilde{\Theta}^{T}\widetilde{\Theta}\), where \(\widetilde{\Theta}\) is the column-normalized version of \(\Theta\), and \((\ )^{T}\) denotes the transposition) are close to 1, with a high probability, and the amplitudes of the non-diagonal elements are small, the matrix meets the RIP condition [39]. To verify the RIP condition, the common Fourier base is used as the sparse basis \(\Psi\). Figure 3 shows the element distribution diagram of the Gram matrix G. The Figure 3 shows that the atmospheric measurement matrix satisfies the RIP condition. We performed the CS simulation experiment using the measurement matrix obtained by the backward MC simulation. The simulation process consists of two parts: the measurement simulation and the reconstruction simulation. In the measurement process, the atmospheric medium was constructed by the MODTRAN model. The position of the laser was changed randomly and a series of single modulation patterns, \(\Phi_{i}\), were generated for different laser propagation directions, which were combined to form the measurement matrix, \(\Phi\). The measurement matrix was multiplied by the original signal, \(x\), to simulate the detection value, \(y\), of the bucket detector. In the reconstruction simulation process, based on the simulated detection values and the selected Fourier sparse basis \(\Psi\), the basic OMP algorithm recovered the original image.