# Compressive Sensing Imaging Based on Modulation of Atmospheric Scattering Medium
## 2 Theoretical Basis and Imaging Model  
Shannon-Nyquist sampling theorem proposes that in order to ensure the sampled signal can restore the complete information of the original signal, the sampling frequency is at least twice the bandwidth of the original signal. In 2004, Candes et al. [4; 24] proposed a new guiding theory of information acquisition called CS. According to this theory, if a high-dimensional signal can be sparsely represented in a projection domain, it can be mapped onto a low-dimensional space. By solving the nonlinear optimization problem, the original signal can be accurately recovered in a way far below the Shannon-Nyquist sampling rate. Its mathematical expression [25] is:  
\[y=\Phi\mathbf{x}=\Phi\;(\Psi\;s)=(\Phi\;\Psi)s=\Theta\;s \tag{1}\]  
where \(y\) is the measured value of the \(M\)-dimensional column vector, \(\Phi\) is the \(M\times N\) (\(M\leq N\)) measurement matrix, \(x\) is the original signal of the \(N\)-dimensional column vector, and \(\Psi\) is a sparse \(N\times N\) basis matrix. The product of \(\Phi\) and \(\Psi\) gives an \(M\times N\) matrix, \(\Theta\). \(s\) is the projection value of \(x\) on the sparse basis \(\Psi\), meaning it has only a small group of coefficients that are nonzero. Since the dimensional relationship of the matrix is \(M<N\), the solution of the signal is ill-conditioned (non-deterministic polynomial hard) [26; 27]. It must be resolved by using a proper reconstruction algorithm. Various solutions have been developed, such as orthogonal matching pursuit (OMP) [28], basic pursuit (BP), and expectation-maximization [29].  
For the CS imaging model, the measurement matrix is generated by the DMD. Popoff et al. have shown that the TM of the optical multiple scattering materials is an excellent candidate for CS [22]. In this study, we will demonstrate through theory, simulation, and experiments that the imaging can be produced using the atmospheric scattering medium as a DMD for CS.  
CS imaging based on modulation of atmospheric scattering medium is shown in Figure 1. The collimated laser beam passes through the atmosphere, over a long distance, to the target. The light field is transmitted through the target and collected by the bucket detector. Based on the analysis above, the measurement matrix, \(\Phi\), of the mathematical expression in Equation (1) is determined by the atmospheric scattering medium. When the laser is transmitted through the atmosphere, it is modulated by the medium. Owing to the stochastic and time-varying characteristics of the scattering medium, the modulation process also changes randomly over time. A series of different modulation patterns are obtained to illuminate the target. The modulation effect becomes more pronounced as the transmission distance increases. Long distances, which are difficult to overcome in scattering imaging,become favorable conditions in transmission imaging. Meanwhile, to further ensure the randomness of the modulation process, we designed the laser in such a way that it enters the atmospheric medium at random angles (expressed as the zenith angle \(\theta_{0}\) and azimuth angle \(\varphi_{0}\) in Figure 1). The total light intensity passing through the target is collected by the barrel detector. The target image is reconstructed using measured intensity values with the proper computer algorithm.