## 2 Materials and Methods
\tag{7}\]  
Therefore, the free path \(l_{R}\) could be determined by attenuation coefficient c. Furthermore, updates to the transmission direction, position and time-of-flight of photon packets are accompanied by each movement. Reflection: the process of which is closely related to the material properties of the target surface and the direction of light incidence [33]. In order to describe the reflected energy distribution of the target, the bidirectional reflectance distribution function (BRDF) model proposed by [PERSON] is used in this study to divide the target surface into microfacets satisfying the Gaussian distribution, and each micro element follows a Fresnel reflection formula [34; 35]. The reflection \(f_{s}\) is expressed as:  
\[f_{s}=\frac{F(\theta_{i},\theta_{r})G(\theta_{i},\theta_{r},\Delta\varphi)}{ \pi\cos\theta_{i}\cos\theta_{r}} \tag{8}\]  
\(F(\theta_{i},\theta_{r})\) is the Fresnel reflection coefficient, and \(G(\theta_{i},\theta_{r},\Delta\varphi)\) represents the geometric attenuation factor that the reflected light is attenuated due to the mutual occlusion of the microfacets, including blocking the incident light and the outgoing light, the following equation is used:  
\[G(\theta_{i},\theta_{r},\Delta\varphi)=\min\left(1;\frac{2\cos\theta_{N}\cos \theta_{r}}{\cos\beta};\frac{2\cos\theta_{N}\cos\theta_{i}}{\cos\beta}\right) \tag{9}\]  
\(\Delta\varphi\) is the angle between the incident light and the reflected light at the microfacet, \(\beta\) is the angle between the incident light and the normal to the micro surface, \(\theta_{i}\), \(\theta_{r}\) and \(\theta_{N}\) are the incident angle, the reflection angle and the angle between the normal direction of the microfacet and the normal direction of the target surface, respectively. The diffuse reflection part \(f_{d}\) is expressed as:  
\[f_{d}=\rho_{d}G_{d} \tag{10}\]  
\(\rho_{d}\) is the light intensity ratio between diffuse and full Lambertian surface scattering. Here, \(R_{i}\) is the Fresnel reflectivity of the surface, which can be calculated as \(R_{i}=\left(r_{s}+r_{p}\right)/2\). \(K_{2}\) is the Fresnel reflection of diffuse light on the material surface and is usually assumed to be approximately equal to \(R_{i}\). \(R_{\infty}\) is the reflectivity coefficient when the surface thickness of the object is seen as infinite, which can be seen as a constant for a given material at a fixed incident wavelength. \(G_{d}\) represents the angular distribution of the Lambert diffuse scattering intensity on the surface of the target [36]. \[\rho_{d}=(1-R_{i})\frac{(1-K_{2})R_{\infty}}{1-K_{2}R_{\infty}} \tag{11}\]  
\[G_{d}=\frac{\cos\theta_{r}}{\pi} \tag{12}\]  
Thus, the residual weight of photon packets in the direction of reflection can be determined according to Equations (8) and (10). The condition for identifying photon packets reach target is whether the projection of the distance on the z-axis exceeds the position of target. Once the condition is met, it means that the target was incident in the last random step, and the current position needs to be corrected to make it return to the target plane. \[ \begin{cases}X=X^{\prime}-l_{Rx}(Z^{\prime}-L)/l_{Rz}\\ Y=Y^{\prime}-l_{By}(Z^{\prime}-L)/l_{Rz}\\ Z=L\end{cases}  \tag{13}\]  
in which \(X^{\prime},Y^{\prime}\) and \(Z^{\prime}\) are the original coordinates while \(X,Y\) and \(Z\) are the corrections in the target plane, whose distance from origin is \(L\). \(l_{Rx}\), \(l_{By}\) and \(l_{Rz}\) are the projection of propagation direction on \(X,Y\) and \(Z\) axis. Gathering: the receiver is placed at \(\left(l_{x},l_{y},l_{z}\right)\) with a specific aperture, focal length and field of view (FOV), and the weights of photon packets are accumulated while the following conditions are met:  
\[ \begin{cases}&x\leq l_{x}\\ &y\leq l_{y}\\ &z\leq l_{z}\\ &u_{z}^{\prime}\geq\cos(\text{FOV})\end{cases}  \tag{14}\]  
Subsequently, the image is obtained according to the calibrated response curve, and the SSIM, MSSIM and PSNR indicators are used as the evaluation standard for it [37,38].