# Research on Multipath Correlated Imaging with the Grayscale Target in Endoscopic Applications
## 2 Model and Theoretical Analysis of Multipath Correlated Imaging  
Multipath can be divided into primary reflection multipath and higher-order multipath. Among them, primary reflection multipath has the strongest energy in multipath signals [33; 34; 35], so primary reflection multipath is considered in the following analysis. Figure 1a shows a multipath model of endoscope application in the closed environment. The signals received by the receiving port consist of one direct signal and several reflected signals [36]. Due to the reflection, the path length of the multipath signal is longer than that of the direct signal. What is more, the energy of multipath signals is usually much weaker than that of the direct signal on account of reflection. For simplicity, we treat all multipath signals as one path and combine it with CI to obtain the model of multipath CI, as shown in Figure 1b. It should be emphasized that when the time delay is larger than the coherence time of the source, there is inter-pulse disturbance. On the contrary, there is intra-pulse disturbance. In this case, the disturbance of multipath effect on the image is the most serious. So, intra-pulse disturbance is discussed in this paper. The classical thermal light source, usually obtained by illuminating a continuous laser into a slowly rotating ground glass, is divided into three beams by a combination of two beam splitters (BS). The three beams travel through a reference arm and two test arms, which are characterized by their impulse response function \(h_{2}(x_{2}\),\(u_{2}\)), \(h_{1}(x_{1}\),\(u_{1})\), \(h_{3}(x_{1}\),\(u_{1})\), respectively. The test arm includes a grayscale object to be imaged, which is located at a distance \(z_{2}\) from the bucket detector (\(D_{1}\)). As for the reference arm, the light propagates freely \(z\) length to a reference detector (\(D_{2}\)) with spatial resolution. Detectors \(D_{1}\) and \(D_{2}\) are used to record the intensity distribution at \(u_{1}\) and \(u_{2}\), respectively. In the test arm, the light may directly reach the object (see the dashed line \(z_{1}\)) or it may be reflected by the reflector and then reach the object (see the dashed line \(z_{3}\)). The reflected light interferes with the direct light during the imaging process, which may affect imaging quality. To get a deeper insight into the influence of multipath effect on CI, two parameters are defined, namely mismatch ratio (\(\eta\)) and reflection ratio (\(\rho\)). The former is defined as \((z_{3}-z_{1})/z_{1}\), and the latter is defined as the ratio of power between the reflected light and the total object light. Figure 1: (**a**) A multipath model of endoscope application in the closed environment; (**b**) Experimental setup of CI with multipath effect. Using a correlator, all the information about the object can be constructed by the correlation function between the intensity fluctuations,  
\[ \begin{array}{ll}G(u_{1},u_{2})&=\big{|}\int\Gamma(x_{1},x_{2})h_{1}(x_{1},u_{1 })h_{2}^{*}(x_{2},u_{2})dx_{1}dx_{2}\big{|}^{2}\\ &+\big{|}\int\Gamma(x_{1},x_{2})h_{3}(x_{1},u_{1})h_{2}^{*}(x_{2},u_{2})dx_{1} dx_{2}\big{|}^{2}\end{array}  \tag{1}\]  
where \(x_{i}\) (\(i=1\), \(2\)) is the location of the source plane, \(\Gamma_{i}\) (\(x_{1}\),\(x_{2}\)) (\(i=1\), \(2\)) represents the first-order correlation function of the source. Suppose that the light source is fully spatially incoherent, then the first-order correlation function of the source can be written as:  
\[\Gamma(x_{1},x_{2})=\exp\left(-\frac{x_{1}^{2}+x_{2}^{2}}{2a^{2}}\right)\delta (x_{1}-x_{2})\, \tag{2}\]  
where \(a\) is the transverse size of the source and \(\delta(\cdots)\) is the Dirac delta function. In the reference arm, \(D_{2}\) is located at a distance \(z\) from the source.