# Research on Multipath Correlated Imaging with the Grayscale Target in Endoscopic Applications
## 2 Model and Theoretical Analysis of Multipath Correlated Imaging
Therefore, the impulse response function can be given by the Fresnel integral,  
\[h_{2}(x_{2},u_{2})=\left(\frac{-i}{\lambda z}\right)^{-1/2}\exp\left[\frac{-i \pi}{\lambda z}(u_{2}-x_{2})^{2}\right]. \tag{3}\]  
As for the test arm, there are two propagation paths \(z_{1}\) and \(z_{3}\), and the corresponding impulse response functions are  
\[h_{1}(x_{1},u_{1})=\left(\frac{-1}{\lambda^{2}z_{1}z_{2}}\right)^{-1/2}\int \exp\left[\frac{-i\pi}{\lambda z_{1}}(y-x_{1})^{2}\right]t(y)\times\exp\left[ \frac{-i\pi}{\lambda z_{2}}(u_{1}-y)^{2}\right]dy, \tag{4}\]  
\[h_{3}(x_{1}-x_{0},u_{1})=\left(\frac{-1}{\lambda^{2}z_{2}z_{3}}\right)^{-1/2} \int\exp\left(\frac{-i\pi}{\lambda z_{3}}[y-(x_{1}-x_{0})]^{2}\right)t(y) \times\exp\left[\frac{-i\pi}{\lambda z_{2}}(u_{1}-y)^{2}\right]dy, \tag{5}\]  
where the coordinate \(x_{0}\) represents the deviation of the pseudo-image relative to the real image at both horizontal and vertical distances. Substituting Equations (2)-(5) into Equation (1), the simplified correlation function of intensity fluctuation \(G(u_{2})\) can be obtained as  
\[ \begin{array}{ll}G(u_{2})&=\frac{\theta_{0}\theta_{2}}{\pi^{3}}\int dy|t(y)| ^{2}\big{[}\beta_{1}h^{\prime}(y,u_{2})+\beta_{3}h^{\prime\prime}(y,u_{2}) \big{]}\,\\ &=G_{1}(u_{2})+G_{2}(u_{2})\end{array}  \tag{6}\]  
where \(\alpha=\frac{1}{a^{2}},\beta_{0}=\frac{\pi}{\lambda z},\beta_{1}=\frac{\pi}{ \lambda z_{1}},\beta_{2}=\frac{\pi}{\lambda z_{2}}\), and \(\beta_{3}=\frac{\pi}{\lambda z_{3}}\). From Equation (6), one can see that the reconstructed image consists of two terms. The first term represents the real image which is from the correlation between the direct light and the reference light, and the second term corresponds to the defocused image formed by correlating the reflected light with the reference light. From Equation (6), the PSF of the multipath CI system can be written as:  
\[ \begin{array}{ll}h(y)&=h^{\prime}(y)+\rho h^{\prime\prime}(y)\\ &=\frac{\pi}{\lambda^{2}}\exp\left[\frac{-y^{2}}{\frac{\lambda^{2}+x}{2a^{2} }}\right]+\rho\frac{\pi}{\frac{1}{\lambda^{2}}\sqrt{1+\frac{\pi^{2}\lambda^{2 }}{\lambda^{2}}\left(\frac{1}{2}-\frac{1}{\lambda_{3}}\right)^{2}}}\.\\ &\times\exp\left[\frac{-y^{2}-22\alpha y-x_{0}^{2}}{\frac{\lambda^{2}-x_{0}^{2} }{2a^{2}}\left[1+\frac{\pi^{2}\lambda^{2}}{\lambda^{2}}\left(\frac{1}{2}- \frac{1}{\lambda_{3}}\right)^{2}\right]}\right]\end{array}  \tag{7}\]  
According to Equation (7), it clearly emerges that there are mainly three parameters that affect imaging quality of the reconstructed image: the path length \(z_{3}\) of reflected light, reflection ratio \(\rho\), and the deviation \(x_{0}\). It should be emphasized that when there exists deviation, the defocused image and the real image will overlap partly or not overlap. While the two images will overlap completely when there is no deviation. In this case, the defocused image has the greatest interference on the reconstructed image. Thus, in the following, we will mainly discuss the influence of \(\eta\) and \(\rho\) on the reconstructed image. Based on Equation (7), the normalized PSF is depicted in Figure 2. It is well known that the resolution is dependent on the width of PSF. The smaller the width is, the higher the resolution is. From Figure 2a, it is shown that the width of PSF is the smallest when there is no multipath effect, and the profile becomes broader with the increase of \(\eta\). From Figure 2b, it is found that the increase of the power of the reflected light results in a decrease of imaging resolution. It is also shown that the reflection ratio has smaller effect than the mismatch ratio by comparing Figure 2a,b.