## 2 Materials and Methods  
### Data Preparation  
The deep learning model requires numerous training pairs for optimal accuracy and performance. The de-blurring process involved working with a carefully curated dataset of blurred and original clear images. The corresponding depth labels associated with the blurry images were used to train the depth estimation model. However, data collection presented practical challenges in acquiring significant training pairs. The depth-dependent point spread function (PSF), which characterizes light scattering in biological tissue at different depths, was implemented to convolve the original clear images, generating the desired blurred images to overcome this limitation. Figure 1 shows the difference between fluorescent and transillumination images with the assumption that the light diffused well in the absorbing object plan. In fluorescent imaging, the light point source is placed inside the scattering medium, as shown in Figure 1a, and the light distribution on the observing surface (dashed orange line) can be mathematically represented by Equation (1) [6]:  
\[\text{PSF}(d,\rho)=C\left(u_{s}^{\prime}+\mu_{a}+\left[\kappa_{d}+\frac{1}{ \sqrt{\rho^{2}+d^{2}}}\right]\frac{d}{\sqrt{\rho^{2}+d^{2}}}\right)\left(\frac {\exp[-\kappa_{d}\sqrt{\rho^{2}+d^{2}}]}{\sqrt{\rho^{2}+d^{2}}}\right) \tag{1}\]  
where \(k_{d}^{2}=3\mu_{a}(\mu_{s}^{\prime}+\mu_{a})\). \(C\), \(\mu_{s}^{\prime}\), \(\mu_{a}\), and \(d\) represent the constants with respect to \(\rho\) and \(d\), the reduced scattering coefficient, the absorption coefficient, and the depth of the light source, respectively. In transillumination imaging, the light source is placed outside the scattering medium, as shown in Figure 1b, and the light distribution on the observing surface (black line) is a collection of the distribution of the light-missing points. The depth-dependent PSF is derived from a light source, so we cannot apply it directly for transillumination imaging. To make the PSF applicable, we invert the distribution of the light-missing point (black line) to have the distribution the same as the distribution of light in fluorescent imaging (dashed orange line), as shown in Figure 1b. The effectiveness of this approach was rigorously evaluated through comprehensive simulations and experimental validations [7; 10; 12]. Convolution images of the original structures with depth-dependent point spread functions were used at different depths to generate the data in this study, as described in Equation (2) and Figure 2.  
\[y=h\otimes x \tag{2}\]  
where \(\otimes\) denotes convolution operation. In this study, the original structure images are images obtained from the image of 12 randomized structures in a transparent medium. These pre-blurred images are specifically designed to emulate the intricate structural characteristics of blood vessels beneath the skin. The blurred images were generated by convoluting the original structures with the PSF given by Equation (1) at different depths with the parameter values of \(\mu_{s}^{\prime}=1.0\) mm\({}^{-1}\) and \(\mu_{a}=0.00536\) mm\({}^{-1}\). These parameters were used in all simulations described hereinafter. These coefficients are pivotal in understanding light-tissue interactions, particularly in optical imaging and spectroscopy. \(\mu_{s}^{\prime}\) signifies the extent of light scattering as it traverses tissue, indicating the likelihood of scattering per unit path length. This is crucial because of diverse tissue structures that cause light to scatter in various directions. \(\mu_{a}\) gauges the extent of light absorbed by tissue during propagation, closely related to light-absorbing constituents such as hemoglobin, lipids, and water. Different tissues and substances possess  
Figure 1: The intensity distribution at the medium surface in fluorescent (**a**) and transillumination (**b**) imaging. Figure 2: Blurred image generation process. varying absorption characteristics at different wavelengths.