## 2 Materials and Methods  
### Data Preparing  
Deep learning requires many pairs of training for the training process to ensure optimal accuracy and performance. Collecting data through experiments is a challenge for optical imaging techniques. Data collection presented practical challenges to acquire a significant quantity of training pairs. However, it has been solved and proven through experiments using depth-dependent PSF, even in the case of imaging of absorbing structures, to produce blurred scattering image data from existing structures [25; 26; 27; 28; 29; 30]. Figure 2 shows a schematic diagram of the light intensity distribution observed on the surface of the scattering medium. In transillumination imaging of light-absorbing structures, homogeneous light is irradiated outside the scattering medium. The scattered light passes through the absorbing structure and projects a shadow on the surface of the scattering object. As shown in Figure 2, the scattering medium was considered an infinitely wide slab; the orange lines show the light distributions of the light source with the same sizes as the absorbing object. In reality, because of the limitation of the observed image, the light distribution was cut out. Thus, the depth-dependent light-source point spread function (PSF), originally derived for a light source, cannot be applied directly to the transillumination images. To overcome this problem, the light distribution in the original clear image will be inverted. The absorption distribution in the clear image becomes the light distribution in the inverted image. Then, the light-source PSF can be expected to apply correctly to the inverted clear image. In this study, the image noise captured by the camera was neglected. The depth-dependent PSF was implemented to convolve clear images of the absorbing structures and the image of illuminated conditions to generate the desired blurred images. Therefore, the simulated image for transillumination imaging by convolving the clear structure image with a light-source PSF can be written as in Equation (2):  
\[y=1-[h\times(1-x)] \tag{2}\]  
where \(x,y,\) and \(h\) represent, respectively, the original structure image, the simulated image, and the depth-dependent light-source PSF and the \(\times\) denotes the convolution operation. In this process, the image's \(x\) and \(y\) were normalized to be 0 to 1. This makes the dataset of transillumination images for deep learning possible and easier. The effectiveness of this approach was rigorously evaluated and validated through tissue-equivalent and small-animal experiments [27; 31]. The dataset in this study was created using the method specifically described in previous studies [27; 31]. Figure 2: Geometry for PSF as light distribution observed at the scattering medium surface in reality for transillumination imaging. The orange circle denotes the light point sources in both cases. For the de-blurring model, a dataset consists of 204,000 pairs of images. These were collected from four subdatasets, each simulating objects of different sizes, namely 0.1, 0.5, 1.0, and 2.0 times the depth range from 0.1 to 100.0 mm, with a step size of 0.1 mm. Each of these datasets comprises 51,000 pairs of images. Each pair of images is augmented by rotating the angle, ranging from 0\({}^{\circ}\) to 360\({}^{\circ}\) with a step of 20\({}^{\circ}\). The dataset was used for training, totaling 3,672,000 image pairs. In this study, the Attention Res-UNet model [31; 32] is used. The output of the model will be a de-blurred image of the light-absorbing structure. In the depth estimation model, the dataset comprises pairs of blurred images and the corresponding depth information. These blurred images are generated by convolving the original images with the PSF. Similarly to the de-blurring model, data augmentation techniques are applied, but exclusively to data with a magnification of 1.0. In this study, the CNN model (DenseNet169) [31; 33] is used.