# Deep Learning of Diffuse Optical Tomography Based on Time-Domain Radiative Transfer Equation
## 3 Results on Single Absorber Detection  
The TD data in our simulations consist of signals at eight detectors. The multi-position of detectors leads to time difference in a picosecond scale. Light propagation simulations by RTE provide precise information on the time difference at detectors. We aim to develop a new method to predict the position of the absorber from the experimental data.  
Here, we consider the case for a laser beam injected from S2. Figure 2 shows the resultant temporal profiles of the absorption measure \(A(t)\), depending on the position of the absorber. For example, the upper left panel shows temporal profiles of \(A(t)\) for an absorber at #3. Since the absorber is very close to detector D8, the curve labeled D8 shows significant absorption. Similarly, the lower left panel (absorber at #0) and the upper right panel (absorber at #26) exhibit strong absorption at the detectors in the vicinity of the absorber. The lower right panel (absorber at #13) shows the result for an absorber near the center of the whole region. In this case, all detectors register significant absorption, and the temporal profiles depend strongly upon the detector position. These features of the absorption measure \(A(t)\) are used for training data in the deep learning method used in this work as described below.  
The first classification is conducted to determine a target group out of eight groups. In this classification, the number of learning epochs is about 40 and the number of training data and the number of test data are 2500 and 500, respectively. The final loss value is 0.03, and the resultant accuracy to determine the target group reaches 99%. As the the second step, the sub-classification is achieved to determine a domain containing an absorber in  
Figure 5: The algorithm of the present deep learning based on the LSTM method.  
the group. For groups composed of four domains, the number of learning epochs is 10 and the number of training data and test data are 1000 and 200, respectively. For groups composed of two domains, the number of learning epochs is 10 and the number of training data and test data are 600 and 100, respectively. As a result, we find that the final accuracy rate to predict the domain reaches 99.9%. The accuracy rate depends on the noise level. If we employ Gaussian noises with \(\sigma=0.01\) that is ten times larger than the fiducial value (\(\sigma=0.001\)), then the accuracy rate is further improved.  
We also study the detection of an absorber shifted from a domain. Figure 6 shows the results for the detection of a shifted absorber. The absorber is shown by a red circle, and the predicted position is indicated by a square. In Figure 6, the upper left panel shows that the model can predict a position near the correct position. The upper right panel shows the results for an absorber at the center of four domains. Again, a near position is predicted. Next, we consider the refinement of domains. In the lower panels of Figure 6, the results for eight domains are shown. The accuracy for the prediction of a position can be improved. These results demonstrate that, by the refinement of domains, we can predict the position of absorber more precisely.