# Improvement in Signal Phase Detection Using Deep Learning with Parallel Fully Connected Layers
## 3 Results and Discussions  
A pixel error rate (PxER) index is introduced to evaluate the proposed method. PxER is defined as the ratio of the number of pixels of incorrect signal data to the total number of signal-data pixels. Evaluation is performed on 48,600 signal-data pixels in the 300-page data acquired for the test data. The results are shown in Figure 5, which show that the proposed method significantly reduces PxER compared with that in the minimum-finding algorithm in a previous study [19]. In particular, at a Nyquist ratio of 1.4, the PxER drops by three orders of magnitude. Moreover, PxER is almost equivalent independent of the number of pixels to be simultaneously classified, even at 32-pixel simultaneous classification. If a network is configured using a single output layer to predict multiple signal phases, not the parallel output layers proposed in this paper, the PxER increases with the number of pixels classified simultaneously. The reason can be considered as follows. When the multiple signal phases are classified using a single output layer, the number of classes within the network \(C\) is expressed as  
\[C=Pb^{N}, \tag{2}\]  
where \(Pb\) is the number of phase levels and \(N\) is the number of pixels to be predicted simultaneously. Equation (2) implies that the number of classes \(C\) increases exponentially as the number of pixels \(N\) increases. Accordingly, the number of parameters of the fully connected layer increases exponentially, and at the same time, the number of training data assigned to each class also decreases exponentially. \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Fully Connected Layers** & **Input Image Size** \\ \hline
1 & 3 \(\times\) 3 data pixels \\
2 & 4 \(\times\) 4 data pixels \\
4 & 4 \(\times\) 6 data pixels \\
8 & 6 \(\times\) 6 data pixels \\
16 & 6 \(\times\) 10 data pixels \\
32 & 10 \(\times\) 10 data pixels \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relationship between the input image size and number of fully connected layers. Figure 4: Network structure with four fully connected layers. In contrast, in our proposed network with parallel output layers, since the number of classes classified in each output layer is equal to the number of phase levels \(Pb\), the total number of classes \(C\) in all output layers can be expressed as  
\[C=N\times Pb. \tag{3}\]  
Therefore, our proposed network can suppress the increase in the number of classes \(C\) within the network compared to the network with a single output layer. As a result, even if the number of simultaneous signal pixels \(N\) is increased, a sufficient number of training data can be maintained for each class, and over-fitting due to a large number of parameters can be prevented. Using another network structure, U-net [33], it is possible to predict all signal phases within a page efficiently. However, even U-net still requires a considerable amount of training data. It has been reported that an error rate of 0.1% was achieved with 9000 training data for 128 \(\times\) 128 signal data pixels [28]. In our method, on the contrary, the required training data do not depend on the input image size but on the number of parallel output signals. Therefore, even if a large number of signal data pixels are included within a page to realize a large data transfer rate, our method can moderately suppress the required training data. Next, we evaluate the computation time to output the signal phase from the input image. The neural network requires a substantial computational cost to output the resultant category. Therefore, deep learning requires a longer phase-evaluation time than the deterministic methods such as the minimum-finding algorithm. Verification is performed 10 times each by detecting 300 pages of test data for seven Nyquist ratios and by measuring the time required for the phase output. Figure 6 shows the results of the phase-output time.