## 2 Material and Methods
### Proposed Model
Then, the CTC loss function is used to train the network. During training, CTC only requires an input sequence and the corresponding transcriptions. For given training data D, CTC minimizes the negative logarithm of the likelihood loss function, formulated as Equation (1). \[l_{CTC}=-log\left(\prod_{(x,z)\in D}p(z/x)\right), \tag{1}\]  
where \(x=x_{1},x_{2},\ldots,x_{\text{T}}\) is the input sequence with length T, \(z=z_{1},z_{2},\ldots,z_{\text{C}}\) is the corresponding target for \(C<T\),and the p(z/x) is computed by multiplying the probability of labels along the path \(\pi\) that contains the output label over all time-steps t, as shown in Equation (2). \[p(\pi/x)=\prod_{t}^{T}p(\pi_{t},t/x), \tag{2}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Network Layers** & **Kernel Size** & **Stride** & **Feature Maps** \\ \hline Convolution & \(3\times 3\) & \(1\times 1\) & 64 \\ Max-Pooling & \(2\times 2\) & \(2\times 2\) & - \\ Convolution & \(3\times 3\) & \(1\times 1\) & 128 \\ Max-Pooling & \(2\times 1\) & \(1\times 1\) & - \\ Convolution & \(3\times 3\) & \(1\times 1\) & 256 \\ BatchNormalization & - & - & - \\ Convolution & \(3\times 3\) & \(1\times 1\) & 256 \\ BatchNormalization & - & - & - \\ Max-Pooling & \(2\times 1\) & \(1\times 1\) & - \\ Convolution & \(3\times 3\) & \(1\times 1\) & 512 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Convolutional network layers of the proposed model and their corresponding parameter values for an input image size \(32\times 128\times 1\). \begin{table}
\begin{tabular}{l l} \hline \hline
**Network Layers (Type)** & **Hidden Layer Size** \\ \hline BLSTM & 128 \\ BLSTM & 128 \\ Soft-Max & No. class = 281 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The recurrent network layers of the proposed model with their corresponding parameter values. The input size of the Long–Short-Term Memory (LSTM) is a squeezed output of the convolutional layers, which is depicted in Table 2.  
where t is the time-step and \(\pi_{\text{t}}\) is the label of path \(\pi\) at t.  
A target label in path \(\pi\) is obtained by mapping the reduction function B that converts a sequence of soft-max outputs for each frame into a label sequence by removing repeated labels and blank (\(\phi\)) tokens of the given sequences. Taking an example from [9], for a given sequence of observation (o) with a length of eighteen, \(o=\phi aa\phi mm\phi hbaa\pi r\rho ic\phi\). Then, the paths are mapped to a label sequence \(l_{\text{s}}=B(o)=B(\phi aa\phi mm\phi hbaa\pi r\rho ic\phi)=B(\phi a\phi m \phi h\phi a\phi r\rho ic\phi)=\text{`amharic'}\), where B is a reduction mapping function which works by first collapsing the repeated tokens and then removing blanks. The target sequence probability y from input sequence x is the sum of the probability of all paths by reducing each path to this label sequence using B, and it is formulated as Equation (3). \[p(y/x)=\sum_{\pi\in\mathcal{B}(y)}p(\pi/x) \tag{3}\]  
Once the probability of label sequence y from an input sequence x is obtained with the CTC forward-backward algorithm proposed by [41], we employ the best path decoding method, fast and simple, to find a character (C) that has the highest score from outputs (i) at every time-step; the final recognized string text can be generated using B without segmentation of the input sequence; this is formulated as Equation (4). \[C_{\text{max}}=B(arg\,max_{i}(y_{i}^{t})),for\ t=1,2...T \tag{4}\]  
In all experiments and results reported below, we used the same network architecture, and the performance of the proposed model is described in terms of Character Error Rate (CER), which is computed by counting the number of characters inserted, substituted, and deleted in each sequence and then dividing by the total number of characters in the ground truth; this can be formulated as Equation (5).