## 2 Materials and Methods
### Connecting Temporal Classification  
The connection temporal classification (CTC) method, proposed within the realm of deep learning, serves as a valuable tool for classifying and labeling sequence data. Specifically designed to handle sequences of diverse lengths, CTC plays a pivotal role in mapping input sequences to output sequences, allowing for the existence of unaligned states between the input and output data [19]. Consequently, CTC offers substantial advantages in addressing sequence-labeling challenges associated with varying sequence lengths. In the domain of recognizing license plate characters, the issue of diverse license plate character sequences arises due to the varying lengths inherent to different license plate types. This discrepancy in sequence length creates a non-uniform alignment relationship between the input and output data. CTC tackles this challenge by enabling each label in the output license plate sequence to correspond to multiple time steps in the input license plate sequence. In this way, it accommodates the inherent variability in sequence lengths observed across different license plate types and character counts. CTC is instrumental in handling ambiguous alignments. By employing the CTC loss function, a license plate recognition model can be trained, enabling the utilization of decoding algorithms to convert the model's output into the definitive license plate character sequence. This methodology has demonstrated remarkable outcomes in various domains, including license plate recognition [20]. Consider an input license plate sequence, denoted as \(X\), with a length of \(T\), a target output license plate character sequence \(Y\), and a predicted output license plate character length of \(U\). The objective is to identify the most probable license plate character output sequence \(Y\) based on the provided input license plate character sequence \(X\). To account for empty characters or gaps between characters, it is essential to establish an intermediate sequence \(C\) with a length of \(T\). Sequence \(C\) encompasses all potential characters found on license plates, including a designated blank character. The CTC loss function is designed to compute the likelihood probability of the output sequence \(Y\). To achieve this, CTC introduces an intermediate variable \(L\), representing conceivable sequences involving repeated characters and blank characters. A neural network is employed to model the conditional probability \(P(L\,|\,X)\). Subsequently, \(P(Y\,|\,X)\) is computed by summing across all feasible label sequences. The CTC formula is expressed as follows:  
\[P(L\,|\,X)=softmax(\alpha(X,L)), \tag{4}\]  
In the given formula, we define the conditional probability \(P(L\,|\,X)\) to represent the likelihood of label sequence \(L\) given the input sequence \(X\). In this context, \(\alpha(X,L)\) signifies the output generated by the neural network. \[P(Y|L)=\prod_{(l\in L)}\ p(l), \tag{5}\]  
In the given context, the conditional probability \(P(Y\,|\,L)\) is defined as the probability of generating the output sequence \(Y\) given the label sequence \(L\). Conventionally, this conditional probability is established through a mapping function, where \(p(l)\) represents the probability of mapping the label \(l\) to an output character. \[P(Y|X)=\sum P(Y|L)\ *\ P(L\,|\,X), \tag{6}\]  
In the given formula, the objective is to compute the sum over all potential label sequences \(L\). The ultimate objective is to determine the most probable output sequence \(Y\) using dynamic programming algorithms. \[Y^{*}=argmax\ P(Y|X), \tag{7}\]  
In license plate recognition tasks, connectionist temporal classification (CTC) plays a pivotal role by modeling the conditional probabilities \(P(L\,|\,X)\) and \(P(Y\,|\,L)\), and employing dynamic programming algorithms to compute the most probable output sequence \(Y^{*}\). This process yields accurate predictions for license plate character information.