## 3 Dynamic Sampling Rate Adjustment Method
A detailed explanation of our dynamic sampling rate adjustment method, based on the actual frame rate of the extracted images when detecting the start and end points of sign language utterances in a mobile environment, is as follows. The most important aspect of the proposed method is the ability to calculate the actual frame rate when extracting images during sign language utterances, considering the performance of the mobile device. The frame rate in mobile environments (\(fps_{mobile}\)) can be calculated using Equation (7). \[fps_{mobile}=\frac{N_{frames}}{T_{utterance}} \tag{7}\]  
It is also necessary to calculate the ratio of the frame rate in mobile environments to the frame rate on the server. Based on this, the sampling rate can be adjusted dynamically. The ratio of the frame rate in mobile environments to the frame rate on the server (\(R_{fps}\)) is calculated using Equation (8). \[R_{fps}=\frac{fps_{mobile}}{fps_{server}} \tag{8}\]  
Finally, in the proposed method, the dynamic sampling rate (\(SR_{mobile}\)) is determined by adjusting the optimal fixed sampling rate used in server environments according to the frame rate ratio, as shown in Equation (9). \[SR_{mobile}=\left\lfloor SR_{server}\times R_{fps}+0.5\right\rfloor \tag{9}\]  
As shown in Equation (9), this ensures that the sampling rate is dynamically adjusted to maintain consistent recognition performance despite variations in frame rates in mobile environments. Algorithm 1 illustrates the process of the proposed dynamic sampling rate adjustment for mobile devices. To clearly illustrate the method for calculating the dynamic sampling rate described in Algorithm 1, consider the following scenario in which the optimal fixed sampling rate in server environments (\(SR_{server}\)) is 5. Additionally, assume that the number of frames extracted in real time (\(N_{frames}\)) is 50, the duration of the sign utterance (\(T_{utterance}\)) is 10 s, and the frame rate in server environments (\(fp_{server}\)) is 30 fps. ```
1:Input:\(SR_{server}\), \(N_{frames}\), \(T_{utterance}\), and \(fp_{server}\)
2:Output:\(SR_{mobile}\)
3:Calculate \(fps_{mobile}\) using Equation (7):
4:\(fps_{mobile}\leftarrow\frac{N_{frames}}{T_{utterance}}\)
5:Calculate \(R_{fps}\), the ratio of mobile to server frame rate, using Equation (8):
6:\(R_{fps}\leftarrow\frac{fps_{mobile}}{fps_{server}}\)
7:Calculate the dynamic sampling rate, \(SR_{mobile}\), using Equation (9):
8:\(SR_{mobile}\leftarrow\left\lfloor SR_{server}\times R_{fps}+0.5\right\rfloor\)
9:return\(SR_{mobile}\)
```  
**Algorithm 1** Dynamic sampling rate adjustment for mobile devices  
Based on these assumptions, we can compute the frame rate in mobile environments (\(fps_{mobile}\)) as shown in Equation (10). The ratio of the mobile frame rate to the server frame rate (\(R_{fps}\)) is calculated as shown in Equation (11). Finally, we can obtain the dynamic sampling rate (\(SR_{mobile}\)) as shown in Equation (12). \[fps_{mobile}=\frac{N_{frames}}{T_{utterance}}=\frac{50}{10}=5\,\mathrm{fps} \tag{10}\]  
\[R_{fps}=\frac{fps_{mobile}}{fps_{server}}=\frac{5}{30}\approx 0.167 \tag{11}\]  
\[SR_{mobile}=\left\lfloor SR_{server}\times R_{fps}+0.5\right\rfloor=\left\lfloor 5 \times 0.167+0.5\right\rfloor=\left\lfloor 1.335\right\rfloor=1 \tag{12}\]  
In this scenario, the dynamic sampling rate in mobile environments (\(SR_{mobile}\)) is set to 1. This means that each extracted frame is sampled and checked to detect the start and end points of a sign language utterance. In summary, the proposed method checks image frames at shorter intervals as the number of extracted frames from a mobile device decreases, in order to improve sign language recognition accuracy. Therefore, the proposed method can maintain consistent recognition performance, even with a lower frame rate compared to server environments. Algorithm 2 provides a method for detecting sign language utterances' start and end points.