# Use of Digital Technology in Integrated Mathematics Education
## 3 Methods
In this regard, if the derivative of a signal is fed into an integrator, the output of the integrator will yield the original signal. If the second derivative of the signal is fed into the integrator, the output will yield the first derivative. Now the simulation scheme of the model can be constructed using Simulink functional blocks. By using the multiplication block, the sine block, and two integration blocks, the equation of motion can be obtained. Having created the simulation scheme, the monitored quantity is connected to an oscilloscope for visualization. Scenario 3--Automatic people detection within an established perimeter  
The proposed scenario for detecting people is integrated into a video surveillance system that performs real-time perimeter monitoring around a firing range and generates an alarm when people are detected inside this monitored area. Thus, the role of this system is to trigger the alarm to notify the human decision-maker responsible for the monitored area. When the alarm is triggered, the system presents the scene where a person was detected. Mathematical solution for Scenario 3  
The people detection will be performed through facial recognition, using Principal Component Analysis (PCA) [33], by reducing the number of variables for representing a face, a technique used in computer vision and pattern recognition. PCA is a method that reduces the dimensionality of data while retaining most of the variation in the dataset. In facial recognition, PCA is used to extract the most important features from a facial image [34]. These features are then compared to a database of known faces to determine a match. The process involves capturing an image of a person's face, identifying key facial features such as eyes, nose, and mouth, and then using PCA to extract the most relevant information from these characteristics. Through this method, the system can accurately detect and identify individuals in real-time, making it a valuable tool for security and surveillance applications. Additionally, PCA can also help improve the speed and accuracy of facial recognition systems, making them more reliable and efficient in various environments [35]. We will consider that we have several images of the same size available for several people in a database, and we can have one or more poses for the same person. We will consider that the person's face is centred in any facial image [36]. Thus, each pixel of an image can be considered a variable, and a face is thus represented as a vector of P variables (where P represents the number of pixels). Given the considerable number of variables involved, we can reduce the complexity of comparing two faces by approximating a face through a smaller number of variables. An image of n pixels can thus be considered a point in image space (n-dimensional space) (Figure 19). Figure 18: Simulation schema of the pendulum in Simulink. The image matrix turns into a line vector such that the line vector \(p_{x}=(i_{1};i_{2};\ldots i_{n})\) where \(i_{1}\),\(i_{2}\), \(\ldots\) represents the grey level of pixels. The vector is formed by transforming the image into a line vector; thus, for a 128 \(\times\) 128 image, we will have the size n = 16,384 (16K):  
Given N images, each of n pixels, the entire set of images can be represented as a \((N\times n)\) matrix D, where each image represents a line:  
\[D= \begin{bmatrix}150&152&\cdots&255&\cdots&252\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 144&171&\cdots&245&\cdots&223\end{bmatrix} ,N\times n \tag{49}\]  
The first step of the PCA method is to move the origin of the data to the middle of the n variables, an operation that is equivalent to subtracting the average value of the variables in each image. In our case, the average image (average face) is calculated by averaging over all the columns of the matrix D. This average face is to be extracted from each image of the dataset D. The obtained difference matrix is denoted by U. In the considered example, the average is  
\[ \begin{bmatrix}120&140&\cdots&230&230&\cdots&240\end{bmatrix}