## 3 The Proposed Method
### Relevant Principles of Image Correction Based on Probabilistic Hough Transform  
The projection of the four corner coordinates obtained from the Progressive Probabilistic Hough Transform is presented. The projection process involves mapping a graphical object from one plane to another using a projection matrix. Projection transformations can be divided into two categories: affine transformations and linear transformations. The relationship between the two subsets of projection transformations is shown in Figure 2b. Linear transformation includes various transformations, including image rotation, skew transformation, scaling, and their combinations. After linear transformation, the origin of the image coordinates remains unchanged, and straight lines retain their straightness and parallel lines their parallelism. Affine transformation, also known as affine projection, is a combination of linear transformation and translation. In geometry, it refers to the process of transforming one vector space into another by applying a combination of translation and linear transformations. An equation can be used to illustrate this concept. Assume \(K\) and \(J\) are two vector spaces, defined as follows:  
\[\left\{ \begin{array}{l}K=(x,y)\\ J=(x^{\prime},y^{\prime})\end{array} \right. \tag{5}\]  
Figure 2: Process for correcting the uneven shape of industrial nameplates using the Probabilistic Hough Transform. **(a)** Find the edge in the image with the greatest length. **(b)** The relationship between projection transformation and its subsets. The double line segment obtained from the transform is used to extract four corners of the nameplate, which are then projected onto a rectangular shape based on their coordinates. The transformation of the vector space from \(K\) to \(J\) is given by the following formula:  
\[J=K\times\omega+b \tag{6}\]  
The equation will be divided to produce:  
\[\left\{ \begin{array}{l}x^{\prime}=\omega_{00}\times x+\omega_{01}\times y+b_{0 }\\ y^{\prime}=\omega_{10}\times x+\omega_{11}\times y+b_{0}\end{array} \right. \tag{7}\]  
Matrix multiplication conversion:  
\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\end{array} \right]=\left[ \begin{array}{ccc}\omega_{00}&\omega_{12 }&b_{0}\\ \omega_{10}&\omega_{11}&b_{1}\end{array} \right]\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right]=M\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right] \tag{8}\]  
The Probabilistic Hough Transform is used to locate the four corners of an industrial nameplate in the image. A parameter matrix \"\(M\)\" is used to implement geometric transformations such as translation, scaling, rotation and mirroring. \(M\) acts as an intermediary between two vector spaces that undergo an affine transformation. Affine transformation is a combination of translation and linear transformation where the vector space is transformed and translated from one space to another. Our approach applies a perspective transformation, also known as projection transformation, to turn the irregular square in the 2D image into a rectangular shape. Using the Probabilistic Hough Transform, the four vertices of the nameplate of the equipment are determined. The projection matrix then projects the irregular quadrilateral onto the chosen plane. The projection transformation is completed through three stages. 1. General perspective transformation formula:  
\[\left[ \begin{array}{c}u^{\prime}\\ v^{\prime}\\ \omega\end{array} \right]=\left[ \begin{array}{c}u\\ v\\ \omega\end{array} \right]\left[ \begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ \omega_{21}&\omega_{22}&a_{23}\\ \omega_{31}&\omega_{32}&a_{33}\end{array} \right] \tag{9}\]  
The parameters \(u\) and \(v\), which define \(\omega=1\) and \(a_{33}=1\) and the image coordinates \((x,y)\) generated via projection transformation, are taken from the original image. When \(x\) and \(y\) are satisfied:  
\[\left\{ \begin{array}{l}x=\frac{x^{\prime}}{\omega^{\prime}}\\ y=\frac{y}{\omega^{\prime}}\end{array} \right. \tag{10}\]  
2.