# Newly Designed Identification Scheme for Monitoring Ice Thickness on Power Transmission Lines
## 3 New Scheme for Iced Transmission Line Identification
### Mathematical Morphology  
There are many kinds of operators in mathematical morphology, among which the most basic ones are dilatation and erosion. The opening and closing operations are two critical secondary operations based on the dilatation and erosion operations. The morphological calculation process from a mathematical perspective is as follows:  
It assumes that \(\Omega\) represents two-dimensional Euclidean space. \(B\) is a mathematical structural element that operates with \(A\), where \(A\) and \(B\) are subsets of \(\Omega\). \(\Phi\) represents empty sets. The erosion operation (sometimes called \"Minkowsky subtraction\") is defined as follows:  
\[A\ \Theta\ B=\{\chi|[(B)x]\ A\}. \tag{19}\]  
Based on Equation (19), the erosion of \(A\) by \(B\) means that \(B\) contains the set of all points x in \(A\) after \(B\) translates \(x\). Corrosion is an operation that shrinks or refines \(A\). The dilation operation (sometimes called \"Minkowsky addition\") is defined as follows:  
\[A\oplus B=\{\chi|\big{[}\big{(}\overline{B}\big{)}x\big{]}\ \cap A\}\
eq\Phi. \tag{20}\]  
Which represents the expansion of \(B\) to \(A\). The operator \(\overline{B}\) is the structural element of B's origin image. The expansion of B to A refers to the set of all displacements \(x\) based on the translation of the image structure element \(x\). Dilation refers to growing or coarsening pixels in a binary image. Opening and closing are critical morphological filters for image smoothing, which is good for smoothing light and dark image features. These smoothed features could be extracted for image analysis. However, based on the characteristics of the transmission line icing image used in this research, it only uses a closing operator to maintain its edge contour. The closing operation for the filter is described as follows:  
\[A\ \circ B=(A\ \Theta\ B)\oplus B. \tag{21}\]  
where the element \(B\) first dilates \(A\), and then \(B\) erodes the result. When the morphology is applied in the image processing, \(A\) is to process the image, and B is a structural element. The gaps and holes are connected to the neighboring objects and smooth objects. In addition, both operations will not significantly change the image area or shape. The binary image obtained after the segmentation threshold contains a transmission line covered with iced targets, a background, and other interference objects. In addition, the segmentation result of the transmission line is still rough, so it is necessary to enhance the object. Mathematical morphology is a nonlinear filtering method that can be applied to simplify image data while maintaining its essential shape characteristics. In most cases, a single structural element realizes the mathematical morphology, which makes edge detection of transmission line effects poor. A single structural element can only detect edge structural element information in the same direction, while the other directions are not edge-sensitive. Morphological operations are applied to segmented images to smooth the image. The mathematical morphology has been enhanced in this study. Figure 7 shows a flowchart for image segmentation refinement using mathematical morphology. The scheme extended a compromise method, which uses multiscale and two structuring elements for alternating sequences of morphological opening and closing filtering to smooth the image and remove the noise. It effectively solves the problems posed by single structural elements. Equation (22) contains a morphological edge detection operator with multiscale elements and two structuring elements, where \(F(x,y)\) is the gray image and \(B(s,t)\) is the structuring element:  
\[E(F)=(F\circ B)\oplus A-(F\bullet B)\Theta A. \tag{22}\]  
where \(A\) and \(B\), respectively, are diamond-type \(5\times 5\) structuring elements and cross-shaped \(3\times 3\) structuring elements. \[A= \begin{bmatrix}0&0&1&0&0\\ 0&1&1&1&0\\ 1&1&1&1&1\\ 0&1&1&1&0\\ 0&0&1&0&0\end{bmatrix} \quad B= \begin{bmatrix}0&1&0\\ 1&1&1\\ 0&1&0\end{bmatrix} .