## 3 Design of Improved Multi-Structural Morphology Filtering Approach  
### Target Vessel Detection Process  
A frame of an image is read from an image sequence for target vessel detection, and then the Hough transform algorithm, together with the OSTU-based adaptive threshold segmentation method, is used to not only eliminate the image's superfluous lines and occlusions but also to convert the image into a binary map [31; 32]. Next, the weighted morphological filtering of multiple groups of structural elements is carried out to separate the target vessel from the port background. Then, based on the geometric characteristics of the vessel, neighbor-based adaptive fast median filtering is performed to filter out the impulse noise on the image. Subsequently, the connected domain labeling algorithm is performed to obtain the target information of the vessel and establish a model of its  
Figure 3: Schematic diagram of the path coherence function.  
\(\theta\in[0,\pi]\) is the angle between \(k\) and distance \(S_{k+1}\) and the distance parameter \(S_{k}\) at the origin point \(P_{i}^{k}\). \(k\) can be parameterized by bias \(S_{k}\) and slope \(\theta\). Additionally, reverse mapping can be performed to translate any valid \((S,\theta_{k})\) pair to a line instance. The line-to-parameters and the inverse mapping are defined as follows:  
\[\left\{\begin{array}{l}S_{k},\;\theta\gets Z(K)\\ k\gets Z^{-1}(S,\;\theta_{k})\end{array}\right. \tag{10}\]  
where \(Z\) and \(Z^{-1}\) represent bijective mapping, and \(S\) and \(\theta_{k}\) are quantized into discrete bins that are processed by computer programs. The quantization can be expressed as follows:  
\[\left\{\begin{array}{l}\hat{S}_{k}=\left[\frac{S_{k}}{\Delta r}\right]\\ \hat{\theta}_{k}=\left[\frac{\theta}{\Delta\theta}\right]\end{array}\right. \tag{11}\]  
where \(\hat{S}_{k}\) and \(\hat{\theta}_{k}\) are the quantized line parameters, while \(\Delta r\) and \(\Delta\theta\) are the quantization intervals for \(S\) and \(\theta_{k}\), respectively. The number of quantization levels \(\Theta\) and \(R\) can be written as follows:  
\[\left\{\begin{array}{l}\Theta=\frac{\pi}{\Delta\theta}\\ R=\frac{\sqrt{(1/2+V^{2}})}{\Delta r}\end{array}\right. \tag{12}\]  
Considering the input image \(Y\), the convolution neural network characteristics \(\mathbf{A}\in\mathbb{R}^{C\times I\times V}\) are first extracted, where \(C\) is the number of channels, while \(U\) and \(V\) are the spatial size. Then, DHT takes \(\mathbf{A}\) as the input image and produces the transformed characteristics, \(\mathbf{B}\in\mathbb{R}^{C\times\Theta\times K}\). The sizes of the transformed characteristics \(\Theta\) and \(R\) are determined using the quantization intervals, as shown in Equation (12) [36; 37]. As displayed in Figure 6a, for any line \(k\) on the image, the characteristics of every pixel along \(k\) are combined to \(\left(\hat{\theta},\hat{S}_{k}\right)\) in the parametric space \(B\), as follows:  
\[\mathbf{B}(\hat{\theta},\;\hat{S}_{k})=\sum_{i\in k}\mathbf{A}(i) \tag{13}\]  
where \(i\) is the positional index. Equation (11) determines \(\hat{\theta}\) and \(\hat{S}_{k}\) based on the characteristics of the line \(k\), which are quantized into discrete grids using Equation (12). According to the number of \(\Theta\) and \(R\) quantization levels, there is a unique line of candidates denoted by \(\Theta\cdot R\)[38]. For all these candidate lines, DHT is performed and their respective characteristics are associated with their corresponding positions in \(\mathbf{B}\). As shown in Figure 6b, the characteristics of neighboring lines in characteristic space are converted into neighboring points in parametric space. In parametric space, a simple \(3\times 3\) convolutional operation makes it easy to capture contextual information for the center line (orange). It should be noted that DHT is order-independent in both characteristic space and parametric space, making it highly parallelizable [39].