# Hierarchical Transmission Tower Detection from High-Resolution SAR Image
## 2 Methodology
### Transmission Tower Detection  
In order to remove false alarm targets in detected candidate transmission tower pixels, such as ground background, road edge lines and surrounding artificial targets, the geometric feature-based filtering algorithm is designed. Based on the imaging characteristics of transmission towers in SAR images, it is found that the constituent pixels of the same transmission tower will inevitably break and thus be divided into several subsets. Surrounding them are also interspersed with independent interfering pixels that are not removed by spatial density detection. Therefore, the candidate transmission tower pixel sets are first grouped by calculating the pixel distances and the interfering pixels will be removed as well. The distance between the pixels of the candidate transmission towers is as,  
\[d_{ii^{\prime}}\triangleq\left\|(x_{i},y_{i})-(x_{i^{\prime}},y_{i^{\prime}}) \right\|_{2} \tag{6}\]  
Setting the distance threshold as \(T_{G}\), candidate transmission tower pixels that satisfy the threshold range will be divided into one group, and each group of pixels is considered as a quasi-transmission tower. Therefore, the quasi-transmission tower pixels can be expressed as \(\mathbf{G}=\{i,\forall i\in\mathbf{C},\exists d_{ii^{\prime}}<T_{G},i^{\prime}\in\mathbf{C}\}\). The distance threshold \(T_{G}\) can be calculated by the following:  
\[T_{G}=\left\lceil\frac{\beta}{2L_{\mathbf{W}_{i}}}\right\rceil \tag{7}\]  
where \(\lceil\cdot\rceil\) is the upward rounding symbol, \(\beta\) denotes the maximum break length between subsets of pixels in one transmission tower, which is determined by counting the transmission tower pixels within the image. \(L_{\mathbf{W}_{i}}\) is the length of window \(\mathbf{W}_{i}\). Furthermore, if the distances to any of the other pixels are larger than \(T_{G}\), it will be judged as a background interference pixel and filtered out. The schematic diagram of candidate transmission tower pixels grouped into several quasi-transmission towers is shown in Figure 4. Observing Figure 4, which contains two sets of candidate transmission tower pixels, A and B. The single pixel marked as #1 in A is considered as interference background pixels, so is #3 in B. Therefore, the two groups of pixels labeled #2 and #4 in A and B constitute two quasi-transmission towers, respectively. The proposed method can effectively filter out the independent interference pixels around the transmission towers. Therefore, the quasi-transmission towers can be denoted as \(\mathbf{G}=\{G_{j},j=1,\ldots,M\}\), where \(G_{j}\) is the set of quasi-transmission tower pixels corresponding to transmission tower \(j\), satisfying \(U_{i=1}G_{j}=\mathbf{G}\); \(G_{j}\subset\mathbf{C}\), \(M\) is the number of transmission towers. Figure 3: Sketch of candidate transmission tower pixels space density calculation. Then, the convex-hull is generated for each quasi-transmission tower \(G_{j}\) which can be denoted as \(\mathbf{H}=\big{\{}H_{j},j=1,\ldots,M\big{\}}\). After that, a MBR is constructed for each convex-hull which is used to define the geometric shape of the transmission tower. The MBRs of convex-hulls can be denoted as \(\mathbf{R}=\big{\{}R_{j};j=1,\ldots,M\big{\}}\). Figure 5a-c show the MBR corresponding to quasi-transmission tower, the MBR superimposed onto the original image and the sketch of the convex-hull. Finally, the aspect ratio is determined in accordance with transmission tower geometry to further filter out quasi-transmission towers that do not satisfy the geometric parameters of transmission towers. That is, let \(R_{j}=\big{\{}(x_{i},y_{i}),l_{i},w_{i},\theta_{i}\big{\}}\), where \((x_{i},y_{i})\) is the coordinate of the center position of the rectangle, \(l_{i}\) is the length of the rectangle, \(w_{i}\) is the width of the rectangle, and \(\theta_{i}\) is the angle between the long side of the rectangle and the horizontal coordinate axis.