# Hierarchical Transmission Tower Detection from High-Resolution SAR Image
## 2 Methodology
### Potential Transmission Tower Pixels Detection  
For a given preprocessed and cropped amplitude SAR image \(z=\{z_{i}(x_{i},y_{i})\);  
\((x_{i},y_{i})\in D,i\in N\}\), where \(\mathbf{D}\) is the image domain, \(i\) is the pixel index, \((x_{i},y_{i})\) is the grid position coordinate of pixel \(i\), \(\mathbf{N}=\{1,\ldots,n\}\) is the pixel index set, \(n\) is the total number of pixels, and \(z_{i}\) is the spectral reflection amplitude of pixel \(i\). The idea behind potential transmission tower pixels detection corresponds to the transmission tower characteristics (1)in Section 2.1. That is, using transmission towers possessing strong scattering characteristics can distinguish potential transmission tower pixels with strong scattering characteristics from background pixels with weak scattering characteristics. In order to judge the scattering ability of pixels more accurately, it is necessary to convert the original SAR image to its SCR image, as the SCR image is more representative of the scattering ability by calculating the spectral values of central pixel and its neighboring pixels. The definition of SCR is motivated by the spatial distribution of potential transmission tower pixels and the sliding detection window, as shown in Figure 1. Each transmission tower presents a number of aggregated pixels. The SCR value of the pixels within each window can be obtained by sliding the window. That is, first, an appropriate detection window \(W\) is established on the SAR image to be detected. By sliding the window, the pixels in the window are arranged in ascending order, the pixel with the largest amplitude is regarded as the target pixel, and the average value of the other smallest 90% of the pixels is regarded as the background pixel, thereby calculating the signal-to-noise ratio of the target pixel. The schematic diagram is shown in Figure 1. Let \(W\) be a detection window and its size be set according to the number of pixels occupied by the transmission tower and the SAR image resolution to minimize the detection of pseudo-pixels while ensuring that the transmission tower pixels are not missed. The detection window with too-large size will miss the transmission tower pixels, otherwise, it will result in too many pseudo pixels. Accordingly, for pixel \(i\), its SCR is calculated as follows,  
\[SCR_{i}=z_{i\max}\left(\frac{1}{n_{\textit{90\%}}}\sum_{i\in N_{\textit{90 \%}}}z_{i^{\prime}}\right)^{-1} \tag{1}\]  
where \(SCR_{i}\) is the SCR value at pixel \(i\), \(z_{i\max}\) is the maximum spectral reflection amplitude of pixels within the detection window central on pixel \(i\), i.e., \(z_{i\max}=\max\{z_{i^{\prime}},i^{\prime}\in N_{i}\}\), \(N_{i}\) is the index set of pixel \(i\) and its neighboring pixels in the detection window, \(N_{\textit{90\%}}\) is the index set of pixels of the lowest amplitude 90% in the detection window, \(n_{\textit{90\%}}\) is the number of pixels in \(N_{\textit{90\%}}\). Therefore, the generated SCR image can be expressed as \(\textit{SCR}=\{SCR_{i},i\in N\}\). To obtain the potential transmission tower pixels, the SCR threshold of the SCR image is set to \(T_{scr}\), i.e., the potential transmission tower pixels can be detected as \(SCR_{i}\geq T_{scr}\). Due to the different background complexity within the image, the statistical distribution of the pixels' amplitudes is prone to different situations such as single-peak distribution or multi-peak distribution, and the corresponding SCR images also show a variety of distributions, as shown in Figure 2a,b. In order to accurately describe the SCR image statistical distribution, a Gaussian mixture model is used,  
\[F(SCR_{i})=\sum_{k}^{K}\Phi_{k}f(SCR_{i};\mu_{k};\sigma_{k}) \tag{2}\]  
Figure 1: Schematic diagram for transmission towers distribution and sliding detection window.