## 2 Method
### Building Feature Set Extraction from SAR Image
#### 2.1.2 G\({}^{0}\) Statistical Texture Parameter  
To better simulate extreme heterogeneous regions in SAR images, [PERSON] [28] proposed the G\({}^{0}\) distribution. It models texture variables using an inverse gamma distribution based on the multiplicative model. The uniformity of the target can be reflected by the texture parameter of the G\({}^{0}\) distribution [29]. This parameter can provide valid information in building extraction.  
For non-uniform regions, the texture features of the object target have a wealth of available information. The SAR image uses the multiplicative model to model it. The SAR images can be expressed as multiplicative products of mutually independent texture variables \(\tau\) and speckle variables X:  
\[Z=\tau\cdot X \tag{11}\]  
where Z represents SAR image data. It may be an intensity image, an amplitude image, a covariance matrix, or a coherence matrix. \(\tau\) represents the texture variable, reflecting the average backscattering coefficient of the scatters in the ground-resolving unit. It models the inverse gamma (\(\overline{\gamma}^{-1}\)) distribution for heterogeneous regions, for example, urban areas. X represents the speckle of the SAR image, which obeys the gamma distribution for the L-look single-polarimetric SAR intensity image [30] and the Wishart distribution for the L-look multi-polarimetric SAR image [31]. According to Equation (11), the SAR image obeys the G\({}^{0}\) distribution, and the probability density distribution of the L-look multi-polarimetric SAR image is  
\[\mathrm{P_{C}(C)}=\frac{\mathrm{L^{dL}}|C|^{L-d}\Gamma(dL+\lambda)(\lambda-1 )^{\lambda}}{\Gamma_{d}(L)|\Sigma|^{L}\Gamma(\lambda)}\times\left(\mathrm{Ltr }\left(\Sigma^{-1}C\right)+\lambda-1\right)^{-\lambda-dL} \tag{12}\]  
where \(\lambda\) is the texture parameter, d is the scattering vector dimension, \(\Gamma(\cdot)\) is the gamma function, \(\Gamma_{d}(L)=\pi^{2d(d-1)}\Gamma(L)\dots\Gamma(L-d+1)\), \(\Sigma\) is the mean, and \(\mathrm{tr}(\cdot)\) is the trace of the matrix.  
Regarding the G\({}^{0}\) texture parameter estimation of L-look multi-polarimetric SAR data, according to the study of [PERSON] [32], we can make \(\mathrm{M}=\mathrm{tr}\left(\Sigma^{-1}C\right)\). Then, the vth moments of M can be uniformly expressed as  
\[\mathrm{E}\{M^{v}\}=\left(\frac{\lambda-1}{L}\right)^{v}\frac{\Gamma(\lambda- v)}{\Gamma(\lambda)}\frac{\Gamma(Ld+v)}{\Gamma(Ld)} \tag{13}\]  
[PERSON] [33] estimates the texture parameter \(\lambda_{C}\) of the G\({}^{0}\) distribution for multi-polarimetric SAR image by using the second-order moment of M.  
\[\lambda_{C}=\frac{2 LVar\{M\}+d(Ld-1)}{LVar\{M\}-d} \tag{14}\]  
According to \(\mathrm{Var}(x)=\mathrm{E}\left(x^{2}\right)-\mathrm{E}^{2}(x)\), the estimation formula of the G\({}^{0}\) distribution texture parameter of the multi-polarimetric SAR image can be expressed as  
\[\lambda=\frac{Ld+1}{Ld\frac{Var\{x\}}{\mathrm{E}\left(x\right)^{2}}-1}+2 \tag{15}\]  
where \(x=\mathrm{tr}\left(\Sigma^{-1}C\right)\).  
The range of \(\lambda\) calculated using Equation (15) is usually too broad, which inconveniences the display of images and the statistics of samples. For this reason, this paper takes the logarithm operation as a new statistical texture feature named G\({}^{0}\)-para  
\[\mathrm{G^{0}}\text{-para}=\log(\lambda) \tag{16}\]