## 2 Related Methods
### Improved Optimum Index Factor (IOIF)  
A basic principle of wavelength selection for HSI is that the information redundancy between wavelengths in the obtained band subset is small and independent. It can be described by the correlation coefficient matrix, which is calculated by the following Equations (1) and (2) used for the correlation coefficient:  
\[R=\left[\begin{array}{cccc}r_{11}&r_{12}&\cdots&r_{1n}\\ r_{21}&r_{22}&\cdots&r_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ r_{n1}&r_{n2}&\cdots&r_{nn}\\ \end{array}\right] \tag{1}\]where \(n\) denotes the total number of bands of the HSI and \(r_{ij}\) is the correlation coefficient between the band \(i\) and \(j\):  
\[r_{ij}=\frac{\sum\limits_{k=1}^{m}{(x_{ik}-\overline{x_{i}})(x_{jk}-\overline{x_ {j}})}}{\sqrt{\sum\limits_{k=1}^{m}{(x_{ik}-\overline{x_{i}})}^{2}\sum\limits_ {k=1}^{m}{(\overline{x_{jk}}-\overline{x_{j}})}^{2}}} \tag{2}\]  
where \(m\) is the total number of samples, that is, the total number of pixels in a band image, \(x_{i}={(x_{i1},x_{i2},\cdots,x_{im})}^{T}\) is the corresponding vector of the spectral image of the band \(i(i=1,2,\cdots,n)\), \(\overline{x_{i}}\) and \(\overline{x_{j}}\) are the mean values of band \(i\) and \(j\). The smaller the correlation between the selected wavelengths, the higher the independence of the information it contains, and the smaller the information redundancy.  
In addition, the selected wavelengths should contain more information. In 1982, [PERSON] et al. proposed the Optimum Index Factor (OIF) method [22] to evaluate the optimal band combination by calculating the ratio of the sum of the standard deviations of the bands to the sum of the correlation coefficients. In 2018, [PERSON] proposed a new band selection method Improved OIF (IOIF) [21] based on the OIF information calculation method. In the band subspace, the Local Band Index method (LBI) was used to filter out high-quality band sets. Then, combine the bands from the different sets of high-quality bands and calculate the IOIF value. This method effectively avoids the phenomenon of centralized selection of bands, reduces the number of band bases involved in the operation, and improves the efficiency of the algorithm.  
The local band index method is an adaptive band selection method that comprehensively considers the information content and local independence of the image. The LBI is calculated as in the following Equation (3):  
\[\text{LBI}_{i}=\frac{\sigma_{i}}{(|r_{i-1,i}|+|r_{i,i+1}|)/2} \tag{3}\]  
where \(r_{i-1,i}\) is the correlation coefficient between band \(i-1\) and band \(i\), \(r_{i,i+1}\) is the correlation coefficient between band \(i\) and band \(i+1\), the average of the two represents the local correlation of band \(i\), and \(\sigma_{i}\) is the mean square deviation of band \(i\).  
IOIF is defined as in Equation (4):  
\[\text{IOIF}=\frac{S_{i}+S_{j}+S_{k}}{|r_{i,j}|+|r_{i,k}|+\left|r_{j,k}\right|} \tag{4}\]  
where \(i,j\) and \(k\) are three high-quality bands from different subspaces, \(S\) is the mean square error, and \(|r|\) is the absolute value of the correlation coefficient.  
The set of bands selected by IOIF is very similar to the graph of the original bands and contains the richest information of the original bands. After verification by the RBF-SVM classifier, the classification accuracy obtained by the IOIF DR method is higher than the traditional OIF and PCA algorithms. However, this method mainly takes the information of wavelength combination as the basis of wavelength selection.  
For the classification of objects with similar colors and close spectrum, a high amount of information is not enough to meet the needs of improving the classification accuracy, and the separability between the selected spectral objects should be further considered.