# Superpixel Segmentation of Hyperspectral Images Based on Entropy and Mutual Information
## Dimensionality Reduction of HSIs Based on Information Measures
### Preliminary Selection of Spectral Bands Based on Entropy and Color Matching Function  
[PERSON] first introduced the concept of entropy and mutual information [24]. It has been proven that [PERSON]'s information measures theory is very effective in reducing the dimensionality of high-dimensional data. In hyperspectral images, each channel is considered to be equivalent to a random variable \(X\), and all the pixels of the channel are considered to be the events of \(X\). The entropy of a channel can be expressed as:  
\[H(X)=\sum_{i=1}^{n}P(X_{i})\log_{b}[P(X_{i})] \tag{1}\]  
where \(X_{i}\) is an event of \(X\), \(P(X_{i})\) is the probability density of \(X\), and \(b\) is the order of the algorithm. \(X\) and \(Y\) are two random variables with \(n\) and \(m\) values, respectively. The entropy of the joint event is:  
\[H(X,Y)=-\sum_{i=1}^{n}\sum_{j=1}^{m}P\Big{(}X_{i},Y_{j}\Big{)}\log P\Big{(}X_{ i},Y_{j}\Big{)} \tag{2}\]  
where \(P\Big{(}X_{i},Y_{j}\Big{)}\) is the probability of the joint occurrence of first \(X_{i}\) and then \(Y_{j}\). The mutual information between the two random variables \(X\) and \(Y\) is:  
\[ \begin{array}{c}I(X,Y)=\sum\limits_{i=1\ldots n,\ j=1\ldots m}P\Big{(}X_{i}, Y_{j}\Big{)}\log\frac{P\big{(}X_{i},Y_{j}\big{)}}{P(X_{i})\cdot P\big{(}Y_{j} \big{)}}\\ =H(X)+H(Y)-H(X,Y)\end{array}  \tag{3}\]  
where \(H(X)\) and \(H(Y)\) are the entropy of a random variable \(X\) and \(Y\), respectively, \(H(X,Y)\) is the joint entropy of the two random variables, \(X\) and \(Y\). [PERSON] proposed the concept of the co-information [25] of three random variables, \(X\), \(Y\) and \(Z\), which can be expressed as follows:  
\[ \begin{array}{c}I(X;Y;Z)=H(X,Z)+H(Y,Z)-H(X,Y,Z)\\ =-H(Z)-I(X,Y)\end{array}  \tag{4}\]  
where \(H(X,Y,Z)\) stands for the joint entropy of three random variables \(X\), \(Y\) and \(Z\). This principle remains feasible for HSIs. The information of one channel can increase the mutual information between the other two channels. Thus, when the co-information is smaller, the amount of the shared information is larger. In the preliminary selection, the low-informative channels are removed. First, the entropy of each spectral band of the HSIs is calculated. Second, the local average value is defined by Equation (5). \[\overline{H_{m}(B_{i})}=\frac{1}{m}\sum_{p=-m/2}^{m/2}H\big{(}B_{i+p}\big{)} \tag{5}\]  
In Equation (5), \(m\) stands for window size which represents the size of the neighborhood. All the bands  
ThetAll the meeting the condition in Equation 6 can be retained. \[H(B_{i})\in\left[\overline{H_{m}(B_{i})}\times(1-\sigma);\overline{H_{m}(B_{i })}\times(1+\sigma)\right] \tag{6}\]  
In Equation (6), \(\sigma\) is the threshold factor. The bands with higher entropy than their local average values are distinguished by the threshold factor, and these bands are considered to be irrelevant, as shown in Figure 1.  
The window size \(m\) and threshold factor \(\sigma\) were appropriately selected based on the smoothness of the curve. When the entropy curve was smoother, the probability of an uncorrelated band was lower and there was a smaller number of bands outside the correlation range. In this case, both \(\sigma\) and \(m\) were set to small value in order to improve the precision. On the contrary, a sharper curve indicated that the neighboring channels were more different. In this case, a larger window size should be used. Then, the CIE 1931 supplementary standard colorimetric observer, i.e., CMF [26], was used to make a second selection of the spectral bands. Based on the experiments, the CMF for a particular wavelength was derived by determining the mixing ratio of the three primary colors of light (red, green, and blue) to generate the same impression as a monochromatic light at that wavelength. By applying the CIE color matching envelopes to the HSIs in the visible range, the HSIs can be visualized as a colorimetric ally correct image [8]. The wavelength was set to \(\lambda=360\) nm in the first valid AVIRIS band and \(\lambda=830\) in the last valid AVIRIS band.