## 2 Methods
### Wavelet-Feature Markov Clustering Algorithm  
Based on the preceding theoretical analysis, the WFCRMCA uses a simulated annealing technique to gradually decrease CR threshold \(T\) through Markov chain decomposition in wavelet feature space at each temperature, obtaining the best clustering centers of the whole space. Supposed that \(c_{i}\) is the class center of class \(i\), \(\mathbf{S}_{ci}\) is the pixel set of class \(i\), \(\mathbf{C}\) is the set of all classes, \(\mathbf{Z}_{ci}\) is the set of class centers absorbed by class \(i\) at the current temperature, \(N_{\text{c}}\) is the class number, \(N_{\text{s}}\) is the number of sampled pixels (initial class centers are sampled pixels \(s_{i},i\in[0,\ N_{s}-1]\)), \(R(c_{1},c_{2})\) is the CR between \(c_{1}\) and \(c_{2}\), \(N(\mathbf{Z}_{c_{i}})\) is the number of class centers absorbed by class \(i\), \(N(\mathbf{S}_{c_{i}})\) is the pixel number in class \(i\), \(T_{start}\) is the initial value of CR \(T\), and \(T_{end}\) is the lowest CR threshold. The detailed process of WFCRMCA is provided in the flow chart in Figure 4. The simulated annealing Markov chain decomposition clustering in wavelet feature space is listed as follows:  
1. Input parameters: \(S_{\text{topx}}\) and \(S_{\text{topy}}\) are the sampling distances along horizontal or vertical directions; \(b\), \(m\), \(n\) are, separately, the band number, column number, and row number of original remotely sensed images; \(S_{\text{cale}}\) is the wavelet transform scale; \(S_{\text{cale2}}\) is the number of minutia scale attending clustering (i.e., \(S_{\text{cale}}-S_{\text{cale2}}-S_{\text{cale}}\) minutia sections)
2. Data preprocessing: delete bands primarily affected by noise and atmosphere, such as the 1-6 th, 33 rd, 107-114 th,153-168 th, and 222-224 th bands of AVIRIS. Multi-spectral images need to expand bands. 3. Apply band-pass \(\mathbf{S}_{\text{cale}}\)-scale wavelet filter (for example, Equation (6) [23; 24]) to all pixels, search extreme points above noise threshold \(T_{\text{peak}}\) between neighbor crossing zero points on each minutia section, and mark upward-maximal point as one and downward-minimal point as two at the corresponding position. \[ \begin{array}{l}\text{H}=\{0.0,0.125,0.375,0.375,0.125,0.000\},\\ \text{G}=\{-0.0061,-0.0869,-0.5798,0.5798,0.0869,0.0061\}\end{array} \] (6)
4. According to \(S_{\text{topx}}\times S_{\text{topy}}\) sampling distance, sample the pixels and create \(N_{s}\) sampled pixels evenly. 5. Apply simulated annealing Markov state decomposition clustering to \(S_{\text{cale}}\)-\(S_{\text{cale}2}\)-\(S_{\text{cale}}\) scale minutia sections of sampled data. 1. Set initial temperature \(T\) as \(T_{\text{start}}\), the clustering signal standard \(T_{\text{signal}}\) (ratio of intra-class sampled pixel number over the number of total sampled pixels) is 1.0, and each pixel is one class center (beginning with \(N_{s}\) class centers). In the end, according to step b-e, apply Markov chain decomposition in state space to the wavelet features of the sampled pixels by gradually depressing the signal size. \[N_{c}=N_{s},\mathbf{C}=\{c_{0},c_{1},\cdots,c_{N_{c}-1}\},\mathbf{S}_{c_{i}}= \{s_{i}\},c_{i}\in\mathbf{C},i\in[0,N_{c}-1]\] 2. Make judgments to all present class centers. If class \(i\) is a significant signal in which the number of pixels is more prominent than \(T_{signal}N_{s}\), move to the next class. Otherwise, search forward one by one for another class \(j\) whose size is smaller than \(T_{signal}N_{s}\), and make clustering judgments between class \(j\) and \(i\). 3. According to Equation (8), if the CR between the centers of two classes (\(i\) and \(j\)) meets the condition \(P_{ij}=r_{ij}-T>0\), then class j is absorbed into class \(i\). Continue this process (b) until the last class is detected. \[