## 2 Methods
### Markov Chain Clustering in Wavelet Feature Space  
The wavelet-feature Markov clustering algorithm, i.e., WFCRMCA, first denoises the original data to make the spectral features more accurate, then uses a band-pass wavelet filter to detect all dot vectors for sharp points, including upward-maximal and downward-minimal points. As a result, simulated annealing Markov chain decomposition in state space, formed by evenly spaced sampled data, could realize the best centers at each temperature and sub-finest centers on the whole scope. According to the peculiarity of simulated annealing Markov clustering, each clustering center is one state, and the space is a definite Markov state chain. If two classes (or states) merge, according to CR, it has nothing to do with other states. For example, for Markov chain \(\mathbf{I}=\left\{1,2,\cdots,n\right\}\) in definite state space \(\left\{\mathbf{X}(n)\right\}\), if any two states communicate, they must be in the same class. Thus, the whole state space (pixels) could be separated into a few isolated classes according to transferred communication. \(T\), which is defined as a threshold value of CR \(r_{ij}\), is used as an annealing temperature to control the clustering process. **Definition 1**. : _If \(P_{ij}^{1}=r_{ij}-T>0\) for states \(i\) and \(j\), they have one-step transferred communication denoted as \(i\overset{1}{\leftrightarrow}j\)._  
**Theorem 1**. : _Communication can be transferred. If \(i\overset{m}{\leftrightarrow}k\) and \(k\overset{T}{\leftrightarrow}j\) (\(p_{ik}^{m}>0\), \(p_{kj}^{n}>0\)), \(i\overset{m+n}{\longleftrightarrow}j\)(\(p_{ij}^{m+n}>0\))._  
**Proof of Theorem 1. ** According to Chapman-Kolmogorov equation:  
\[p_{ij}^{m+n}=\sum_{g\in I}p_{ig}^{m}\cdot p_{gl}^{n}\geq p_{ik}^{m}\cdot p_{ kj}^{n}>0.\text{i.e., }i\overset{m+n}{\longleftrightarrow}j.\]  
\(\Box\)  
**Definition 2**. : _If feature \(i\) in wavelet characteristic space has \(p_{ij}=1\), then \(i\) is an absorptive state, forming a single-dot set \(\left\{i\right\}\)._  
**Theorem 2**. : _After Markov clustering, all wavelet features in wavelet feature space are frequently returned states._  
**Proof of Theorem 2. ** (i). A single-dot set is an absorptive, frequently returned state. (ii). As simulated annealing clustering causes \(T\) to be reduced gradually, the \(k\)+1 th clustering iteration is supposed to create a non-single dot set. For instance, \(m\) pixels \(\left\{1,2,\ldots,m\right\}\) are absorbed into one class. \(T^{k}\) is the CR threshold of the \(k\)th iteration, and \(c_{1}\), \(\ldots\), \(i\), \(\ldots\), \(\ldots\), and \(c_{n}\) are the created clustering centers of the \(k\)th iteration. Thus, \(r_{ij}<T^{k}\). During the \(k\)+1 th iteration, \(T^{k+1}=T^{k}-T_{step}\), where \(T_{step}\) is the depressed step of \(T\) at each iteration. If \(T^{k+1}<r_{ij}<T^{k}\), then \(p_{ij}=r_{ij}-T^{k+1}=r_{ij}+T_{step}-T^{k}>0\) and \(p_{ij}<T^{k}-T^{k+1}=T_{step}\), so \(i\xrightarrow{i}j\), then \(i\) and \(j\) are merged together. If \(T_{step}\) is small enough (i.e., the temperature is reduced slowly), and \(i,j,l\) are absorbed in \(k\)+1 th iteration, \(p_{ij}=p_{il}=p_{jl}\approx T_{step}\) so that it could be supposed as in Figure 3 that \(p_{ii}=x,\ p_{ij}=(1-x)/(m-1),i\
eq j,i,j\in\{1,2,\cdots,m-1\}\). As \(m\) states communicate with each other, other \(m-1\) states could be seen as one state \(j\). Let \(p_{ii}=p_{\bar{j}}=x,p_{\bar{j}}=1-x\); thus  
\[f_{ii}=x+(1-x)^{2}+(1-x)^{2}x+\cdots+(1-x)^{2}x^{n}+\cdots=x+(1-x)^{2}\Big{(}1 +x+x^{2}+\cdots\Big{)}=1\]  
So state \(i\) is a frequently returned state. As \(m\) states communicate, the merged \(m\) states are frequently returned. The sufficient and necessary condition of closed set \(C\) is that, for arbitrary elements \(i\in C,j\
otin C\), there exists \(p_{ij}^{(n)}=0,n\geq 1\) (referring [21,22]).