## 2 Methods  
Although the spectral curves of the same objects under different conditions are somewhat different, they have the same feature points (upward, downward, maximum, and minimum) at the same spectral positions (Figure 1). The WFCRMCA could detect abrupt signals through band-pass wavelet transform, such as crossing zero and extreme points. But crossing the zero point cannot be ensured to be a pulse signal, and perhaps is a smoothly changed signal, so the extreme points between adjacent zero points are much more critical.  
Figure 1: Five points at different spatial positions within the same class have the same features at the exact spectral locations.  
The signs in spectral vector format are classified according to the priority of importance from low to high: downward, upward, protruding, and concave (Figure 2).  
Figure 2 is the result of four kinds of abrupt signals processed by \(\psi(t)\) (Equation (1), [19], 1 st derivative Gauss function \(\theta(t)\)). For some remotely sensed images affected by too many mixed pixels, the position of critical points will probably deviate or have many little fluctuations, so WFCRMCA could eliminate unimportant signals by setting a maximum threshold and only clustering the partial minutia at a high-level scale.  
\[\theta(t)=\frac{1}{\sqrt{2\pi}}e^{-t^{2}/2},\psi(t)=\frac{d\theta}{dt}=-\frac {1}{\sqrt{2\pi}}te^{-t^{2}/2} \tag{1}\]  
Wavelet feature clustering algorithms only analyze minutia data by detecting and determining the positions of abrupt signals. Using a fast binary Mallet wavelet algorithm [20] in Equation (2) to extract wavelet coefficients, WFCRMCA can mark the upward-maximal points (Figures 2a and 2a') and downward-minimal points (Figures 2b and 2b') along the spectrum. WFCRMCA will overlook the weak signals if \(T_{\text{peak}}\) is large enough, leading to a failure in identifying some valuable signs among hidden objects.  
\[\begin{cases}c_{j,k}=\sum_{n}h_{n-2k}c_{j-1,n}\\ d_{j,k}=\sum_{n}g_{n-2k}c_{j-1,n}\end{cases},j=1,2,\ldots,S_{cale}(\leq\lfloor log _{2}b\rfloor) \tag{2}\]  
The WFCRMCA uses \(r_{ij}\) (correlation ratio), which works like a distance but not Euclidean distance as clustering criteria, to evaluate the difference between two spectral vectors on partial minutia. Equation (3) uses \(S_{cale2}\)-scale minutia of \(S_{cale}\)-scale wavelet coefficients to cluster, \(t_{ik}\) is the \(k\)th feature of its sampled vector, \(N(\cdot)\) is the number of feature positions that match criteria. WFCRMCA could use binary values to mark whether the position is valuable enough to attend clustering. When \(S_{cale2}=S_{cale}=\lfloor log_{2}b\rfloor\), the bit number attending clustering comparison is \(b\sum_{i=1}^{\lfloor log_{2}b\rfloor}1/2^{i}\approx b-1\).  
\[\begin{split}& r_{ij}=\frac{N(t_{ik}=t_{ik},k\in\Omega)}{N \left(t_{ik}\
eq t_{jk},k\in\Omega\right)+N\left(t_{ik}=t_{jk},k\in\Omega \right)}\\ & if\ S_{cal2}=S_{cale},\Omega=\left[0,b\sum_{k=1}^{S_{cale}} \frac{1}{2^{k}}-1\right]\\ & if\ S_{cal2}<S_{cale},\Omega=\left[b\sum_{k=1}^{S_{cale}-S_{ cal2}}\frac{1}{2^{k}},b\sum_{k=1}^{S_{cal}}\frac{1}{2^{k}}-1\right]\end{split} \tag{3}\]  
Figure 2: The wavelet band-pass filter and four kinds of abrupt signals. (**a**–**d**) are the four critical signals: upward-maximal point, downward-minimal point, protruding crossing zero point, and concave crossing zero point. \(\psi(t)\) is the band-pass wavelet filter, (**a**–**d**) are the output of the four signals through the wavelet filter.