## Theory Description
### Adaptive Signal Decomposition Based on Reassignment Vector  
Reassignment method (RM) realizes the TF representation of multicomponent signals in two directions of time and frequency scale. In other words, RM can accurately position the TF signature of the interested modes. Based on the denoising operation in STFT domain defined in Section 2.1, the group delay estimation \(\hat{t}_{f}(t,\xi)\) and instantaneous frequency estimation \(\hat{\omega}_{f}(t,\xi)\) can be calculated as:  
\[\hat{\tau}_{f}(t,\xi)=t+\Re\left\{\frac{V_{f}^{fg}(t,\xi)}{V_{f}^{\delta}(t,\xi) }\right\} \tag{6}\]  
\[\hat{\omega}_{f}(t,\xi)=\xi-\frac{1}{2\pi}\Im\left\{\frac{V_{f}^{\delta^{ \prime}}(t,\xi)}{V_{f}^{\delta}(t,\xi)}\right\} \tag{7}\]  
where \(V_{f}^{\delta}\) and \(V_{f}^{\delta^{\prime}}\) are corresponding to the denoised STFT operation using sparse low-rank matrix approximation of multicomponent signal \(f\) calculated by the window function \(tg(t)\) and \(g^{\prime}(t)\). \(\Re\{\cdot\}\) and \(\Im\{\cdot\}\) is illustrated as the real and imaginary part of the complex number. Therefore, RM algorithm can be described as:  
\[RM(t,w)=\int_{-\infty}^{+\infty}\left|V_{f}^{g}(\mu,\eta)\right|\cdot\delta(w- \hat{w}_{f}(\mu,\eta))\cdot\delta(t-\hat{t}_{f}(\mu,\eta))d\mu d\eta \tag{8}\]  
According to the definition above, Reassignment Vector (RV) can be expressed as follows [25; 26]:  
\[\text{RV}(t,\xi)=\big{(}\hat{\tau}_{f}(t,\xi)-t,\hat{\xi}_{f}(t,\xi)-\xi\big{)} \tag{9}\]  
From the formula (9), we find that the RV is also considered in terms of time and frequency. If \(f=\delta_{t_{0}}\) is satisfied, \(\text{RV}(t,\xi)=(t_{0}-t,0)\), which indicates that the component is existing along the time axis. Similarly, if \(f\) is a pure harmonic signal such as \(f=e^{2\pi\xi\xi t}\), \(\text{RV}(t,\xi)=(0,\xi_{0}-\xi)\), which demonstrates a component only along the frequency directions. More generally, a linear chirp signal is described with phase function \(\phi(t)\) and window function \(g(t)=e^{-\sigma\pi t^{2}}\), thus we can obtain the RV as follows:  
\[\text{RV}(t,\xi)=\frac{(\xi-\phi^{\prime}(t))}{\sqrt{\sigma^{2}+\phi^{\prime \prime}(t)^{2}}}\big{(}-\phi^{\prime\prime}(t),\sigma^{2}\big{)} \tag{10}\]  
Generally speaking, RV points to that characteristic curve in the TF plane following the direction \(\big{(}-\phi^{\prime\prime}(t),\sigma^{2}\big{)}\). Theoretically, on the basis of the projection of RV in a specific direction, contour points (CPs) representing the TF curve or intrinsic component can be obtained for the subsequent mode function extraction. Obviously, RV will exhibit a strong variation along the orientation by crossing the ridge curve. To determine the projection of RV in a specific direction, the key step is to get the angle \(\theta\). For the convenience of discretization and computation, RV can be regarded as a displacement on a grid. Then, the STFT is computed at frequencies \(\frac{p}{N}\), where \(p=0,...,N-1\) and \(N\) is the number of frequency bins. By the way above, the grid is indexed by \((k,p)\) and \(k\) is corresponding to the time instant. Through the above theoretical definition, the CPs can be computed by projecting this vector following the inner product operation:  
\[\alpha(k,p):=\left\langle\text{RV}_{r}\Big{(}k,\frac{p}{N}\Big{)}\,v_{\theta_{ r}(k,p)}\,\text{mod}\,\pi\right\rangle=0 \tag{11}\]  
where \(v_{\theta}\) is denoted as the vector along the direction \(\theta\). Based on it, a local projection angles (LPAs) algorithm published in [25] is used to achieve a robust estimation of TF signatures for a wide class of multicomponent signal. Using the just estimated CPs, each corresponding mode \(f_{i}(t)\) can be reconstructed through:  
\[f_{i}(t)=\frac{1}{g(0)}\int_{(t,\xi)\in B_{i}}V_{f}^{g}(t,\xi)dw \tag{12}\]  
where \(B_{i}\) is the basin of interested components corresponding to the mode \(i\). Figure 4: The time-frequency (TF) mode decomposition by the proposed method. Figure 3: The TFR obtained by short-time Fourier transform (STFT) and reassignment vector (RV) method.