Extraction of ridges and skeleton Another related approach is the extraction of ridges and skeleton. This method extracts from either the STFT or the continuous wavelet transform (CWT) some particular sets of curves deduced from the stationary points of their phase (see [Fla93] for more information about the stationary phase principle). Indeed, applying the stationary phase theorem to the signal reconstruction formula of the CWT $T_x(t,a;\Psi)$ expressed in the frequency domain : $$X(\nu)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \sqrt... ... H(a \nu)\ T_x(t,a;\Psi)\ e^{-j2\pi \nu t}\ dt\ \frac{da}{a^2}$$ leads to particular points such that $$\hat{t}(x;t,a)=t-\Phi'_h(\nu_0)\ \ \mbox{ and }\ \ \hat{a}(x;t,a)=a,$$ (4.25) with $\Phi_h(\nu)=\arg\{H(\nu)\}$, and which constitute a set of curves called the horizontal ridges of the representation. Similarly, applying the stationary phase principle to the signal reconstruction formula of the CWT expressed in the time domain leads to particular points such that $$\hat{t}(x;t,a)=t\ \ \mbox{ and }\ \ \hat{a}(x;t,a)=a\ \frac{\nu_0}{\phi'_h(0)},$$ (4.26) with $\phi_h(t)=\arg\{h(t)\}$, and which constitute a set of curves called the vertical ridges of the representation. These relations between the ridges and the reassignment operators suggest to extract the ridges of any reassigned distribution by a straightforward generalization of expressions (4.25), (4.26). For example, let us extract the ridges from the spectrogram of the previous signal (see fig. 4.39): >> [tfr,rtfr,hat]=tfrrsp(sig); >> ridges(tfr,hat); Figure 4.39: Extraction of ridges from the spectrogram The result is interesting : apart from some ``gaps'' present in particular on the sinusoidal frequency modulation, this method concentrates and localizes nearly ideally the signal in the time-frequency plane, even when there are two components present at the same time (or at the same frequency).
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