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We prove that the wavelet modulus maxima with a complex-valued wavelet

can detect and characterize singularities.  This can be regarded as an extension

of the previous wavelet modulus work with a real wavelet.   The ridges

of wavelet transforms are the places in the time-frequency plane where the local

energies of a signal are mostly concentrated. We show that based on the ridges

of the wavelet transforms, the osciallating singularities are better located and

their oscillating components are better characterized than they are based on the

general maxima of real-valued wavelet transforms.  We demonstrate potential

applications of these techniques.

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Key Words: Continuous Wavelet Transform, Wavelet Modulus Maxima, Singularity, Instantaneous Frequencies

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