Wavelet transform modulus maxima method: Difference between revisions

Revision as of 15:06, 12 April 2011

Revision as of 15:07, 12 April 2011

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<math>G' (t,a,b) = \frac{a}{(2\pi)^{-1/2}}(t - b) e^{[\frac{-(t-b)^2}{2a^2}]}</math>

Once a "mother wavelet" is chosen, the continuous wavelet transform is carried out as a continuous, [[square-integrable [http://en.wikipedia.org/wiki/Square-integrable_function] function]] that can be scaled and translated. Let <math>a > 0</math> be the scaling constant and <math>b\in\mathbb{R}</math> be the translation of the wavelet along the signal:

:<math>X_w(a,b)=\frac{1}{\sqrt{a}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt</math>

where <math>\psi(t)</math> is a continuous function in both the time ___domain and the frequency ___domain called the mother wavelet and <math>^{\ast}</math> represents the operation of complex conjugate [http://en.wikipedia.org/wiki/[Complex_conjugate]].

By calculating <math>X_w(a,b) </math> for subsequent wavelets that are derivatives of the mother wavelet, singularities can be identified. Successive derivative wavelets remove the contribution of lower order terms in the signal, allowing the maximum <math>h_i</math> to be detected. (Recall that when taking derivatives, lower order terms become 0.) This is the "modulus maxima".