Continuous wavelet transform

From Academic Kids

In mathematics, the continuous wavelet transform (CWT) is a wavelet transform defined by

<math>\gamma(\tau, s) =
 \int_{-\infty}^{+\infty} x(t) \frac{1}{\sqrt{s}} \psi^{*} \left( \frac{t - \tau}{s} \right) dt

<math> where <math>\tau<math> represents translation, <math>s<math> represents scale and <math>\psi(t)<math> is the mother wavelet.

The original function can be reconstructed with the inverse transform

<math>x(t) =
 \frac{1}{C_\psi} \int_{-\infty}^{+\infty}
   \int_{-\infty}^{+\infty} \gamma(\tau, s)
     \psi\left( \frac{t - \tau}{s} \right) d\tau \frac{ds}{|s|^2}

<math> where

<math>C_\psi = \int_{-\infty}^{+\infty}
 \frac{\left| \hat \Psi(\zeta) \right|^2}{\left| \zeta \right|} d\zeta

<math> is called the admissibility constant and <math>\hat{\Psi}<math> is the Fourier transform of <math>\psi<math>. For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:

<math>C_\psi < +\infty<math>.

Note also that the admissibility condition implies that <math>\hat \Psi(0) = 0<math>, so that a wavelet must integrate to zero. For reference, the relationship between the so-called mother wavelet and the daughter wavelets is as follows:

<math>\psi_{s,\tau}(t) = \frac{1}{\sqrt{s}} \psi \left( \frac{t-\tau}{s} \right) <math>.

Continuous wavelets

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