# Continuous wavelet transform

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

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


[itex] where [itex]\tau[itex] represents translation, [itex]s[itex] represents scale and [itex]\psi(t)[itex] is the mother wavelet.

The original function can be reconstructed with the inverse transform

[itex]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}


[itex] where

[itex]C_\psi = \int_{-\infty}^{+\infty}
 \frac{\left| \hat \Psi(\zeta) \right|^2}{\left| \zeta \right|} d\zeta


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

[itex]C_\psi < +\infty[itex].

Note also that the admissibility condition implies that [itex]\hat \Psi(0) = 0[itex], 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:

[itex]\psi_{s,\tau}(t) = \frac{1}{\sqrt{s}} \psi \left( \frac{t-\tau}{s} \right) [itex].

### Continuous wavelets

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