Chemistry - Continuous wavelet transform

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

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

where $τ$ represents translation, $s$ represents scale and $ψ(t)$ is the mother wavelet.

The original function can be reconstructed with the inverse transform

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

where

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

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

$C_\psi < +\infty$ .

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

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

Continuous wavelets

Mexican hat wavelet
Hermitian wavelet
Hermitian hat wavelet
Complex Mexican hat wavelet
Morlet wavelet
Modified Morlet wavelet
Addison wavelet
Hilbert-Hermitian wavelet

Continuous wavelet transform

Continuous wavelets

Further reading