# Available Wavelet Families

There are two tiers of wavelet types in this package. The most abstract is the `ContWave`

type, representing a class of wavelets. This is split into several strictly continuous wavelets, and a `ContOrtho<:ContWave`

type, which is a supertype of continuous versions of the orthogonal wavelets defined in Wavelets.jl.

`ContinuousWavelets.ContWave`

— Type`ContWave{Boundary,T}`

The abstract type encompassing the various types of wavelets implemented in the package. The abstract type has parameters `Boundary<:WaveletBoundary`

and `T<:Number`

, which gives the element output type. Each has both a constructor, and a default predefined entry. These are:

`Morlet`

: A complex approximately analytic wavelet that is just a frequency domain Gaussian with mean subtracted.`Morlet(σ::T) where T<: Real`

.`σ`

gives the frequency domain variance of the mother Wavelet. As`σ`

goes to zero, all of the information becomes spatial. Default is`morl`

which has $\sigma=2\pi$.$\psi\hat(\omega) \propto \textrm{e}^{-\frac{\sigma^2}{2}}\big(\textrm{e}^{-(\sigma - \omega)^2} -\textrm{e}^{\frac{\omega^2-\sigma^2}{2}}\big)$

`Paul{N}`

: A complex analytic wavelet, also known as Cauchy wavelets.`pauln`

for n in`1:20`

e.g.`paul5`

$\psi\hat(\omega) \propto \chi_{\omega \geq 0} \omega^N\textrm{e}^{-\omega}$

`Dog{N}`

: Derivative of a Gaussian, where N is the number of derivatives.`dogn`

for`n`

in`0:6`

. The Sombrero/mexican hat/Marr wavelet is`n=2`

.$\psi\hat(\omega) \propto \omega^N\textrm{e}^{-\frac{\omega^2}{2}}$

`ContOrtho{OWT}`

. OWT is some orthogonal wavelet of type`OrthoWaveletClass`

from Wavelets.jl. This uses an explicit construction of the mother wavelet for these orthogonal wavelets to do a continuous transform. Constructed via`ContOrtho(o::W)`

where`o`

is from Wavelets.jl. Alternatively, you can get them directly as`ContOrtho`

objects via:`cHaar`

Haar Wavelets`cBeyl`

Beylkin Wavelets`cVaid`

Vaidyanathan Wavelets`cDbn`

Daubhechies Wavelets. n ranges from`1:Inf`

`cCoifn`

Coiflets. n ranges from`2:2:8`

`cSymn`

Symlets. n ranges from`4:10`

`cBattn`

Battle-Lemarie wavelets. n ranges from`2:2:6`

Above are examples of every mother wavelet family defined in this package; the only analytic and/or complex wavelets are the `Morlet`

and the `Paul`

wavelet families. Once you have chosen a type of wavelet, this is used to construct the more specific CWT, which specifies more details of the transform, such as frequency spacing, whether to include an averaging filter or not, a frame upper bound, etc.