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.ContWaveType
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.