Wavelets
Specific interests ...
Comparison of the resolution of various continuous wavelets in the frequency- and space- domains
Kirby (2005)
compares the radially-averaged Fourier power spectrum against the
global scalogram for seven continuous,
two-dimensional wavelets: Derivative of Gaussian, Halo, Morlet, Paul, Perrier
and Poisson wavelets.
It also describes a new wavelet based on a superposition of rotated
Morlet wavelets, named the 'fan' wavelet.
Only the fan, Halo and Morlet wavelets reproduce the Fourier spectrum exactly.
The Poisson wavelet reproduces it very poorly in the wavenumber domain, but gives the
best resolution in the space domain.
Fortran95 source code is available to compute the 2D CWT with these wavelets upon
request.
Scalograms of a superposition of sine waves: Halo (left) and Poisson (right) wavelets:
Isotropic wavelet phase of 2D signals
The CWT with real (isotropic) wavelets yields wavelet coefficients that are real-valued only, and hence are not able to reveal the wavelet phase.
And while complex wavelets do return a non-zero imaginary component in the coefficients, they are anisotropic, only resolving features in a certain
direction and hence only giving an anisotropic phase.
Kirby (2005) describes the construction of an isotropic complex wavelet that gives the isotropic phase of a 2D
signal, and can hence be used in flexural isostatic modelling (Kirby and Swain, 2004;
Kirby and Swain, 2006).
Shown below is the isotropic fan wavelet in the wavenumber domain (top), and its real (bottom left) and imaginary (bottom right) parts in the space domain:
The anisotropic fan wavelet in the wavenumber domain (top), and its real (bottom left) and imaginary (bottom right) parts in the space domain:
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