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I have a very specific application which requires to use the complex Morlet wavelet (or Gabor wavelet, if that's the name you use). I currently use the convolution theorem to compute wavelets transforms, but I'm reading everywhere that direct wavelet transforms are more computer efficient, because of the lifting scheme.

On the other hand, I searched for lifting scheme, and it's so much of a different paradigm that I have no idea how to use it with my wavelet, or whether it's even possible.

My question is: is it possible to use the lifting scheme with the Morlet wavelet in 1D and 2D? If so, can you recommend some litterature explaining the algorithm?

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The Lifting scheme is special algorithm that allows a very efficient computation of the discrete wavelet transform (DWT), via special filters (QMF) and down/up-samplers. In order to compute a DWT the wavelet has to fulfill orthogonality constraints under translation and dilation.

The Morlet wavelet does not fulfill these contraints. Thus, you cannot compute a DWT with it und consequently you cannot use the lifting scheme. The Morlet wavelet is only applicable for computing a continuous wavelet transform. If you need a fast approximation you can have a look here: Fast Gabor-Like Windowed Fourier and Continuous Wavelet Transforms

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  • $\begingroup$ Thanks, this fully answers my question! In my case, the precision I get from the continuity and redundancy is more important than a fast computation. I'll stick with convolutions through FFTs for now! $\endgroup$
    – PhilMacKay
    Commented Mar 19, 2013 at 19:04

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