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2. The Multifractal Pumping sub-menu

The most obvious way to increase the local Hölder regularity by an amount d is simply to multiply all the wavelet coefficients at scale j by 2^(-dj). This roughly amounts to performing a fractional integration of order d (indeed, the local Hölder exponent is related to a notion of local fractional derivative). This sub-menu does exactly this. You just have to select the Analyzed signal and to adjust a Spectrum shift value, either by entering a value or by using the arrows. The spectrum shift value is the parameter d mentioned above (the justification for this denomination is that all exponents are increased by d, thus the whole multifractal spectrum is shifted to the right by an amount of d), and hit Compute. The output will be called den_signal#, and will be a regularized version of your original (1D or 2D) signal. It is interesting to note that you can input negative values for d, so that you may decrease the regularity of your signal. Also, this procedure is fully reversible, except for numerical round-offs. Try a large positive value for d (e.g. 4). The result will be a very blurred signal, which seems to contain almost no information. Since the blurred signal is the currently selected one in the Variables list, hit Refresh so that the denoising algorithm knows that you now want to process another signal. Select -d as a spectrum shift. You will recover your original signal.

Note finally that typical values for the shift in this case are around 0.5.


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