forcing nlevels

[tdr1991]

Hi everybody, I have 1D data, the length is 8, level set 3, there is a warning: Forcing nlevels = 2 Warning: required level (3) is greater than the maximum possible level for haar (2) on a 8x1 ima ge.

when the length is 5, level is 2, the warning is ： Forcing nlevels = 1 Warning: required level (2) is greater than the maximum possible level for haar (1) on a 5x1 image.

How can I deal with it，thank you。

[pierrepaleo]

Hi @tdr1991 pypwt automatically limits the number of decomposition levels. As the data size is halved at each level, "level" cannot be set arbitrarily. The maximum number for "level" is log2(input_size/(filter_size-1)), it does not seem to be correctly implemented in the underlying PDWT library.

So you are right, maximum level for Haar should be 3 on data with 8 samples.

Thank you for the report, I will fix this.

[tdr1991]

Hi @pierrepaleo , there is another question: The higher the level, the more like the original data, or the opposite？ my experiment is the opposite.

[pierrepaleo]

The Wavelets Transform maps a signal to two sets of coefficients: "approximation coefficients" and "details coefficients". The first gives a coarse approximation as the input signal ; the second somehow indicates how to "compensate" for the approximation error.

The approximation coefficients give a coarser representation as the parameter "level" is increased. So if I understood the question correctly, your insight is right.

NB: In your figures, I see that the wavelets coefficients vector has the same length as the input data, which would mean that you used a stationary transform (do_swt=1).

[tdr1991]

The approximation coefficients will be decomposed more “details coefficients” as the parameter "level" is increased，so it will more like the input signal，right？ I indeed used a stationary transform(do_swt=1) 。

[pierrepaleo]

No, when you increase "level", the approximation coefficient are coarser, i.e less like the input signal. Having more detail coefficients doesn't mean that the representation is more precise. The input signal can always be perfectly reconstructed (up to numerical error) from its coefficients regardless of the value of "level".

• In the case of regular DWT (do_swt=0), there are actually exactly as many coefficients (approx + detail) as samples in the input signal. The output space of DWT has the same dimensionality as the input space.
• In the case of SWT (do_swt=1), there are more coefficients than samples in the input signal. This is due to the way SWT is computed (through "undecimated wavelet transform").

Regarding the "similarity" between the input signal and the coefficients, you can make an analogy with a succession of "smoothings". In the case of Haar wavelet :

• The approximation coefficients at level 1, "A1", are obtained by averaging neighboring coefficients of the input signal. The approx. coefficients at level 2, "A2", are obtained by averaging neighboring coefficients of "A1", and so forth. By applying this series of smoothing operation, the approximation signal becomes coarser, and less like the original signal.
• The detail coefficients at level 1, "D1", are obtained by differentiating the input signal. The detail coefficients at level 2, "D2", are obtained by differentiating A1, etc.

[tdr1991]

Ok, thank you very much.

[pierrepaleo]

@tdr1991 The bug should be fixed now (version 0.8.1).

[tdr1991]

Ok, thank you.