All examples use variants of Morlet wavelets. The wavelet transform modulus are added 5% of noise before being sent to the reconstruction algorithm.
The figure on the left of each line represents the original signal (blue) and its reconstruction (red). On the three next figures are three components of the wavelet transform modulus (blue) along with their reconstructions (red).
Realization of a Gaussian process
Signals
Low frequencies
Middle frequencies
High frequencies
Line from an image
Signals
Low frequencies
Middle frequencies
High frequencies
Sums of sinusoids (successful)
Signals
Low frequencies
Middle frequencies
High frequencies
Sums of sinusoids (unsuccessful)
The difference between the ground true wavelet transform modulus and the reconstructed one is larger than the amount of additional noise.
Signals
Low frequencies
Middle frequencies
High frequencies
Audio signals
All examples use variants of Morlet wavelets. The wavelet transform modulus are added 1% of noise before being sent to the reconstruction algorithm.
The quality of reconstructed signals depends on the wavelets. Here, we still use variants of Morlet wavelets, with eight wavelets per octave, and a varying Q-constant.
For all the algorithms, the reconstruction is good when the constant is small (the wavelets have a large overlap), and becomes worse when the constant increases.
Note that, in this example, the signal reconstructed with a traditional algorithm presents an echo from Q=14, while this phenomenon appears only at Q=17 with our algorithm.
The difference between the initial and reconstructed wavelet transform modulus is in general of the same order as the input noise, or smaller. However, the difference between the initial and reconstructed signals can be much larger.
We give an example of reconstruction for three levels of noise.
1 - Input noise: 5%
Error over the modulus: 3.5%
Error over the signal: 17.1%
Signals
Low frequencies
Middle frequencies
High frequencies
2 - Input noise: 10%
Error over the modulus: 8.0%
Error over the signal: 53.3%
Signals
Low frequencies
Middle frequencies
High frequencies
3 - Input noise: 15%
Error over the modulus: 11.0%
Error over the signal: 81.2%
In this case, the relative difference between initial and reconstructed wavelet transform modulus is 0.47% (for an input noise of 1%). However, the relative difference between initial and reconstructed signals is 95%.
We plot below the difference in phase between initial and reconstructed wavelet transforms. Black points correspond to places where the modulus is too small so that the phase can be significant.
The phase varies slowly in both time and frequency, compared to the scalogram, and the variation is all the more slow that the modulus is large.
