Wavelet Transformation & Scattering Transformation

• Wavelet Transformation & Scattering Transformation
  • Wavelet Transform (WT)
    • Introduction to Continuous Wavelet Transform (CWT)
    • Focus on Mother Wavelet
    • Implement by bandpass filter
    • DWT & Resolution
    • Octacve
  • Implement Wavelet Transform
    • Wavelet
    • Convolution with wavelet
    • Scalogram
  • Try MATLAB Built-in Functions
  • Comparison with Synchrosqueezing Transforms (SST)
  • Scattering Transformation (ST)
    • Main idea
  • Reference

Wavelet Transform (WT)

Introduction to Continuous Wavelet Transform (CWT)

The basic idea in wavelet analysis is to start with a function $\psi (t)$, known as the mother wavelet. Then, a dilated wavelet is
$${\psi }_{\lambda }(t)=\frac{1}{\lambda }\psi \left(\frac{t}{\lambda }\right)$$
Note that if $\lambda$ is larger then the oscillation frequency is lower. Consider the following function
$${\psi }_{a,b}(t)=\frac{1}{a}\psi \left(\frac{t-b}{a}\right)$$
The parameters $a$ and $b$ are responsible for scaling and translating the function. Hence, we could use above wavelet to plot scalogram, with scaling axis and translating axis by
$$X(a,b)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t){\psi }_{a,b}(t{)}^{\ast }dt$$

The higher scale factor is, the lower frequency will be captured. Hence, CWT($a$, $b$) could be viewed as capturing the (frequency, time) at $(\frac{{\omega }_0}{a},b)$, where ${\omega }_0$ is origninal frequency of mother wavelet and $\psi$ center at $t=0$.

Sometimes, if define ${\psi }_{a,b}(t)=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$, then there is an factor $\frac{1}{a}$ in the above transformation.

• Refer to S. L. Brunton & J. Nathan Kutz and their video .
• Refer to S. Qian & D. Chen .

Focus on Mother Wavelet

A complex Morlet wavelet $\psi$ can be defined as the product of a complex sine wave and a Gaussian window
$$\psi (t)=e^{2i\pi f_{0}t}{e}^{-\frac{t^2}{2{\sigma }_t}}=e^{i{\omega }_0}{e}^{-\frac{t^2}{2{\sigma }_t}}$$
where $f_0$ is frquency and ${\sigma }_t$ is the bandwidth of the Gaussian in time domain, Actually we have 3 way to determine the Morlet wavelet

1. Gaussian bandwidth (${\sigma }_t$) in time domain
2. Gaussian bandwidth (${\sigma }_{\omega }$) in frequency domain
3. number of cycle in bandwidth in time domain

In fact, ${\sigma }_t$ and ${\sigma }_{\omega }$ is satisfied the uncertainty principle.

Moreover, the number of cycle ($n$) and ${\sigma }_t$ follow the following relation
$${\sigma }_t=\frac{n}{2\pi f}$$
On the other hand, Refer to M. X Cohen (2018) or refer to their video .
For instance, let $f_0=10$, determine the wavelet by bandwidth in frequency domain.

The time frequency trade-off can be seen.

Implement by bandpass filter

Now, apply Fourier transformation on wavelet, say $\hat{\psi }(\omega )$, where $\omega =2\pi f$. Clearly, $\hat{\psi }(\omega )$ is a Gaussian pulse with bandwidth ${\sigma }_{\omega }$ and center at $\omega$. It could be viewed as bandpass filter.

Now, implement CWT as bandpass filter.
$$X(a,b)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\omega )\hat{\psi }(\omega )e^{i\omega b}dt$$
Refer to MATLAB help center .

Please refer to M. X Cohen video .

DWT & Resolution

Since the uncertainty principle, the resolution should be as following figure.

First, focus at time domain. If low frequency, the bandwidth ${\sigma }_{\omega }$ in frequency domain is small. Hence, the bandwidth ${\sigma }_t$ in time domain is large, so the time resolution at low frequency is small. And vice versa.

Second, focus at frequency domain. If high frequency, the bandwidth ${\sigma }_{\omega }$ in frequency domain is large. Hence the frequency resolution at high frequency is small. And vice versa.

According to above figure, we could see that

On the other hand, the time frequency trade-off can be seen as following figure.

According to above figure, we could let $a=2^j$ and $b=k{2}^j$. The discrete wavelet transformation is as
$${\psi }_{j,k}(t)=\frac{1}{2^j}\psi \left(\frac{t-k{2}^j}{2^j}\right)$$

Note that if $a$ is large then frequency is small. Meanwhile, the space between two time grids is large.

Octacve

In the above equation $j$ might not be integer. We could divide ''octave'' to be finer. The parameter $Q$ is often referred to as the number of ''voices per octave''.
$${\psi }_{j,k}(t)=\frac{1}{2^{j/Q}}\psi \left(\frac{t-k{2}^{j/Q}}{2^{j/Q}}\right)$$
where let new $\psi (t)$ satisfy $\hat{\psi }(\omega )$ center at frequency ${\omega }_0$ and with bandwidth $\frac{{\sigma }_{\omega }}{Q}$.

For instance, if $Q=1$, and ${\sigma }_{\omega }=\frac{{\omega }_0}{2}$,

If $Q=2$, the bandwidth is half as long as original bandwidth. Moreover, it requires two step to reach next octave.

The reason $Q$ is referred to as the number of voices per octave is because increasing the scale by an octave (a doubling) requires $Q$ intermediate scales. Refer to MATLAB help center .

Implement Wavelet Transform

Wavelet

Take $Q=2$. Let mother wavelet with $f_0=1$ (Hz) and bandwidth ${\sigma }_{\omega }=1/4=\frac{f_0}{2Q}$ in frequency domain. Let wavelet center at $\mathrm{\Lambda }=\{2^{j/Q}{\}}_{j=-4}^{12}$. The following shows that $j$ is even case.

Convolution with wavelet

The first is the original signal.Each row shows that signal convolution with different wavelet.

Note that resolution is different.

Scalogram

Use interpolation to plot the scalogram.

Try MATLAB Built-in Functions

Use cwt.

Comparison with Synchrosqueezing Transforms (SST)

Scattering Transformation (ST)

Main idea

Extend the graph mentioned. The blue nodes are created by the first WT and the purple nodes are created by the second WT.

Focus on second layer (blue). There are 3 nodes, which are created by the first wavelet transformation. However, each node contains different information. Hence, they convole with different number of wavelet.

Reference

1. MATLAB Help Center, Continuous and Discrete Wavelet Transforms.
2. MATLAB Help Center, Continuous Wavelet Transform as a Bandpass Filter.
3. Chun-Lin Liu, A Tutorial of the Wavelet Transform, NTU, (2010).
4. Steven L. Brunton & J. Nathan Kutz, Data Driven Science & Engineering: Machine Learning, Dynamical Systems, and Control, UW, (2017).
5. Steven L. Brunton, Youtube Channels.
6. Kevin E Alexander, Time Frequency Analysis Plugin for EEGLAB, Air Force Research Laboratory.
7. Kevin E Alexander, Youtube Channels.
8. Jian-Jiun Ding, Lecture Note of Time-Frequency Analysis and Wavelet Transform, NTU, (2021).
9. Chia-Feng Lu, Lecture Note of MATLAB Programming for Medical Signal Analysis, NYCU, (2021).
10. Rami Khushaba, Youtube Channels, USYD.
11. Shie Qian, Dapang Chen, Joint Time-Frequency Analysis: Method and Application.
12. Mike X Cohen, Youtube Channel, Radboud University Medical Center.
13. Ingrid Daubechies, Ten Lectures on Wavelets.