Visualization of wavelet

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Many wavelet does not have functional form, but defined by the MRA coefficient.

The visualization of wavelet can be done by using wavelet construction.

\displaystyle v_{j+1,k} = \sum_{n} g_0(k-2n)v_{j,n} + g_1(k-2n) w_{j,n}

For scaling function, we can define v_0 = {1} and w_0 = {0} .

\displaystyle v_{1,k} = \sum_{n} g_0(k-2n)v_{0,n} = g_0(k)

Similarly, the wavelet can be started with v_0 = {0} and w_0 = {1} .

\displaystyle v_{1,k} = \sum_{n} g_1(k-2n)w_{0,n} = g_1(k)

Then build by iteration,

\displaystyle v_{j+1,k} = \sum_{n} g_0(k-2n) v_{j,n}

From last post on the scaling coefficient, i calculated and plot the wavelet for m = 4.

Capture.PNG

we can see the wavelet becomes the Haar wavelet as the free parameter goes to 1. In fact, it becomes a shifted Haar wavelet when the free parameter goes to 0, as we can imagine.

When the free parameter is 0.683013, it is the Daubechies-2 wavelet. Notes that some people will absorbed a factor $latex 1/ \sqrt{2} $ into the coefficient, so that their free parameter is 0.683013/\sqrt{2} = 0.482963.

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Orthonormal Scaling Coefficient

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A multi-resolution analysis is defined by scaling function and the corresponding wavelet. From the scaling relations

\displaystyle \phi_{j,k}(x) = \frac{1}{\sqrt{2}} \sum_{l} g_0(l) \phi_{j+1,2k+l}(x)

\displaystyle \psi_{j,k}(x) = \frac{1}{\sqrt{2}} \sum_{l} g_1(l) \phi_{j+1,2k+l}(x)

the scaling function and wavelet can be defined from the scaling coefficient g_0, g_1

The coefficients are constrained due to the properties of wavelet and scaling function.

\displaystyle \int \phi(x) dx = 1

\displaystyle \int \phi_{j,k}(x) \phi_{j,k'}(x) dx = \delta_{kk'}

\displaystyle \int \psi(x) dx = 0

\displaystyle \int \psi_{j,k}(x) \psi_{j,k'}(x) dx = \delta_{kk'}

\displaystyle \int \psi_{j,k}(x) \phi_{j,k'}(x) dx = 0

These properties lead to

\displaystyle \sum g_0(l) = 2

\displaystyle \sum g_1(l) = 0

\displaystyle \sum_{l,n} g_0(l) g_0(l+2n) = \begin{matrix} 2, & n=0 \\ 0, & else  \end{matrix}

\displaystyle \sum_{l,n} g_1(l) g_1(l+2n) = \begin{matrix} 2, & n=0 \\ 0, & else  \end{matrix}

\displaystyle \sum_{l,n} g_0(l) g_1(l+2n) = 0

The 3rd and 4th constrains requires the numbers of non-zero element in g_0, g_1 are even.

One of the solution is setting

g_1(k) = (-1)^k g_0 (1-k)

so that we don’t need to worry g_1 and the 4th constrain becomes the 3rd constrain, and the 5th constrain is always satisfied. Now, only the 1st, 2nd, and 3rd constrains are needed. This is equivalent to 1+m/2 equations with number of non-zero elements in g_0 is m.

m 1 + \frac{m}{2} Degree of Freedom
2 2 0
4 3 1
6 4 2
8 5 3

For size of 4, the solution is

\displaystyle g_0 = \left(a,  \frac{1-\sqrt{1+4a-4a^2}}{2},  1-a,  \frac{1+\sqrt{1+4a-4a^2}}{2}\right)

In fact, the coefficient for g_0 can be grouped as even and odd, so that

\displaystyle \sum g_0(2k) = \sum g_0(2k+1) = 1

and the constrain 3rd can lead to,

\displaystyle (\sum g_0(2k))^2 + (\sum g_0(2k+1)^2 = 2 ,

which is automatically fulfill.

 

Wavelet Analysis or MRA

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Although the Fourier transform is a very powerful tool for data analysis, it has some limit due to lack of time information. From physics point of view, any time-data should live in time-frequency space. Since the Fourier transform has very narrow frequency resolution, according to  uncertainty principle, the time resolution will be very large, therefore, no time information can be given by Fourier transform.

Usually, such limitation would not be a problem. However, when analysis musics, long term performance of a device, or seismic survey, time information is very crucial.

To over come this difficulty, there a short-time Fourier transform (STFT) was developed. The idea is the applied a time-window (a piecewise uniform function, or Gaussian) on the data first, then FT. By applying the time-window on difference time of the data (or shifting the window), we can get the time information. However, since the frequency range of the time-window  always covers the low frequency, this means the high frequency  signal is hard to extract.

To improve the STFT, the time-window can be scaled (usually by 2). When the time window is shrink by factor of 2, the frequency range is expanded by factor of 2. If we can subtract the frequency ranges for the time-window and the shrink-time-window, the high frequency range is isolated.

To be more clear, let say the time-window function be

\phi_{[0,1)}(t) = 1 , 0 \leq t < 1

its FT is

\hat{\phi}(\omega) = sinc(\pi \omega)

Lets also define a dilation operator

Df(t) = \sqrt{2} f(2t)

the factor \sqrt{2} is for normalization.

The FT of D\phi(t) has smaller frequency range, like the following graph.

Capture.PNG

We can subtract the orange and blue curve to get the green curve. Then FT back the green curve to get the high-frequency time-window.

We can see that, we can repeat the dilation, or anti-dilation infinite time. Because of this, we can drop the FT basis Exp(-2\pi i t \omega), only use the low-pass time-window to see the low-frequency behaviour of the data, and use the high-pass time-window to see the high-frequency behaviour of the data. Now, we stepped into the Multi-resolution analysis (MRA).

In MRA, the low-pass time-window is called scaling function \phi(t) , and the high-pass time-window is called wavelet \psi(t).

Since the scaling function is craetd by dilation, it has the property

\phi(t) = \sum_{k} g_{0}(k) \phi(2t-k)

where k is integer. This means the vector space span by {\phi(t-k)}_{k}=V_0 is a subspace of the dilated space DV_0 =V_1. The dilation can be go one forever, so that the whole frequency domain will be covered by V_{\infty}.

Also, the space span by the wavelet, {\psi(t-k)}=W_0, is also a subspace of V_1. Thus, we can imagine the structure of MRA is:

Capture.PNG

Therefore, any function f(t) can also be expressed into the wavelet spaces. i.e.

f(t) = \sum_{j,k} w_{j,k} 2^{j/2}\psi(2^j t - k)

where j, k are integers.

I know this introduction is very rough, but it gives a relatively smooth transition from FT to WT (wavelet transform), when compare to the available material on the web.