MSRI Mathematical Graphics Workshop, 2005

Quick Look at Wavelets

Fourier Transform v.s Wavelets

Fourier transform decompose signal into periodic stationary sine functions. Fourier transform does not provide time-localization.
Sin x


Sin 3x+30


Sin x + Sin 3x+30

Wavelet transform use a wavelet function that can be dilated and translated to to decompose signal. Above is Haar wavelets, you can drag the point to see how it translate or dilate.

Information concerning time-localization e.g. high frequency bursts cannot be read off easily from Fourier Transform. Wavelet is one way to meet the such time localization.

Wavelet Transform

We can see that the wavelet transform define as .
which can be viewed as an inner product of The function with a family of fucntion indexed by two labels . The function are called wavelets and is sometimes called mother wavelet.

Continues Wavelet Transform

Here the dilation and transformation parameters vary continuosly over Real line except at zero. And a function can be reconstructed from its wavelets transform by using resolution identity formula.

Discrete Wavelet Transform and Multiresolution Analysis (MRA)

For very special choices of and particular way to discretize the transform, using i.e. and , we will found that the function we choose will gives us an orthonormal bases. The Haar function shows in the previous applet, is one of the example.
The ilustration below is to give a proof that Haar family constitute an orthonormal bases. This also will gives us an idea about the MRA.
There are two things to prove here. The first is to see that the wavelet function that we got are orthonormal. This one can be easily seen from the applet below, by translated and dilate one of the Haar function over the other. The function with the same scale will never overlaps each other. The function with the smaller scale may be overlap with the bigger scaled function, but the smaller one always in the interval which gives the value of the bigger scaled function has a constant value.

The second one is to see how well a function can be approximated by linear combination of Haar Wavelets.
The approximation function should be a piecewise constant on the same lenght intervals, . The next step, approximate the approximation function, which is piecewise constant on intervals that twice as large as originally. And find the difference between that two approximation function. Infact that difference function is a linear combination of translated and dilated Haar function with the same scale of the first approximation function. Let us see the ilustration below. The first diagram shows the first approximation from the original function.

From the rest diagrams below we can see a recursive approximation of function to a larger interval. The red graphs are the difference between the blue graphs from the two approximation respectively. Notice that this implies that the original function can be approximated by the sum of the red ones which are the linear combination of the Haar fuction.

If we define as a space of a piecewise constant function on intervals with length , then we have introduced a ladder of spaces that has the properties below:

Generally, for any such ladder that has those properties and there exist a scaling function such that is a bases for , then the ladder is an MRA. The MRA also gives that there exist a wavelet function that gives an orthonormal bases for .