Wavelet Analysis |
The discrete wavelet transform decomposes a function as a sum of basis functions called wavelets. These basis functions have the property that they can be obtained by dilating and translating two basic types of wavelets known as the scaling function, or father wavelet , and the mother wavelet . These translations and dilations are defined as follows:
Conversely, if you know the detail and scaling coefficients at level , then you can obtain the scaling coefficients at level by using the relationship
Suppose that you have data values
at equally spaced points . It turns out that the values are good approximations of the scaling coefficients . Then, by using the recurrence formula, you can find and , . The discrete wavelet transform of the at level consists of the scaling and detail coefficients at level . A technical point that arises is that in applying the recurrence relationships to finite data, a few values of the for or might be needed. One way to cope with this difficulty is to extend the sequence to the left and right by using some specified boundary treatment.
Continuing by replacing the scaling coefficients at any level by the scaling and detail coefficients at level yields a sequence of coefficients
This sequence is the finite discrete wavelet transform of the input data . At any level the finite dimensional approximation of the function is
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