There are many kind of wavelet transform, and I think the names are quite confusing.
For instance, there are continuous and discrete wavelet transforms, in which, the “continuous” and “discrete” are for the wavelet parameters, not for the “data” itself. Therefore, for discrete data, there are “continuous” and “discrete” wavelet transforms, and for function, there are also “continuous” and “discrete” wavelet transforms.
In here, we will focus on discrete wavelet transform for function first. This discrete wavelet transform is also called as wavelet series, which express a compact support function into series of wavelet.
For simplicity, we also focus on orthonormal wavelet.
As the wavelet span the entire space, any compact function can be expressed as
where are integer.
Now, we move to discrete data discrete wavelet transform. The data is discrete, we can imagine only points are known with finite .
the integration becomes a finite sum.
Without loss of generality, we can set , and then the time axis becomes an integer number axis. We found that as the wavelet can only be expand, not shrink. Because there are finite number of data point, i.e. , .
However, this double summation for each is very time consuming. There is a Fast Discrete Wavelet Transform. Before we continuous, we must study the wavelet.
From the last post, we know that the scaling function that generate a MRA must be:
, where are integer. The set of shifted scaling function span a space . For the wavelet,
The set of shifted wavelet span a space , so that , so that
Since the wavelet is generated from the scaling function, we expect the coefficient of and are related. In fact, the relationship for orthonormal scaling function and wavelet is
For discrete data , it can be decomposed into the MRA space. We start by the largest space, where the wavelet is most shrunken.
to decompose to the and space. We can use the nested property of the MRA space, can be decomposed into and ,
where (given that and $\latex \psi(t)$ are orthonormal ),
Therefore, using the coefficient of and , the wavelet coefficient can be decomposed to
in graphic representation
This is a fast discrete wavelet transform.
Due to the nested space of MRA, we also expect that the coefficient and are related to . For orthonormal wavelet,
Since the is finite, the are all finite. That greatly reduce the computation cost of the discrete wavelet transform.
To reconstruct the discrete data , we don’t need to use
using the nested space of MRA, ,
in graphical representation,
I attached the wavelet transfrom class for Qt, feel free to modify.
in the code, the data did not transform to MRA space. The code treats the data already in the MRA space. Some people said this is a “crime”. But for the seek of “speed”, it is no need to map the original discrete data into MRA space. But i agree, for continuous function, we must map to MRA space.