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

$\displaystyle f(t) = \sum_{j,k} \left \psi_{j,k}(t)$

$\psi_{j,k}(t) = 2^{j/2} \psi(2^j t - k)$

where $j, k$ are integer.

Now, we move to discrete data discrete wavelet transform. The data is discrete, we can imagine only $t_n = t_0 + n \Delta$ points are known with finite $n$.

$\displaystyle f_n = f(t_n) = \sum_{j,k} \left \psi_{j,k}(t_n)$

the integration becomes a finite sum.

Without loss of generality, we can set $t_0 = 0, \Delta = 1$, and then the time axis becomes an integer number axis. We found that $j \leq 0$ as the wavelet can only be expand, not shrink. Because there are finite number of data point, i.e. $n < \infty$, $-Log_2(n) < j \leq 0$.

However, this double summation for each $f_n$ 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:

$\displaystyle \phi(t) = \sum_{k} g_0(k) \phi(2t-k)$

$\left<\phi(t-k) | \phi(t-k') \right> = \delta_{kk'}$

, where $k$ are integer. The set of shifted scaling function span a space $V_0$. For the wavelet,

$\displaystyle \psi(t) = \sum_{k} g_1(k) \psi(2t-k)$

$\left<\psi(t-k) | \psi(t-k') \right> = \delta_{kk'}$

The set of shifted wavelet span a space $W_0$, so that $W_0 \perp V_0$, so that

$\left<\phi(t-k)|\psi(t-k') \right> = 0$

Since the wavelet is generated from the scaling function, we expect the coefficient of $g_0(k)$ and $g_1(k)$ are related. In fact, the relationship for orthonormal scaling function and wavelet is

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

For discrete data $x_i$, it can be decomposed into the MRA space. We start by the largest $V_0$ space, where the wavelet is most shrunken.

$\displaystyle x_i = \sum_{k} v_{0,k} \phi(i-k)$

to decompose to the $V_{-1}$ and $W_{-1}$ space. We can use the nested property of the MRA space, $\phi(2t)$ can be decomposed into $\phi(t-k)$ and $\psi(t-k)$,

$\displaystyle \psi(2t-l) = \sum_{k} h_0(2k-l) \phi(t-k) + h_1(2k-l) \psi(t-k)$

where (given that $\phi(t)$ and $\latex \psi(t)$ are orthonormal ),

$h_0(2k-l) = \left< \phi(2t-l) | \phi(t-k) \right>$

$h_1(2k-l) = \left< \phi(2t-l) | \psi(t-k) \right>$

Therefore, using the coefficient of $h_0$ and $h_1$, the wavelet coefficient $v_{0,k}$ can be decomposed to

$\displaystyle v_{s-1,k} = \sum_{l} h_0(2k-l) v_{s,l}$

$\displaystyle w_{s-1,k} = \sum_{l} h_1(2k-l) v_{s,l}$

in graphic representation

This is a fast discrete wavelet transform.

Due to the nested space of MRA, we also expect that the coefficient $h_0$ and $h_1$ are related to $g_0$. For orthonormal wavelet,

$\displaystyle h_0(k) = \frac{1}{2} g_0(-k)$

$\displaystyle h_1(k) = \frac{1}{2} (-1)^{k} g_0 (k+1)$

Since the $g_0$ is finite, the $g_1, h_0, h_1$ are all finite. That greatly reduce the computation cost of the discrete wavelet transform.

To reconstruct the discrete data $x_i$, we don’t need to use

$\displaystyle v_{s+1,l} = \sum_{k} v_{s,k} \phi(l - k) + w_{s,k} \psi(l-k)$

using the nested space of MRA, $\psi(t) = \sum_{k} g_1(k) \psi(2t-k)$,

$\displaystyle v_{s+1,l} = \sum_{k} g_0(l-2k) v_{s,k} + g_1(l-2k) w_{s,k}$

in graphical representation,

I attached the wavelet transfrom class for Qt, feel free to modify.