Wavelet Toolbox

Wavelets and Associated Families

In the one-dimensional context, we distinguish the wavelet from the associated function , called the scaling function. Some properties of and are

• The integral of is zero,
  
  , and is used to define the details.
• The integral of is 1,
  
  , and is used to define the approximations.

The usual two-dimensional wavelets are defined as tensor products of one-dimensional wavelets: is the scaling function and

are the three wavelets.

The following figure shows the four functions associated with the two-dimensional coif2 wavelet.

Figure 6-2: Two-Dimensional coif2 Wavelet

To each of these functions, we associate its doubly indexed family, which is used to

• Move the basic shape from one side to the other, translating it to position b (see the following figure).
• Keep the shape while changing the one-dimensional time scale a () (see Figure 6-4,).

So a wavelet family member has to be thought as a function located at a position b, and having a scale a.

In one-dimensional situations, the family of translated and scaled wavelets associated with is expressed as follows.

Translation	Change of Scale	Translation and Change of Scale
(x-b)

Figure 6-3: Translated Wavelets

Figure 6-4: Time Scaled One-Dimensional Wavelet

In a two-dimensional context, we have the translation by vector and a change of scale of parameter .

Translation and change of scale become

In most cases, we will limit our choice of a and b values by using only the following discrete set (coming back to the one-dimensional context):

Let us define

We now have a hierarchical organization similar to the organization of a decomposition; this is represented in the example of Figure 6-5, Wavelets Organization. Let k = 0 and leave the translations aside for the moment. The functions associated with j = 0, 1, 2, 3 for (expressed as _j,0) and with j = 1, 2, 3 for (expressed as _j,0) are displayed in the following figure for the db3 wavelet.

Figure 6-5: Wavelets Organization

In Figure 6-5, Wavelets Organization, the four-level decomposition is shown, progressing from the top to the bottom. We find _0,0; then 2^1/2_1,0, 2^1/2_1,0; then 2_2,0, 2_2,0; then 2^3/2_3,0, 2^3/2_3,0. The wavelet is db3.

Wavelet Shapes

Wavelet Transforms: Continuous and Discrete