General Concepts

This section presents a brief overview of wavelet concepts, focusing mainly on the orthogonal wavelet case. It includes the following sections:

• Wavelets: A New Tool for Signal Analysis
• Wavelet Decomposition: A Hierarchical Organization
• Finer and Coarser Resolutions
• Wavelet Shapes
• Wavelets and Associated Families
• Wavelet Transforms: Continuous and Discrete
• Local and Global Analysis
• Synthesis: An Inverse Transform
• Details and Approximations

Wavelets: A New Tool for Signal Analysis

Wavelet analysis consists of decomposing a signal or an image into a hierarchical set of approximations and details. The levels in the hierarchy often correspond to those in a dyadic scale.

From the signal analyst's point of view, wavelet analysis is a decomposition of the signal on a family of analyzing signals, which is usually an orthogonal function method. From an algorithmic point of view, wavelet analysis offers a harmonious compromise between decomposition and smoothing techniques.

Wavelet Decomposition: A Hierarchical Organization

Unlike conventional techniques, wavelet decomposition produces a family of hierarchically organized decompositions. The selection of a suitable level for the hierarchy will depend on the signal and experience. Often the level is chosen based on a desired low-pass cutoff frequency.

At each level j, we build the j-level approximation A_j, or approximation at level j, and a deviation signal called the j-level detail D_j, or detail at level j. We can consider the original signal as the approximation at level 0, denoted by A₀. The words approximation and detail are justified by the fact that A₁ is an approximation of A₀ taking into account the low frequencies of A₀, whereas the detail D₁ corresponds to the high frequency correction. Among the figures presented in Reconstructing Approximations and Details, one of them graphically represents this hierarchical decomposition.

One way of understanding this decomposition consists of using an optical comparison. Successive images A₁, A₂, A₃ of a given object are built. We use the same type of photographic devices, but with increasingly poor resolution. The images are successive approximations; one detail is the discrepancy between two successive images. Image A₂ is, therefore, the sum of image A₄ and intermediate details D₄, D₃:

• 

Finer and Coarser Resolutions

The organizing parameter, the scale a, is related to level j by . If we define resolution as 1/a, then the resolution increases as the scale decreases. The greater the resolution, the smaller and finer are the details that can be accessed.

j	10	9	...	2	1	0	-1	-2
Scale	1024	512	...	4	2	1	1/2	1/4
Resolution	1/210	1/29	...	1/4	1/2	1	2	4

From a technical point of view, the size of the revealed details for any j is proportional to the size of the domain in which the wavelet or analyzing function of the variable x,

is not too close to 0.

Wavelet Shapes

One-dimensional analysis is based on one scaling function and one wavelet . Two-dimensional analysis (on a square or rectangular grid) is based on one scaling function and three wavelets.

Figure 6-1 shows and for each wavelet, except the Morlet wavelet and the Mexican hat, for which does not exist. All the functions decay quickly to zero. The Haar wavelet is the only noncontinuous function with three points of discontinuity (0, 0.5, 1). The functions oscillate more than associated functions. coif2 exhibits some angular points; db6 and sym6 are quite smooth. The Morlet and Mexican hat wavelets are symmetrical.

Figure 6-1: Various One-Dimensional Wavelets

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 approximations.

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

are the three wavelets.

Figure 6-2 shows the four functions associated with the 2-D 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 of 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 Transforms: Continuous and Discrete

The wavelet transform of a signal s is the family C(a,b), which depends on two indices a and b. The set to which a and b belong is given below in the table. The studies focus on two transforms:

• Continuous transform
• Discrete transform

From an intuitive point of view, the wavelet decomposition consists of calculating a "resemblance index" between the signal and the wavelet located at position b and of scale a. If the index is large, the resemblance is strong, otherwise it is slight. The indexes C(a,b) are called coefficients.

We define the coefficients in the following table. We have two types of analysis at our disposal.

Continuous Time Signal Continuous Analysis	Continuous Time Signal Discrete Analysis

Next we will illustrate the differences between the two transforms, for the analysis of a fractal signal (see Figure 6-6).

Figure 6-6: Continuous Versus Discrete Transform

Using a redundant representation close to the so-called continuous analysis, instead of a nonredundant discrete time-scale representation, can be useful for analysis purposes. The nonredundant representation is associated with an orthonormal basis, whereas the redundant representation uses much more scale and position values than a basis. For a classical fractal signal, the redundant methods are quite accurate.

• Graphic representation of discrete analysis: (in the middle of Figure 6-6, Continuous Versus Discrete Transform) time is on the abscissa and on the ordinate the scale a is dyadic: 2¹, 2², 2³, 2⁴, and 2⁵ (from the bottom to the top), levels are 1, 2, 3, 4, and 5. Each coefficient of level k is repeated 2^k times.
• Graphic representation of continuous analysis: (at the bottom of Figure 6-6, Continuous Versus Discrete Transform) time is on the abscissa and on the ordinate the scale varies almost continuously between 2¹ and 2⁵ by step 1 (from the bottom to the top). Keep in mind that when a scale is small, only small details are analyzed, as in a geographical map.

Local and Global Analysis

A small scale value permits us to perform a local analysis; a large scale value is used for a global analysis. Combining local and global is a useful feature of the method. Let us be a bit more precise about the local part and glance at the frequency domain counterpart.

Imagine that the analyzing function or is zero outside of a domain U, which is contained in a disk of radius : . The wavelet is localized. The signal s and the function are then compared in the disk, taking into account only the t values in the disk. The signal values, which are located outside of the domain U, do not influence the value of the coefficient

• 

The same argument holds when is translated to position b and the corresponding coefficient analyzes s around b. So this analysis is local.

The wavelets having a compact support are used in local analysis. This is the case for Haar and Daubechies wavelets, for example. The wavelets whose values are considered as very small outside a domain U can be used with caution, as if they were in fact actually zero outside U. Not every wavelet has a compact support. This is the case, for instance, of the Meyer wavelet.

The previous localization is temporal, and is useful in analyzing a temporal signal (or spatial signal if analyzing an image). The good spectral domain localization is a second type of a useful property. A result (linked to the Heisenberg uncertainty principle) links the dispersion of the signal f and the dispersion of its Fourier transform , and therefore of the dispersion of and . The product of these dispersions is always greater than a constant c (which does not depend on the signal, but only on the dimension of the space). So it is impossible to reduce arbitrarily both time and frequency localization.

In the Fourier and spectral analysis, the basic function is . This function is not a time localized function. The support is R. Its Fourier transform is a generalized function concentrated at point .

The function f is very poorly localized in time, but is perfectly localized in frequency. The wavelets generate an interesting "compromise" on the supports, and this compromise differs from that of complex exponentials, sine, or cosine.

Synthesis: An Inverse Transform

In order to be efficient and useful, a method designed for analysis also has to be able to perform synthesis. The wavelet method achieves this.

The analysis starts from s and results in the coefficients C(a,b). The synthesis starts from the coefficients C(a,b) and reconstructs s. Synthesis is the reciprocal operation of analysis.

For signals of finite energy, there are two formulas to perform the inverse wavelet transform:

• Continuous synthesis:

• 

• where is a constant depending on .

• Discrete synthesis:

• 

Of course, the previous formulas need some hypotheses on the function. More precisely, see What Functions Are Candidates to Be a Wavelet? for the continuous synthesis formula and Why Does Such an Algorithm Exist? for the discrete one.

Details and Approximations

The equations for continuous and discrete synthesis are of considerable interest and can be read in order to define the detail at level j:

1. Let us fix j and sum on k. A detail is nothing more than the function

• 

2. Now, let us sum on j. The signal is the sum of all the details:

• 

The details have just been defined. Take a reference level called J. There are two sorts of details. Those associated with indices correspond to the scales which are the fine details. The others, which correspond to j > J, are the coarser details.

We group these latter details into

• 

which defines what is called an approximation of the signal s. We have just created the details and an approximation. They are connected. The equality

• 

signifies that s is the sum of its approximation A_J and of its fine details. From the previous formula, it is obvious that the approximations are related to one another by

• 

For an orthogonal analysis, in which the _j,k is an orthonormal family,

• A_J is orthogonal to D_J, D_J-1, D_J-2, ...

• s is the sum of the two orthogonal signals: A_J and
  
  

• 

• A_J is an approximation of s. The quality (in energy) of the approximation of s
  by A_J is

• 

• 

The following table contains definitions of details and approximations.

Definition of the detail at level j
The signal is the sum of its details
The approximation at level J
Link between AJ-1 and AJ	AJ-1 = AJ + DJ
Several decompositions

From a graphical point of view, when analyzing a signal, it is always valuable to represent the different signals (s, A_j, D_j) and coefficients (C(j,k)).

Let us consider Figure 6-7. On the left side, s is the signal; a5, a4, a3, a2, and a1 are the approximations at levels 5, 4, 3, 2, and 1. The best approximation is a1; the next one is a2, and so on. Noise oscillations are exhibited in a1, whereas a5 is smoother.

On the right side, cfs represents the coefficients (for more information, see Wavelet Transforms: Continuous and Discrete), s is the signal, and d5, d4, d3, d2, and d1 are the details at levels 5, 4, 3, 2, and 1.

The different signals that are presented exist in the same time grid. We can consider that the t index of detail D₄(t) identifies the same temporal instant as that of the approximation A₅(t) and that of the signal s(t). This identity is of considerable practical interest in understanding the composition of the signal, even if the wavelet sometimes introduces dephasing.

Figure 6-7: Approximations, Details, and Coefficients

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Fast Wavelet Transform (FWT) Algorithm

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