Introduction to Wavelets and Their Applications

Wavelets are a powerful mathematical tool, used in recent years in many applications. For example, wavelets play a fundamental role in defining the JPEG-2000 and MPEG-4 standards.

Let’s see what they are and how they’re used in applications.

Wavelets in a Nutshell

A wavelet is a real function $f: \mathbb{R} \rightarrow \mathbb {R}$ which represents a wavelike oscillation localized in a limited range of its domain.

Here are some examples:

Given a mother wavelet $f$ we can define a set of child wavelets $f_ {a, b}$ through the parameters $(a, b)$

$$\displaystyle f_ {a, b } = \frac {1} {\sqrt {a}} f \left (\frac {x-b} {a} \right) \qquad \text{with } a,b \in \mathbb{R}\text{ and } a>0$$

Parameter $a$ scales the function while $b$ shifts it. In applications it is common to take into account a discrete set of pairs $(a, b)$ so you can index child functions with discrete parameters $f_ {i, j}$ with $i, j \in \mathbb{N}$.

The general idea behind wavelets is that a function $g(x)$ can be represented as a linear combination of child wavelets:

$$\displaystyle g(x) = \sum_{i, j} A_ {i, j} \cdot f(x) _ {i, j}$$

Compression of Signals

The function $g(x)$ could be, for example, the sound of a musical instrument or the signal of a seismograph or electrocardiogram.

At first, the signal is registered sampling at a certain frequency. In practice for each sampling interval the value of the function $g(x)$ is recorded. If the sampling frequency is high, a signal stored in such a way can occupy a lot of memory.

Through wavelets it is possible to store the signal using only the values of the main coefficients $A_{i,j}$ of the wavelet expansion.

The truncation of the wavelet series results in a little loss of precision in representing the $g(x)$ function, but it also results in a huge saving in the amount of information to be stored, also called compression.

The JPEG-2000 and MPEG-4 Standards

In the JPEG-2000 and MPEG-4 standards, images and videos are represented through a wavelet expansion. In addition to data compression, the main advantage of using wavelets in this field is to manage different resolutions of the image with a single file.

Once an image is saved as a wavelet expansion, if you want to create a low-resolution preview of the same image, it is sufficient to use fewer elements of the summation.

Different image resolutions are obtained by simply truncating the wavelet series at different depths.

Wavelets vs Fourier

Experienced readers will have noticed the similarity between the wavelet decomposition and discrete Fourier transforms.

The Fourier transforms have many properties that make them interesting from a theoretical point of view. However, wavelets have some significant advantages in applied mathematics.

1) Personalization: Fourier transforms always make use of sine and cosine functions. On the other hand, depending on the particular application, you can choose the wavelets that better adapt to deal with that problem.

2) Localization: signals that are analyzed in applications often consist of several blocks of information separated by intervals of near-zero signal (for example, in the case of the electrocardiogram). As a consequence, it’s more natural to decompose this kind of signal through wavelets that represent localized waves.

3) More control over Gibbs phenomenon: Fourier transforms present some problems in describing discontinuous signals. I’m referring to the so-called Gibbs phenomenon.

The classic example is that of a square wave that alternately takes the values 0 and 1. The discrete Fourier expansion of this signal presents a peak near the discontinuity with a value of about 1.09.

The left image shows the approximation using 25 harmonics while the right image using 125 harmonics. The height of the peak remains stable, even increasing the terms of the Fourier series!

This is somewhat counterintuitive, because you would expect the series to converge to the function and so to the value 1.

Also, the wavelet expansion exhibits this kind of phenomenon, but to a lesser extent compared to the discrete Fourier transform.

Some Geometrical Considerations for More Advanced Readers

Another difference between wavelets and Fourier transform is geometric. The sine and cosine functions used in Fourier series form a basis of the space of functions $\mathrm{L} ^ 2 ([0,1])$.

This means they are linearly independent vectors that span the whole space of functions.

Often, wavelets used in applications are frames rather than bases. A frame is a set of vectors that span the vector space but that are not linearly independent.

As a consequence we have that the decomposition of a vector in terms of wavelets is not unique. This feature, which might seem to be a problem, represents instead a further computational advantage, contributing to an improved numerical stability of wavelets with respect to Fourier transform.