Wavelets for feature detection : theoretical background

TLDR: Wavelet theory is a relatively new tool for signal analysis and is still one of the most important tool for signals analysis and Fourier analysis plays an important role in wavelet analysis.

Abstract: Wavelet theory is a relatively new tool for signal analysis. Although the rst wavelet was derived by Haar in 1909, the real breakthrough came in 1988 when Daubechies derived her famous wavelet design. Since then a lot of wavelets have been designed for many applications. A wavelet is a function that goes to zero at the bounds and between these bounds it behaves like a wave. The word wavelet originates from a combination of wave and the French word for small wave, ondelette. This small wave is convoluted with a signal. This expresses the amount of the overlap of the wavelet as it is shifted over the signal. In other words, where the signal resembles the wavelet, the resulting function will have high magnitude and where it has a totally dierent shape it will have low magnitude. How well the wavelet resembles the signal locally can be calculated by shifting the small wave over the entire signal. By not only comparing the signal with shifted wavelets but also comparing wavelets that are dierently dilated, something can be said about the scale (frequency) content of the signal. There are many dierent wavelets with a verity of shapes and properties. Therefore it is a much broader tool for signal analysis compared to the Fourier Transformation where the signal is only compared to sinusoids. However, Fourier analysis plays an important role in wavelet analysis and is still one of the most important tool for signal analysis. Therefore in Chapter 3 a short introduction is given about signal analysis in general and about the decomposition of signals. The Fourier Transformation and the Short Time Fourier Transformation are introduced as two possible analyzing methods. Thereafter, in Chapter 4, the Continuous Wavelet Transformation is introduced and two examples are presented. The Continuous Wavelet Transformation is in general considered redundant because it uses continuous signals and therefore needs to be made discrete before it can be used in an application. This makes the Continuous Wavelet Transform inecient. This can be overcome by using the Discrete Wavelet transform. It is very efficient if it is applied through a lter bank, which is an important part of the Discrete Wavelet Transform. The Discrete Wavelet Transform is discussed in Chapter 5. In Chapter 6 its most important properties are explained. In addition a number of issues related with the DWT are discussed. Finally the most important applications are explained in Chapter 7, whereafter in Chapter 8 some conclusions are presented.