What do Some Wavelets Look Like?

Wavelet transforms comprise an infinite set. The different wavelet families make different trade-offs between how compactly the basis functions are localized in space and how smooth they are.

Some of the wavelet bases have fractal structure. The Daubechies wavelet family is one example (see Figure 3).

Fig. 3. The fractal self-similiarity of the Daubechies mother wavelet. This figure was generated using the WaveLab command: wave=MakeWavelet(2, -4, 'Daubechies', 4, 'Mother', 2048). The inset figure was created by zooming into the region x=1200 to 1500.

Within each family of wavelets (such as the Daubechies family) are wavelet subclasses distinguished by the number of coefficients and by the level of iteration. Wavelets are classified within a family most often by the number of vanishing moments. This is an extra set of mathematical relationships for the coefficients that must be satisfied, and is directly related to the number of coefficients (1). For example, within the Coiflet wavelet family are Coiflets with two vanishing moments, and Coiflets with three vanishing moments. In Figure 4, I illustrate several different wavelet families.

Fig. 4. Several different families of wavelets. The number next to the wavelet name represents the number of vanishing moments (A stringent mathematical definition related to the number of wavelet coefficients) for the subclass of wavelet. Note: These figures were created using WaveLab, by typing:

wave = MakeWavelet(2,-4,'Daubechies',6,'Mother', 2048);
wave = MakeWavelet(2,-4,'Coiflet',3,'Mother', 2048);
wave = MakeWavelet(0,0,'Haar',4,'Mother', 512);
wave = MakeWavelet(2,-4,'Symmlet',6,'Mother', 2048);