We can see the wavelet transform (WT) as a decomposition of a signal
f(
x) onto a set of basis functions called wavelets to obtain a series expansion of the signal. So far there are two kinds of WT, the first-generation wavelets (
Mallat, 1989;
Rioul and Vetterli, 1991;
Unser and Blu, 2003) and the second-generation wavelets (
Sweldens, 1998). In the first-generation wavelets, the basis functions are obtained from a single mother wavelet by dilations and translations. Then, the signal
f(
x) is directly projected onto the basis functions by taking the inner product between
f(
x) and the functions. If a set of basis
functions is obtained from dilating and translating the mother wavelet, the function becomes spread out in time, then the corresponding projection onto the set of basis functions takes only the coarse resolution structure of
f(
x) into account. This implies that this set of basis functions composes a coarse space. Conversely, if a set of basis functions is obtained from contracting and translating the mother wavelet, the fine structure of
f(
x) will be taken. It means that this set of basis functions composes a fine space.