第四章 1BasicsofWavelets.PDF

ISyE8843A, Brani Vidakovic Handout 20 1 Basics of Wavelets The first theoretical results in wavelets are connected with continuous wavelet decompositions of func- tions and go back to the early 1980s. Papers of Morlet et al. (1982) and Grossmann and Morlet (1985) were among the first on this subject. Let be a family of functions defined as translations and re-scales of a single function (1) Normalization by ensures that is independent of and The function (called the wavelet function or the mother wavelet) is assumed to satisfy the admissibility condition, (2) where is the Fourier transformation of The admissibility condition (2) implies Also, if and for some then Wavelet functions are usually normalized to “have unit energy”, i.e., For any function , the continuous wavelet transformation is defined as a function of two variables Here the dilation and translation parameters, and , respectively, vary continuously over Resolution of Identity. When the admissibility condition is satisfied, i.e., it is possible to find the inverse continuous transformation via the relation known as resolution of identity or Calderon’s reproducing ´ identity, If is restricted to wh