7. Applications
Society for Industrial and Applied Mathematics eBooks · 1997
The wavelet transform has the capability of bringing out certain special features of a function under investigation. For instance, isolated discontinuities of the mth-order derivative of a function can be easily detected by the integral wavelet transform by using an analyzing wavelet with m + 2 vanishing moments. If a function represents an acoustic signal, the wavelet transform separates the signal into different octaves and identifies both time locations and magnitudes of the acoustic notes in each frequency octave band. If the function represents a two-dimensional image, then the wavelet transform brings out the spatial and spectral redundancies of the image, again in each octave band but in each of the horizontal and vertical directions. When a differential or integral equation is considered, the function under investigation is the solution of the equation. The wavelet transform can then be used to identify the components of the solution with least significance, so that ignoring these components greatly accelerates numerical computation of the solution. This chapter is intended to highlight these aspects of the wavelet transform. 7.1. Detection of singularities and feature extraction. It has been pointed out in section 6.5 of Chapter 6 that if a function ƒ(t) has a Taylor expansion at as given by (6.5.1), then its integral wavelet transform (IWT) with analyzing wavelet ψ(t) at with scale is the same as the IWT of the remainder of this Taylor expansion, provided that ψ(t) has m vanishing moments and that the support (i.e., time duration) of is a subset of the interval . Since is usually relatively smaller than ƒ(t) in this interval, we see that the value of ( Wψ ƒ) ( t0 ,a) 7.1.1 is usually negligible. Now let be the radius of the Taylor expansion at . That is, the mth-order derivative of ƒ(t) has a discontinuity at .