Wavelets

The word ‘multiscale’ can mean many things. However, in this book we are generally concerned with the representation of objects at a set of scales and then manipulating these representations at several scales simultaneously.

One main aim of this book is to explain the role of wavelet methods in statistics, and so the current chapter is necessarily a rather brief introduction to wavelets. More mathematical (and authoritative) accounts can be found in Daubechies (1992), Meyer (1993b), Chui (1997), Mallat (1998), Burrus et al. (1997), and Walter and Shen (2001). A useful article that charts the history of wavelets is Jawerth and Sweldens (1994). The book by Heil and Walnut (2006) contains many important early papers concerning wavelet theory.

Statisticians also have reason to be proud. Yates (1937) introduced a fast computational algorithm for the (hand) analysis of observations taken in a factorial experiment. In modern times, this algorithm might be called a ‘generalized FFT’, but it is also equivalent to a Haar wavelet packet transform, which we will learn about later in Section 2.11. So statisticians have been ‘doing’ wavelets, and wavelet packets, since at least 1937!