Wavelet Transforms and Multirate Filtering

Abstract

One of the most fascinating developments in the field of multirate signal processing has been the establishment of its link to the discrete wavelet transform. Indeed, it is precisely this link that has been responsible for the rapid application of wavelets in fields such as image compression. The objective of this chapter is to provide an overview of the wavelet transform and develop its link to multirate filtering. The birth of the field of wavelet transforms is now attributed to the seminal paper by Grossman and Morlet (1984) detailing the continuous wavelet transform or CWT. The CWT of a square integrable function is obtained by integrating it over regions defined by translations and dilations of a windowing function called the mother wavelet. The idea of representing functions or signals in terms of dilations can be found even in engineering articles dating back by several years, for example, Helstrom (1966). However, Grossman and Morlet’s formulation was more complete and was motivated by potential application to modeling seismic data. The next step of significance was the discovery of orthogonal wavelet basis functions and their role in defining multi-resolution representations (Daubechies 1988; Meyer 1992). Daubechies also provided a method for constructing compactly supported wavelets. Mallat (1989) established the fact that coefficients of orthogonal wavelet expansions can be obtained through multirate filtering which paved the way for widespread investigation of using wavelet transforms in signal and image processing applications. The objective of the chapter is to provide an overview of the relationship between multirate filtering and wavelet transformation. We begin with a brief account of the CWT, then go through the discrete wavelet transformation (DWT) followed by derivation of the relationship between the DWT and multirate filtering. The chapter concludes with an account of selected applications in digital image processing.