chapter  1
34 Pages

The Wavelet Transform: A Surfing Guide

WithAkram Aldroubi

This chapter presents the various wavelet transforms and their connections to each other. It introduces the continuous wavelet transform in one and several variables. The chapter reviews the dyadic wavelet transform (DWT) in more detail and introduces the redundant discrete wavelet transforms. It describes the multiresolution approximations (MRA) of L2 and their related MRA-type wavelets. In particular, the chapter examines the orthogonal, biorthogonal and semi-orthogonal wavelet bases. It is devoted to the design of wavelet bases, and presents specific constructions of some useful wavelet bases. The chapter provides an overview of the applications of the wavelet transform and some generalizations. Finally, it is also devoted to the general theory of frames. A more advance in the theory of wavelets, the lifting scheme, allows the construction of wavelets on curves and surfaces, and their custom-design to suit needs of specific applications. Multiresolutions and wavelets have been generalized to frame MRAs and frame wavelets.