Chapter 1 Continuous and discrete reproducing systems that arise from translations . Theory and applications of composite wavelets

Reproducing systems of functions such as the wavelet and Gabor systems have been particularly successful in a variety of applications from both mathematics and engineering. In this chapter, we review a number of recent results in the study of such systems and their generalizations developed by the authors and their collaborators. We first describe the unified theory of reproducing systems. This is a simple and flexible mathematical framework to characterize and analyze wavelets, Gabor systems and other reproducing systems in a unified manner. The systems of interest to us are obtained by applying families of translations, modulations and dilations to a countable set of functions. As the reader will see, we can rewrite such systems as a countable family of translations applied to a countable collection of functions. Building in part on this approach, we define the wavelets with composite dilations, a novel class of reproducing systems which provide truly multidimensional generalizations of traditional wavelets. For example, in dimension two, the elements of such systems are defined not only at various scales and locations, as traditional wavelet systems, but also at various orientations. The shearlet system is a special case of a composite wavelet system which provides optimally sparse representation for a large class of bivariate functions. This is useful for a number of applications in image processing, such as image denoising and edge detection. Finally, we discuss some related issues about the continuous wavelet transform and the continuous analogues of composite wavelets. Demetrio Labate North Carolina State University, Raleigh, NC 27695, USA e-mail: [email protected] Guido Weiss Washington University in St. Louis, St. Louis, MO 63130, USA e-mail: [email protected]