Abstract

The main goal of the talk is to explain what are wavelets and framelets and why they are useful in applications. A wavelet transform is often used to study a complicated structure/function by breaking it into a combination of simple building blocks. Two of the widely used transforms are the Fourier transform and the wavelet transform. Because many problems/phenomena are multiscale in nature, being the mainstream multiscale transform and linked to the Fourier transform, wavelets and framelets can represent many types of data by effectively capturing their multiscale structure/features with many successful applications in mathematics, engineering, computer sciences and industry. In this talk, we shall provide an elementary tutorial/introduction to wavelets/framelets and the fast wavelet/framelet transforms. Then we shall explain the main desirable features of wavelets and framelets that are of interest in both theory and applications. To illustrate the usefulness of wavelets and framelets, we shall provide several applications of wavelets and framelets including signal/image processing using directional tight framelets, curve and surface generation in computer graphics (used in animation movies), and numerical solutions to differential equations using derivativeorthogonal wavelets.
