Understanding of the design methods and the properties of wavelet bases in continuous and discrete-time (or space). Understanding of the algorithmic aspects of wavelets and related methods. Introduction to advanced wavelet theory using the fundamental connection with splines. Knowledge of some important applications of these tools (e.g., compression and biomedical imaging).
In recent years, techniques developed in different fields—namely, multiscale bases in applied mathematics, subband coding in digital signal processing, multiresolution techniques in computer vision, and splines in approximation theory—have converged to form a unified theory. Wavelets provide an interesting alternative to Fourier and short-time Fourier transform methods, mainly because of self-similarity properties and the fact that good orthonormal bases do exist. The key concept here is successive approximation in a multiresolution setting: a signal can be seen as a "coarse" version plus added "details". This notion is intuitive and yet very powerful; it also leads to interesting applications. This course presents an overview of filter banks and wavelets, their construction and properties, their relation with splines as well as some generalizations. The point of view that is emphasized is expansion into orthogonal and biorthogonal bases. Applications to image/video compression, and biomedical imaging are discussed.