Instructor: Gregory Beylkin
The purpose of this course is to introduce modern multiresolution
techniques in signal processing and numerical analysis.
The ideas behind multiresolution analysis have appeared
independently in different fields of mathematics, electrical
engineering, physics and computer science. In part, these ideas arose
to address limitations of the Fourier transform as the main tool for
analyzing signals and images in applications, and operators in
mathematics and physics. Before the 1980s, many difficulties in these
fields could be traced to a limited variety of orthonormal bases
that were available for applications. The situation changed during the
80s and 90s with the introduction of wavelets and other bases with
controlled localization in the time-frequency domain.
The course addresses several key developments in multiresolution
methods and has three main components.
First, we will discuss various constructions of bases of functional
spaces. We will consider the Fourier integrals and the Fourier series,
the Haar basis, the Malvar-Coifman-Meyer local
cosine bases, multiwavelets, compactly supported
Daubechies' bases, prolate spheroidal wave functions (or
Slepian's
functions) and several other constructions. All these bases have
different properties whose understanding is critical for their
successful application.
Second, we will turn our attention to signal and image processing
and will consider fast algorithms that use multiresolution
constructions. In particular, we will discuss compression of pictures
and sound (and other measured data), and relevant tools and algorithms
such as wavelet packets, "best
basis", etc.
The last part of the course addresses multiresolution techniques
in numerical analysis, sparse representation of operators and the
resulting
fast algorithms. We will consider some aplications (e.g.,
tomographic imaging) as examples.
Although mathematics is central in this course, the emphasis will be
on how to construct and use mathematical tools rather than
on the details of proofs and
derivations. By design this course is accessible to students in
engineering and physics.
Some familiarity (at the lower graduate level ) with elements of the
Fourier Analysis as it is used in PDEs, or Numerical
Analysis, or Signal Processing, or instructor's
consent.