Instructors: Dave Larson with the help of Ken Dykema, Keri Kornelson and Eric Weber.
This VIGRE seminar was offered in conjunction with our NSF-funded
Summer
REU program .
To motivate wavelets, we begin with a few real-world examples:
Filtering.
A sound signal is often corrupted by noise (i.e., frequencies
different from those in the desirable parts of the signal). Signal
analysis can be used to filter out this unwanted noise. A Dolby
filter, which filters out tape-hiss on cassette tapes, is an example
along these lines.
Data Compression.
Digitized audio and video signals are usually quite large, and are
difficult to transmit electronically. Efficient transmission of these
signals often requires compression, a process that eliminates the less
significant parts of a signal. Compression is used, for example, in
transmitting fingerprints from a police squad car to FBI Headquarters
(in Washington DC) to identify crime suspects.
Detection.
Signals often have some feature that the user wants to detect. For
example, the sound made by a mechanical device often changes when it
does not operate correctly. A device that detects this change would be
useful to the machine operator.
Fourier analysis and wavelets are two of the basic tools used, in
signal analysis, to address the above issues.
Fourier Analysis.
A Fourier series decomposes a signal f into its trignonometric
components, which vibrate at various frequencies.
The Fast Fourier transform
(FFT) is an efficient algorithm for calculating approximate values for
the Fourier coefficients.
The Fourier coefficients can then be
manipulated according to the desired goal. If noise is to be
filtered-out, then the Fourier coefficients corresponding to the
unwanted frequencies can be eliminated. If the signal is to be
compressed, then the Fourier coefficients that are smaller (in
absolute value) than some specified tolerance can be discarded.
Problems in detection can be addressed by matching a subset of the
Fourier coefficients of f to a known profile of the type of
signal to be detected.
Wavelets.
One disadvantage of Fourier series is that the building blocks, sines
and cosines, are periodic waves that continue forever. While this
approach may be quite appropriate for filtering or compressing signals
that have time-independent wave-like features, other signals may have
more localized features that sines and cosines do not model very well.
For example, suppose an isolated noisy ``pop'' to a sound signal is to
be filtered-out. The graphs of sines and cosines do not resemble the
pop's graph, an isolated bump. A different set of building blocks,
called wavelets, are better suited to this type of signal. In
a rough sense, a wavelet resembles a wave that travels for one or more
periods and is nonzero only over a finite interval -- instead of
propagating forever as do sines and cosines.
A wavelet can be translated forward or backwards in time. It also can
be stretched or compressed, by scaling, to obtain low and high
frequency wavelets. Once a wavelet function is constructed, it can be
used to filter or compress signals in much the same manner as Fourier
series. A given signal is first expressed as a sum of translations and
scalings of the wavelet, and then the coefficients corresponding to
the unwanted terms are removed or modified.
Care must be taken in the construction of a wavelet to ensure that its
translates and rescalings satisfy orthogonality relationships --
analogous to those of sines and cosines -- so that efficient
algorithms can be found for the computation of wavelet coefficients
of a given signal.
Focus of this Program. There is a fascinating interplay between wavelets and operator theory (i.e. the theory of linear maps between vector spaces). This interplay will be the key topic of investigation during this summer program. We shall first investigate the role of wandering vectors for unitary systems, both in finite and infinite dimensions. Related topics include introductory ideas in operator algebras, again both in finite and infinite dimensions. In addition, we shall study minimally supported frequency wavelets. This special class of wavelets has an interesting internal structure, but also provides concrete examples of important ideas that will be discussed. Finally, we will investigate several interesting intrinsic problems dealing with wavelet sets.
Proposed Research Problems: