Steve Mann, Media Lab, MIT Guest

Posted: Thu Oct 01, 1992 3:27 am Subject: new preprints




new preprints
``The Chirplet" Submitted to IEEE SP special issue on wavelets.
ftp 18.85.0.2
Name (18.85.0.2:loginname): anonymous
ftp> cd pub
ftp> cd steve
ftp> get README
The README file lists ``The Chirplet" as well as some other related
papers, and tells what the filenames are, along with a brief summary
of each.
In short, a ``chirplet" is a piece of a chirp, in the same sense that
a wavelet is, loosely speaking, a piece of a wave. In fact, it was
using this loose definition (which really embodies timefrequencyscale
transforms as opposed to just timescale transforms) that led to the
extension to timefrequencyscaleshear transforms, or ``chirplets".
The ``wavelet of constant shape" (or just the `wavelet', using the
modern definition) is a member of a family of functions formed from
one primitive element  the mother chirplet  whereby each member
represents an AFFINE change of coordinates, imposed on the mother
chirplet. Thus, for simplicity of discussion, let us refer to the
wavelet as the ``physical affinity"  for 1D wavelets, that means
2 degrees of freedom. Affine coordinate changes in the physical domain
are characterized by time dilation (or contraction) ``a" and time
shift (translation) ``b".
The chirplet, is the family of timefrequencyaffinites: the family
is generated from one mother chirplet by affine coordinate changes in
the timefrequency plane. Thus we have translation in time (``b" above)
and translation in frequency, and dilationintime. These form a
TFS space, but there can also be shears and rotations in the TF plane
which lead to chirping when viewed in the physical domain. Just like
putting the data window over a linear FM (chirp) function. Strictly
speaking, the dilationintime and dilationinfrequency impose an
areapreserving (symplectic) geometry, though a philosophical framework
for varying the timebandwidth product is presented, so that the bases
of the underlying group representation may be conceptualized as a
TFaffinity, in contrast to the physical (e.g. time) affinity of the
usual wavelet transform. This form of the chirplet has been successfully
applied to detection of floating iceberg fragments in radar, and has
given very good detection probablities, better than the other more
traditional methods. Others have also built upon this early work on
radar.
The paper also includes some of the more recent extensions to a family
of chirplets of the form g((ax+b)/(cx+1)) where c is the chirpiness.
This generalization of the wavelet is different than the TFaffinities
but the two generalizations are combined in the paper, and explanation
is given as to when subgroup transforms of the former are more appropriate
versus when the latter is the more appropriate. 
