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Local Feature Extraction and Its Applications Using a Library of Bases, Ph.D.Thesis (Advisor: Prof. Ronald R. Coifman), Department of Mathematics, Yale University, Dec. 1994.

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Abstract

Extracting relevant features from signals is important for
signal analysis such as compression, noise removal, classification, or
regression (prediction).
Often, important features for these problems, such as edges, spikes, or
transients, are characterized by local information in the time (space) domain
and the frequency (wave number) domain.

The conventional techniques are not efficient to extract features
localized simultaneously in the time and frequency domains.
These methods include:
the Fourier transform for signal/noise separation,
the Karhunen-Loeve transform for compression,
and the linear discriminant analysis for classification.
The features extracted by these methods are of global nature
either in time or in frequency domain so that the interpretation
of the results may not be straightforward.
Moreover, some of them require solving the eigenvalue systems
so that they are fragile to outliers or perturbations
and are computationally expensive, i.e., *O(n^3)*, where *n* is a
dimensionality of a signal.

The approach explored here is guided by the *best-basis paradigm* which
consists of three steps: 1) select a "best" basis (or coordinate
system) for the problem at hand from a *library of bases* (a fixed yet
flexible set of bases consisting of wavelets, wavelet packets,
local trigonometric bases, and the autocorrelation functions of wavelets),
2) sort the coordinates (features) by "importance" for the problem
at hand and discard "unimportant" coordinates, and 3) use these
survived coordinates to solve the problem at hand.
What is "best" and "important" clearly depends on the problem:
for example, minimizing a description length (or entropy) is
important for signal compression whereas maximizing class separation (or
relative entropy among classes) is important for classification.

These bases "fill the gap" between the standard Euclidean
basis and the Fourier basis so that they can capture the local features
and provide an array of tools unifying the conventional techniques.
Moreover, these tools provide efficient numerical algorithms, e.g.,
*O(n [log n]^p)*, where *p=0,1,2*, depending on the basis.

In present thesis, these methods have been applied usefully to a variety of
problems:
simultaneous noise suppression and signal compression,
classification, regression, multiscale edge detection and representation,
and extraction of geological information from acoustic waveforms.

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