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.


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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