This dissertation develops a new wavelet design technique that produces a wavelet that matches a desired signal in the least squares sense. The Wavelet Transform has become very popular in signal and image processing over the last 6 years because it is a linear transform with an infinite number of possible basis functions that provides localization in both time (space) and frequency (spatial frequency). The Wavelet Transform is very similar to the matched filter problem, where the wavelet acts as a zero mean matched filter. In pattern recognition applications where the output of the Wavelet Transform is to be maximized, it is necessary to use wavelets that are specifically matched to the signal of interest. Most current wavelet design techniques, however, do not design the wavelet directly, but rather, build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a multiresolution analysis that is generated by a scaling function which has a specific corresponding wavelet. In this dissertation, an algorithm for finding both symmetric and asymmetric matched wavelets is developed. It will be shown that under certain conditions, the matched wavelets generate an orthonormal basis of the Hilbert space containing all finite energy signals. The matched orthonormal wavelets give rise to a pair of Quadrature Mirror Filters (QMF) that can be used in the fast Discrete Wavelet Transform. It will also be shown that as the conditions are relaxed, the algorithm produces dyadic wavelets which when used in the Wavelet Transform provides significant redundancy in the transform domain. Finally, this dissertation develops a shift, scale and rotation invariant technique for detecting an object in an image using the Wavelet Radon Transform (WRT) and matched wavelets. The detection algorithm consists of two levels. The first level detects the location, rotation and scale of the object, while the second level detects the fine details in the object. Each step of the wavelet matching algorithm and the object detection algorithm is demonstrated with specific examples.
Library of Congress Subject Headings
Signal detection; Image processing--Digital techniques; Wavelets (Mathematics)
Department, Program, or Center
Chester F. Carlson Center for Imaging Science (COS)
Chapa, Joseph, "Matched wavelet construction and its application to target detection" (1995). Thesis. Rochester Institute of Technology. Accessed from
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