Graduation Year

2010

Document Type

Thesis

Degree

M.A.

Degree Granting Department

Mathematics and Statistics

Major Professor

Catherine Bénéteau, Ph.D.

Keywords

Daubechies, Cascade algorithm, Scaling functions, Multiresolution analysis, Wavelet

Abstract

Wavelets are functions used to approximate data and can be traced back to several different areas, including seismic geology and quantum mechanics. Wavelets are applicable in many areas, including fingerprint and data compression, earthquake prediction, speech discrimination, and human vision. In this paper, we first give a brief history on the origins of wavelet theory. We will then discuss the work of Daubechies, whose construction of continuous, compactly supported scaling functions resulted in an explosion in the study of wavelets in the 1990's. These scaling functions allow for the construction of Daubechies' wavelets. Next, we shall use the algorithm to construct the Daubechies D4 scaling filters associated with the D4 scaling function. We then explore the Cascade Algorithm, which is a process that uses approximations to get possible representations for the D2N scaling function of Daubechies. Lastly, we will use the Cascade Algorithm to get a visual representation of the D4 scaling function.

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