# Statistical Idealities and Expected Realities in the Wavelet Techniques Used for Denoising

2010

Thesis

M.A.

Mathematics

## Major Professor

Catherine Bénéteau, Ph.D.

## Committee Member

Arthur A. Danielyan, Ph.D.

## Committee Member

Yuncheng You, Ph.D.

## Keywords

expected value, filters, Fourier series, mean square error, wavelet shrinkage

## Abstract

In the field of signal processing, one of the underlying enemies in obtaining a good quality signal is noise. The most common examples of signals that can be corrupted by noise are images and audio signals. Since the early 1980's, a time when wavelet transformations became a modernly defined tool, statistical techniques have been incorporated into processes that use wavelets with the goal of maximizing signal-to-noise ratios. We provide a brief history of wavelet theory, going back to Alfréd Haar's 1909 dissertation on orthogonal functions, as well as its important relationship to the earlier work of Joseph Fourier (circa 1801), which brought about that famous mathematical transformation, the Fourier series. We demonstrate how wavelet theory can be used to reconstruct an analyzed function, ergo, that it can be used to analyze and reconstruct images and audio signals as well. Then, in order to ground the understanding of the application of wavelets to the science of denoising, we discuss some important concepts from statistics. From all of these, we introduce the subject of wavelet shrinkage, a technique that combines wavelets and statistics into a "thresholding" scheme that effectively reduces noise without doing too much damage to the desired signal. Subsequently, we discuss how the effectiveness of these techniques are measured, both in the ideal sense and in the expected sense. We then look at an illustrative example in the application of one technique. Finally, we analyze this example more generally, in accordance with the underlying theory, and make some conclusions as to when wavelets are an effective technique in increasing a signal-to-noise ratio.

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