Wavelet Decomposition of a Function

Thumbnail Image
Schauf, Michael
Date of Issue
Subject Keywords
wavelets, Fourier series, Haar wavelet, Mexican Hat wavelet
Series/Report No.
Wavelet Decomposition of a Function
Other Titles
Computers today play a major role in practically every aspect oflife, especially in the area of communications and signal processing. From e-mail and faxes to grocery market scanners and televisions, computers are relied upon to transfer and interpret almost every type information. One such example is the transfer of fingerprints by the FBI (Brislawn 1278). Today the FBI has some 200 million fingerprint records, stored in the form of inked impressions on paper cards. As part of a modernizing process, the FBI began to digitize the records as 8-bit grayscale images. But fingerprints are so detailed that some 10 megabytes per card are required, making the current archive about 2000 terabytes in size. (Remember a 3.5" floppy disk holds only 1.5 megabytes). In addition to the current cards, the FBI also receives around 30,000 new cards per day. Therefore the FBI has joined the search with others to find a more simplified way of storing and transmitting complicated data. The use of wavelets to encode the digitized fingerprints requires significantly less stored data. Since communication is often times composed of complex phenomena, engineers seek processes that will ease the transfer of information. They do this most commonly by decomposing the complex functions into simpler ones. This process originated in the early 1800s with Joseph Fourier (Haberman 75). Fourier found that complex phenomena could be represented as a sum of sines and cosines of various frequencies and amplitudes. Even though the trigonometric sines and cosines represent a decent basis and are well understood, they do have their drawbacks. Oftentimes a Fourier series requires a large number of coefficients to be stored, along with the fact that it has troubles representing functions that have discontinuities. The difficulty arises from attempting to use smooth sine or cosine functions to model a discontinuous function. In the past couple of decades a new method has been discovered by Jean Morlet and Alexander Grossman that further simplifies complex data (Graps 4). It is known as wavelet theory, and it uses the idea of a simplified orthonormal basis to represent complex functions. These basis functions are obtained by dilating and translating a particular starting function known as the "mother wavelet." Therefore they have the ability to focus in on specified parts of complex data. Like the Fourier series, wavelets can then be combined in a linear combination to represent the original complex data. Two such examples of "mother wavelets" are the Haar wavelet and the Mexican Hat wavelet. In this paper I provide a description of the two wavelets. To show how they are used, I have written programs in Mathematica to break down a function into its sum of Haar or Mexican Hat wavelets. I then compare the wavelet sums with the original function to show the strengths and weaknesses of each approach. The goal was to see whether particular families of wavelets can accurately approximate localized or discontinuous functions by using significantly fewer coefficients than a Fourier series would require. In this way, I hoped to test the claims that wavelets offer a more economical method for encoding such functions.
Degree Awarded
Mathematics, Engineering & Computer Science