A Mathematical Analysis of Wavelet Theory and Applications to Sound Filtering, 2-Dimensional Image Compression, and Continuous Feed Image Compression

Abstract

In today’s world of information technology there is a heavy emphasis upon the efficiency and speed of computers. With this new age, an abundance of new methods have emerged attempting to save time and computer space by compressing both 2- dimensional pictures and continuous feed multi-media, such as movies and music. Recently, wavelet analysis has emerged as the best tool to perform these tasks. In the world of mathematics, wavelet theory has also become a hotbed for both cutting edge research and applications to real-world situations. My paper begins by exploring the mathematics behind wavelet analysis and the connections wavelets have with vector spaces. An exploration of the applications of wavelets to different forms of real-world phenomena follows. I first utilize a program written in Maple to show how wavelets can be applied to the fields of sound filtering and signal compression. Mathematica is then used to illustrate wavelet applications to the compression of 2-dimensional images. An analysis of different compression ratios further demonstrates the effectiveness of the wavelet compression. Finally, a variation of the Mathematica computer program is utilized to perform compression of a continuous feed movie. An analysis to determine the optimal length of segments extracted from a continuous feed signal to send through the wavelet algorithm is then explored.