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Session 3265

Wavelet Transforms on The Letter N 1

John Schmeelk Department of Mathematical Sciences Virginia Commonwealth University College of Design Arts Doha, Qatar

Abstract

This paper studies the two-dimensional wavelet transform applied to two-dimensional images. The classical technique oftentimes implements the Fourier transform. This paper offers a brief discussion regarding the comparison of the two transforms on a single alphabet, N. It provides a comparison of the global properties present in the Fourier transform technique verses a more localized analysis when the wavelet transform is applied to the same image. The wavelet selected in this study is the derivative of the Gaussian since in some sense offers a nice comparison to the Fourier method.

I. Introduction

The applications of wavelet and wavelet transforms to discrete data are so plentiful that they have emerged as the most promising techniques in the past decade. For example the current research by the Federal Bureau of Investigation (FBI) in establishing an appropriate wavelet transform to be applied to its 30 million criminal fingerprints now stored in filing cabinets illustrates the application importance. The advantage will be to compress the data and accelerate the matching techniques. These topics are discussed in Strange44.

Our present implementation of the wavelet transform will be to apply it to a two dimensional image and to be able to extract critical and pertinent information. The literature on wavelet transforms in the one-dimensional case is very extensive. This is due in part to the fact that a signal captured from a piece of hardware can in many situations be obtained in a one-dimensional fashion. Images by their very nature require two or three dimensions and the literature is somewhat less available. However some research has been conducted in the multivariable cane and can be found in references 9,14,26,42 . We will use these developments extensively in our investigations whereby the transform will be implemented on an alphabet and its reflection in the following sequel. The overall procedures will entail a detailed analysis of a two-dimensional “mother” wavelet implemented within a wavelet transform on the alphabet, N, together with a comparison to a Fourier transform. All graphics presented in this paper have been conducted on a MATLAB platform. A preliminary mathematical review is provided to reacquaint the reader with the mathematical analysis of wavelet theory in both the one and two-dimensional case.

1 Funded by the Qatar Foundation, Doha, Qatar

“Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition Copyright © 2003, American Society for Engineering Education”

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Schmeelk, J. (2003, June), Wavelet Transforms On The Letter N Paper presented at 2003 Annual Conference, Nashville, Tennessee. 10.18260/1-2--12382