30-11-2012, 03:09 PM
Application of Wavelet Transform and its Advantages Compared to Fourier Transform
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ABSTRACT
Wavelet analysis is an exciting new method for solving difficult problems in
mathematics, physics, and engineering, with modern applications as diverse as wave
propagation, data compression, signal processing, image processing, pattern
recognition, computer graphics, the detection of aircraft and submarines and other
medical image technology. Wavelets allow complex information such as music,
speech, images and patterns to be decomposed into elementary forms at different
positions and scales and subsequently reconstructed with high precision. Signal
transmission is based on transmission of a series of numbers. The series representation
of a function is important in all types of signal transmission. The wavelet representation
of a function is a new technique. Wavelet transform of a function is the improved
version of Fourier transform. Fourier transform is a powerful tool for analyzing the
components of a stationary signal. But it is failed for analyzing the non stationary
signal where as wavelet transform allows the components of a non-stationary signal
to be analyzed. In this paper, our main goal is to find out the advantages of wavelet
transform compared to Fourier transform.
Introduction
In 1982 Jean Morlet a French geophysicist, introduced the concept of a `wavelet'.
The wavelet means small wave and the study of wavelet transform is a new tool
for seismic signal analysis. Immediately, Alex Grossmann theoretical physicists
studied inverse formula for the wavelet transform. The joint collaboration of Morlet
and Grossmann [5] yielded a detailed mathematical study of the continuous
wavelet transforms and their various applications, of course without the
realization that similar results had already been obtained in 1950's by Calderon,
Littlewood, Paley and Franklin. However, the rediscovery of the old concepts
provided a new method for decomposing a function or a signal. For details one
can see Morlet et al. [8], Debnath [4].
Storing Fingerprint Electronically Using Wavelet
We now discuss how the FBI (Federal Bureau of Investigation) in the USA uses
wavelets as a tool to store fingerprints electronically.
For many years, the FBI stored their fingerprints in paper format in a highly secured
building in Washington; they filled an area which had the same size as football field.
If one needed to compare a fingerprint in San Francisco with the stored fingerprints,
one had to mail it to Washington. Furthermore, comparison of the fingerprints was
done manually, so it was a quite slow process. For these reasons, the FBI started to
search for ways to store the fingerprints electronically; this would facilitate
transmission of the information and the search in the archive.
We can consider a fingerprint as a small picture, so a natural idea is to split
each square-inch into, say, 256× 256 pixels, to which we associate a grey-tone on a
scale from for example 0 (completely white) to 256 (completely black). This way we
have kept the essential information in the form of a sequence of pairs of numbers,
namely, the pairs consisting of a numbering of the pixels and the associated greytone.
Fingerprint Verification
Fingerprint verification is one of the most reliable personal identification methods
and it plays a very important role in forensic and civilian applications. However,
manual fingerprint verification is so tedious, time-consuming and expensive in that
it is incapable of meeting today's increasing performance requirements. Hence, an
automatic fingerprint identification system (AFIS) is widely needed. Here we
mentioned one real example of Fingerprint verification:
In Singapore, a new security system was introduced in Hitachi Tower (a 37-
storey office building) in 2003 : now, the 1500 employees get access to the building
by scanning their fingers. The scanner uses infrared rays to trace the hemoglobin in
blood in order to capture the vein patterns in the finger; these patterns determine the
person uniquely. After comparing with the scanned data in an electronic archive, it is
decided whether the person can get in or not Christensen [1], Meyer [7].
Comparison Wavelet Transform with Fourier Transform
The wavelet transform is often compared with the Fourier transform. Fourier
transform is a powerful tool for analyzing the components of a stationary signal (a
stationary signal is a signal where there is no change in the properties of signal). For
example, the Fourier transform is a powerful tool for processing signals that are
composed of some combination of sine and cosine signals (sinusoids) Mallat [6].
The Fourier transform is less useful in analyzing non-stationary signal (a
non-stationary signal is a signal where there is change in the properties of signal).
Wavelet transforms allow the components of a non-stationary signal to be analyzed.