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5. F. T. Chau and A. K. M. Leung, “Application of wavelet transform in processing chromatographic data,'' in Wavelets in Chemistry, B. Walczak, ed., Elsevier Science, Amsterdam, 2000, pp. 205-223.
6. F. T. Chau and A. K. M. Leung, “Application of wavelet transform in spectroscopic studies,'' in Wavelets in Chemistry, B. Walczak, ed., Elsevier Science, Amsterdam, 2000, pp. 241-261.
7. C. K. Chui, An Introduction to Wavelets, Academic Press, Boston, 1992.
8. A. Cohen, I. Daubechies, and J. C. Feauveau, ‘‘Biorthogonal bases of compactly supported wavelets,'' Commun. PureAppl. Math. 45:485-560 (1992).
9. I. Daubechies, ‘‘Orthonormal bases of compactly supported wavelets,'' Commun. PureAppl. Math. 41:909-996 (1988).
fundamentals of wavelet transform
10. I. Daubechies, Ten Lectures on Wavelets, Vol. 61 of CBMS-NSF Reg. Conf. Ser. Appl. Math., SIAM Press, Philadelphia, 1992.
11. J. B. Gao, F. T Chau, T. M. Shih, and C. K. Chan, “Application of the fast wavelet transform method to compress ultraviolet-visible spectra,'' Appl. Spectrosc. 50:339-349 (1996).
12. T M. Shih, F. T Chau, J. B. Gao, and J. Wang, “Compression of infrared spectral data using the fast wavelet transform method,'' Appl. Spectrosc. 51:649-659 (1997).
13. F. T. Chau, J. B. Gao, and T. M. Shih, “Application of the fast wavelet transform method to compress ultraviolet-visible spectra,'' Appl. Spectrosc. 20:85-90 (1996).
14. S. Mallat, “Multiresolution approximations and wavelet orthogonormal bases of L2(R),” Trans. Amer, Math. Soc., 315:69-87 (1989).
15. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.
16. Y Meyer, ‘‘Principe d'incertitude, bases hilbertiennes et algebres d'operateurs,'' in Seminaire Bourbaki, Vol. 662, Paris, 1986.
APPLICATION OF WAVELET TRANSFORM IN CHEMISTRY
The word ‘‘transform’’ is a mathematical term, but it is frequently used in chemical signal processing. This is because a suitable transform could make difficult calculations become easier and complex signals simpler. Traditionally, Fourier transform (FT) plays a very important role in analytical chemistry. The technique involves a mathematical transformation of signals from one form to another one and is commonly used in analytical instrumentation and computational chemistry for data processing. For example, without the FT technique, it is impossible for chemists to have instruments such as FT-IR, and FT-NMR, as well as some of the signal processing methods mentioned in previous chapters.
As mentioned in Chapter 4, wavelet transform (WT), just as any other mathematical transform, aims at transforming a signal from the original domain to another one in which operations on the signal can be carried out more easily, and the inverse transform reverses the processes. In some respects, the WT resembles the well-known Fourier transform in which the sine and cosine are the analyzing functions. The analyzing function of WT is the wavelet, which is a family of functions derived from a basic function, called the wavelet basis, by dilation (or scaling) and translation. Therefore, unlike FT, which is localized in the frequency domain but not in the time domain, WT is well localized in both the time (or position) domain and the frequency (or scale) domain. Furthermore, compared with FT, a large number of basis functions are available with WT. Owing to these differences, one of the main features of WT is that it may decompose a signal into its components directly according to the frequency. With proper identification of the scales with frequency, higher-frequency signals can be separated from lower ones, in the sense that it has zoomin and zoomout capability at any frequency. Since WT can focus on any smaller part of a signal, it can be called a mathematical “microscope.” Another feature of WT is that the development of signals into the frequency domain can be
Chemometrics: From Basics To Wavelet Transform, edited by Foo-tim Chau, Yi-zeng Liang, Junbin Gao, and Xue-guang Shao. Chemical Analysis Series, Vol. 164.
ISBN 0-471-20242-8. Copyright © 2004 John Wiley & Sons, Inc.
application of wavelet transform in chemistry
constituted with a flexible choice of waveforms as a basis rather than with just the trigonometric functions like FT.
WT became a popular topic in chemistry and other fields of science starting from late 1980s, after the publication of the important papers by I. Daubechies  in 1988 and S. G. Mallat [2,3] in 1989, in which compactly supported orthonormal wavelets and fast calculation algorithms were proposed. Several reference books on WT were published in 1992 and afterwards, such as Ten Lectures on Wavelets , An Introduction to Wavelets , Wavelets: A Tutorial in Theory and Application , Wavelets: Theory, Algorithms and Applications , and Wavelets: A Mathematical Tool for Signal Processing . These books provided general information in wavelet theory, algorithms, and applications. More recently, reference books were also published to introduce applications of WT in various fields of chemistry [9-12].