Wavelet Transform Background

What FT does is to project the signal f t) onto the set of sine and cosine basis functions of infinite duration represented by the complex exponential function (Rioul and Vetterli 1991). The transformation (named analysis) is reversible and the recovering of the original function (named synthesis) is done by summing up all the Fourier components multiplied by their corresponding basis function, that is. 

From the mathematical definition of the Fourier transform, it is advisable that the process determines the frequency content of the signal by obtaining the contribution of each sine and cosine function at each frequency. However, if a transient phenomenon occurred in the signal or it is non-stationary, then the Fourier transform fails either in localizing the anomaly in time domain or representing the signal by the summation of the periodic functions. 

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Fig. 6 The ROIs shown in Fig. 2 denoised by discrete wavelet transform, (a) background, (b)...

M. Chaoxiong and S. Xueguang, Continuous wavelet transform applied to removing the fluctuating background in near-infrared spectra, J. Chem. Inf. Comput. Sci., 44, 907-911 (2004). 

Depezynski, U., Jetter, K., Molt, K and Niemoller, A. (1997) The fast wavelet transform on compact intervals as a tool in chemometrics. I. Mathematical background. Chemom. Intell. Lab. Syst., 39, 19-27. 

U. Depczynski, K. Jetter, K. Molt and A. Niemoller, The Fast Wavelet Transform on Compact Intervals as a Tool in Chemometrics. 1. Mathematical Background, Chemometrics and Intelligent Laboratory Systems, 39 (1997), 19-27. 

Abstract. We discuss the use of a wavelet transform for the subtraction of nonuniform background and for filtering images obtained with the coded-mcisk X-ray telescope ART-P aboard the Granat spacecraft. 

Kobatake et al. (1999) used an adaptive filter called the iris filter to detect regions with low contrast to their background. Multiresolution and multiorientation wavelet transforms have been used for image pre-processing (Li et al. 1997), as well as mass detection (Li et al. 1997) earlier. 

Fig. 6 Results of the (a) differential pulse stripping voltammetric signal of the Zn(II)-KCl system and (bj the background current signal of KCl. solution that was de-noised with the combined B-spline wavelet and Fourier transform analysis. Reproduced from reference [25] with kind permission of Science in China Press.

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