Wavelet transforms smoothing

X.Q. Lu and J.Y. Mo, Methods of Handling Discrete Data for Deconvolution Voltammetry (I) Wavelet Transform Smoothing, Chemical Journal of Chinese University, 18 (1997), 49-51 (in Chinese). 

A wavelet defined as above is called a first-order wavelet. From Eq. (21) we conclude that the extrema points of the first-order wavelet transform provide the position of the inflexion points of the scaled signal at any level of scale. Similarly, if i/ (f) = d it)/dt, then the zero crossings of the wavelet transform correspond to the inflexion points of the original signal smoothed (i.e., scaled) by the scaling function, tj/it) (Mallat, 1991). 

The factor Vl/ 2 is introduced to keep the intensity of the signal unchanged. The 8 first wavelet transform coefficients are the a or smooth components. The last eight coefficients are the d or detail components. In the next step, the level 2 components are calculated by applying the transformation matrix, corresponding to the level on the original signal. This transformation matrix contains 4 wavelet filter... 

Barclay, V.J., Bonner, R.F., and Hamilton, I.P., Application of wavelet transform to experimental spectra smoothing, denoising, and data set compression, Anal. Chem., 69, 78, 1997. 

V.J. Barclay, R.F. Bonner and I.P. Hamilton, Application of Wavelet Transforms to Experimental Spectra Smoothing, De-noising, and Data Set Compression, Anal. Chem. 69 (1997), 78-90. 

In this chapter, compression is achieved by assuming that the data profiles can be approximated by a linear combination of smooth basis functions. The bases used originate from the fast wavelet transform. The idea that data sets are really functions rather than discrete vectors is the main focus of functional data analysis [12-15] which forms the foundation for the generation of parsimonious models. 

Here, we will outline the scheme they have proposed. In [52] the wavelet transform was used both to analyse the image prior to segmentation, enabling feature selection, as well as to provide spatial frequency-based descriptors as features for segmenting textures (see Fig. 27). Smooth and textured images can easily be distinguished from each other by examining their wavelet transforms. 

In order to successfully discriminate between smooth and textured images using the energy of the different channels of their wavelet transforms. Porter and Canagarajah [52] have further grouped the ten channels into low... 

Because of the recursive quadrisection process by which these sampling patterns are built, we can build fast hierarchical transforms. These generalize the fast wavelet transform to the surface setting. While it is much harder to prove smoothness and approximation properties in this more general setting, first re-... 

Wavelet transforms use a single function t/f Z.2(R) (wavelet) and its dilated and shifted versions to analyze data. Wavelets are associated with scaling functions 0eZ2(M). The scaling function usually serve to represent a smooth part of data and it is obtained as a solution of scaling equation... 

ABSTRACT This paper provides a short review of recent developments in crash pulse analysis methods and a short review of wavelet based data processing methods. A discrete wavelet transform can he performed in 0 n) operations, and it captures not only a frequency of the data, but also spatial informations. Moreover wavelet enables sparse representations of diverse types of data including those with discontinuities. And finally wavelet based compression, smoothing, denoising, and data reduction are performed by simple thresholding of wavelet coefficients. Combined, these properties make wavelets a very attractive tool in mary applications. Here, a noisy crash signals are analyzed, smoothed and denoised by means of the discrete wavelet transform. The optimal choice of wavelet is discussed and examples of crash pulse analysis are also given. 

Traditional transforms like the wavelet transform provide efficient representations for smooth images but suffer from curve discontinuities. Therefore, it is desirable to have a compression method that provides efficient representation of edges by exhibiting singularities along arbitrarily shaped curves, which may lead to improvements in compression performance. The compression procedure using the ripplet transform is formulated as follows [22] ... 

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