Wavelet transforms data compression

The variable selection methods discussed above certainly do not cover all selection methods that have been proposed, and there are several other methods that could be quite effective for PAT applications. These include a modified version of a PLS algorithm that includes interactive variable selection [102], and a combination of GA selection with wavelet transform data compression [25]. 

T. Fearn and A.M.C. Davies, A comparison of Fourier and wavelet transforms in the processing of near-infrared spectroscopic data part 1. Data compression, J. Near Infrared Spectrosc., 11, 3-15 (2003). 

Although the Fourier compression method can be effective for reducing data into frequency components, it cannot effectively handle situations where the dominant frequency components vary as a function of position in the spectrum. For example, in Fourier transform near-infrared (FTNIR) spectroscopy, where wavenumber (cm-1) is used as the x-axis, the bandwidths of the combination bands at the lower wavenumbers can be much smaller than the bandwidths of the overtone bands at the higher wavenumbers.31,32 In any such case where relevant spectral information can exist at different frequencies for different positions, it can be advantageous to use a compression technique that compresses based on frequency but still preserves some position information. The Wavelet transform is one such technique.33... 

Fearn, T. and Davies, A.M.C., A Comparison of Fourier and Wavelet Transforms in the Processing of Near-Infrared Spectroscopic Data Part 1. Data Compression /. Near Infrared Spectrosc. 2003, 11, 3-15. 

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. 

The advantage of this transform is that its kernel f(s, x, x) is left unspecified. The discrete wavelet transform was invented by Haar125, used by petroleum geologists to extract meaningful data from noisy seismograms, and later utilized in JPEG2000 pixel compression. 

Wavelet-transformed RDF descriptors can enhance or snppress typical features of descriptors — even filtered, or compressed, ones. This behavior also covers the diversity and similarity of molecnles in a data set. The experiment in Figure 6.14 shows results from a single data, compiled from two types of componnds 100 benzene derivatives followed by 100 monocyclic cyclohexane derivatives. The distribn-tion of deviation of the individnal descriptors to the ASD indicates the diversity of the two data sets. 

The reduction of the descriptor size (i.e., the decrease in resolution) usually has a profound influence on the ability of the descriptor to characterize a molecule. Even though compressed, or filtered, wavelet transforms of descriptors have a reduced size, they preserve the similarity information well and in a much more efficient way. Figure 6.15 shows results from an experiment where a Kohonen neural network classifies the same data set (100 benzene derivatives plus 100 monocyclic cyclohexane derivatives) according to ring type. 

Generally, the DWT is used for data compression and the CWT for signal analysis. Wavelet transforms are applied in several fields, such as molecular dynamics, quantum-chemical calculations, optics, image analysis, DNA and protein sequence analysis, and sismic geophysics. 

One commonly used cost function particularly in data compression is entropy. If we let Wj t j denote the ith wavelet packet coefficient band(j, t) of the wavelet packet transform, then the entropy-like criterion for band(j, t) is defined as follows ... 

The wavelet transform can help to solve such problems. It allows transformation of the data from the original domain (i.e. from the time or space domain) into the frequency-time, or the scale-time domain. New features, i.e. wavelet coefficients, describe local phenomena in a very efficient way and, moreover, allow data compression. As the reader may have already noticed,... 

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. 

Up to December 1998, more than 30 publications have reported spectroscopic studies with the use of a WT algorithm [9,10], Within this work, WT has been utilized in three major areas that include data denoising, data compression, and pattern recognition. Two classes of wavelet algorithm namely discrete wavelet transform (DWT) and wavelet packet transform (WPT), have been commonly adopted in the computation. The former one is also known as the fast wavelet transform (FWT). The general theory on both FWT and WPT can be found in other Chapters of this book and some chemical journals [16-18], and is not repeated here. In the following sections, selected applications of WT in different spectral techniques will be described. 

S.N. Qian and H. Sun, Data-compression Method Based on Wavelet Transformation for Spectral Information, Spectroscopic. Spectra Analysis (Beijing), 16 (1996), 1-8. 

F-T. Chau, J.B. Gao, T.M. Shih and J. Wang, Compression of Infrared Spectral Data Using the Fast Wavelet Transform Method, Applied Spectro.scopy, 51 (1997), 649-659. 

There are few possible strategies of library compression. Each of them has its own advantages and drawbacks. The most efficient method of data set compression, i.e. Principal Component Analysis (PCA), leads to use of global features. As demonstrated in [15] global features such as PCs (or Fourier coefficients) are not best suited for a calibration or classification purposes. Often, quite small, well-localized differences between objects determine the very possibility of their proper classification. For this reason wavelet transforms seem to be promising tools for compression of data sets which are meant to be further processed. However, even if we limit ourselves only to wavelet transforms, still the problem of an approach optimally selected for a particular purpose remains. There is no single method, which fulfills all requirements associated with a spectral library s compression at once. Here we present comparison of different methods in a systematic way. The approaches A1-A4 above were applied to library compression using 21 filters (9 filters from the Daubechies family, 5 Coiflets and 7 Symmlets, denoted, respectively as filters Nos. 2-10, 11-15 and 16-22). 

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. 

Choosing optimal wavelet bases. Unlike the Fourier transform there are several basis functions to select from when using wavelet transforms. This means there must be a criterion for choosing the optimal wavelet. A reasonable criterion is to use the compression ability of the analysing wavelet. This means that the optimal wavelet is defined to be the one that produces the smallest number of coefficients needed to describe the data. The following algorithm is here used ... 

The difference and transform compression schemes described above attempt to decorrelate the image data at a single resolution only. With the advent of multiresolution representations (pyramids, wavelet transforms, subband... 

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. 

Compression of Infrared Spectral Data Using the Fast Wavelet Transform Method. 

---