Wavelet compression

The basic idea of WT is to correlate any arbitrary function f t) with the set of wavelet functions obtained by dilation and translation. A stretched wavelet correlates with low frequency characteristics of the signal, while a compressed wavelet correlates with high frequency characteristics (Blatter 1988). Technically, we can say that scale parameter v relates the spectral content of the function f t) at a different positions X (translation parameter, see Figure 9.9). The correlation process described is the Continuous Wavelet Transform (CWT) of a signal, mathematically described as... 

In most solids, the sound speed is an increasing function of pressure, and it is that property that causes a compression wave to steepen into a shock. The situation is similar to a shallow water wave, whose velocity increases with depth. As the wave approaches shore, a small wavelet on the trailing, deeper part of the wave moves faster, and eventually overtakes similar disturbances on the front part of the wave. Eventually, the water wave becomes gravitationally unstable and overturns. 

For a shock wave in a solid, the analogous picture is shown schematically in Fig. 2.6(a). Consider a compression wave on which there are two small compressional disturbances, one ahead of the other. The first wavelet moves with respect to its surroundings at the local sound speed of Aj, which depends on the pressure at that point. Since the medium through which it is propagating is moving with respect to stationary coordinates at a particle velocity Uj, the actual speed of the disturbance in the laboratory reference frame is Aj - -Ui- Similarly, the second disturbance advances at fl2 + 2- Thus the second wavelet overtakes the first, since both sound speed and particle velocity increase with pressure. Just as a shallow water wave steepens, so does the shock. Unlike the surf, a shock wave is not subject to gravitational instabilities, so there is no way for it to overturn. 

Compression may be achieved if some regions of the time-frequency space in which the data are decomposed do not contain much information. The square of each wavelet coefficient is proportional to the least-squares error of approximation incurred by neglecting that coefficient in the reconstruction ... 

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

Fio. 19. Reconstruction of compressed signal from the wavelet decomposition of Fig. 18... 

Fig. 20. Performance of data compression techniques (a) orthonormal wavelet (b) backward slope (c) boxcar.

B. Walczak and D.L.Massart, Tutorial Noise suppression and signal compression using the wavelet packet transform. Chemom. Intell. Lab. Syst., 36 (1997) 81-94. 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

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). 

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

In cases involving many variables (such as voltammetric data), it is possible to apply LDA and QDA-UNEQ following a preliminary reduction in the variable number, for instance, by PCA or wavelet compression. 

Pattern recognition can be classified according to several parameters. Below we discuss only the supervised/unsupervised dichotomy because it represents two different ways of analyzing hyperspectral data cubes. Unsupervised methods (cluster analysis) classify image pixels without calibration and with spectra only, in contrast to supervised classifications. Feature extraction methods [21] such as PCA or wavelet compression are often applied before cluster analysis. 

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... 

In general, wavelet functions are chosen such that they and their compressed representations are orthogonal to one another. As a result, the basis functions in Wavelet compression, like those in PCA and Fourier compression, are completely independent of one another. Several researchers have found that representation of spectral data in terms... 

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. 

Sinha and Tewfik, 1993] Sinha, D. and Tewfik, A. H. (1993). Low bit-rate transparent audio compression using adapted wavelets. IEEE Trans. Acoust,. Speech, and Signal Processing, 41(12) 3463-3479. 

---