Wavelet thresholding

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

The multiscale basis functions capture the fast changes in coefficients corresponding to the fine-scale basis functions, while the slower changes are captured by the coarse-scale basis functions. Thus, the wavelet thresholding method adapts its resolution to the nature of the signal features and reduces the contribution of errors with minimum distortion of the features retained in the rectified signal. 

FIGURE 10.25 Denoising of the wavelet components of the noisy signal in Figure 10.1 using an entropy threshold and db-7 wavelets. Thresholds for the decomposition are shown as dashed lines. Wavelet coefficients contained inside the thresholds are set to zero in this form of denoising. 

The measurements in each window are filtered by the wavelet thresholding approach described in the previous section [4]. This simple approach is very effective compared to the single scale techniques as shown by the theoretical analysis presented next and the illustrative example. It retains the benefits of the wavelet decomposition in each moving window, while allowing each measurement to be filtered on-line. 

M. Neumann and R. Von Sachs, Wavelet Thresholding Beyond the Gaussian LTD. Situation, Lecture Notes in Statistics, Anestis Antoniadis and Georges Oppenheim, Issue Description Wavelets and Statistics. New York, 103 (1995), 103. 

Signal with deterministic changes. In this example, the noise-free signal is deterministic with some sudden changes in the mean. The variables are contaminated by iid Gaussian error of standard deviation 0.5, and the results are summarized in Fig. 11. Wavelet thresholding of the result of maximum... 

Fig. 11 Data rectification of signal with deterministic features. Dashed line is noisy data, (a) Original and noisy data, (h) Wavelet thresholding, (c) Wavelet thresholding after maximum likelihood rectification, (d) Multi.scale Bayesian rectification.

Heteroscedastic backgrounds can confound signals in a way that simple wavelet thresholding routines become ineffective at removing their influences. The changing variance of this noise allows the noise to move from one... 

The hard-thresholding filter, fjj, selects wavelet coefficients that exceed a certain threshold and sets the others to zero ... 

The soft-thresholding filter, Fl, is similar to the hard-thresholding filter, but it also shrinks the wavelet coefficients above the threshold,... 

Figure 10.16. Filtered approximation of Y through wavelet denoising using hard-thresholding with one level of decomposition (left) and two levels of decomposition.

Fl (1) Fh (1) Soft-thresholding and hard-thresholding wavelet filters Fw d) Wiener wavelet filter... 

---