Hard thresholding

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,... 

The soft-thresholding is often preferred as the hard-thresholding has discontinuities that introduce artifacts to the denoised signal. The next step is to determine the threshold value, t. 

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... 

For compression, other threshold selection criteria include the compression ratio (CR), the mean-square error (MSE), and the local point-wise error of approximation, which are discussed in more detail in [13]. In this chapter, the universal threshold given in Eq. (13) will be used for both filtering and compression. Two thresholding techniques have been studied Hard thresholding and Soft thresholding [4,6,22] ... 

Hard thresholding. For the wavelet coefficients at scale j, dj k, and a threshold value, t, the thresholded coefficients, dj.k, are determined as. 

Hard thresholding can lead to better reproduction of peak heights and discontinuities, but at the price of occasional artifacts that can roughen the appearance of the signals estimate [4]. 

Chau and his co-workers have proposed some wavelet-based methods to compress UV-VIS spectra [24,37]. In their work, a UV-VIS spectrum was processed with the Daubechies wavelet function, Djfi. Then, all the Cj elements and selected Dj coefficients at different) resolution levels were stored as the compressed spectral data. A hard-thresholding method was adopted for the selection of coefficients from Dj. A compression ratio up to 83% was achieved. As mentioned in the previous section, the choice of mother wavelets is vast in WT, so one can select the best wavelet function for different applications. Flowever, most workers restrict their choices to the orthogonal wavelet bases such as Daubechies wavelet. Chau et al. chose the biorthogonal wavelet for UV VIS spectral data compression in another study [37]. Unlike the orthogonal case, which needs only one mother wavelet (p(t), the biorthogonal one requires two mother wavelets. (p(t) and (p(t), which satisfy the following biorthogonal property [38] ... 

Figure 14 WT-based denoising has four steps (1) Transform the signal, (2) isolate the small-amplitude wavelet coefficients corresponding to the noise components, (3) zero-out or reduce these coefficients, and (4) apply a reverse WT to the signal. Compare this with the smoothing routine illustrated in Figure 13. The isolation of small-amplitude coefficients in step 2 was achieved by using a progressive reduction hard-thresholding approach, which reduces the elimination threshold for each lower frequency band.

FIGURE 19.16 Typical activation functions (a) hard threshold unipolar, (b) hard threshold bipolar, (c) continuous... 

The training process starts usually with values of all weights set to zero. This learning rule can be used for both soft and hard threshold neurons. Since desired responses of neurons is not used in the learning procedure, this is the unsupervised learning rule. The absolute values of the weights are usually proportional to the learning time, which is undesired. 

The single-layer recurrent network was analyzed by Hopheld (1982). This network, shown in Fig. 19.29, has unipolar hard threshold neurons with outputs equal to 0 or 1. Weights are given by a symmetrical square matrix Wwith zero elements (= 0 for i = j) on the main diagonal. The stabflity of the system is usually analyzed by means of the energy function... 

Hopfield (1984) extended the concept of his network to autoassociative memories. In the same network structure as shown in Fig. 19.29, the bipolar hard-threshold neurons were used with outputs equal to —1... 

Perceptron Network with hard threshold neurons. 

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