Soft thresholding

Donoho, D.L., De-noising by soft thresholding, IEEE Trans. Inf. Theory, 41, 613,... 

The similarity threshold for ligand pairs that are considered in calculating the discontinuity score is set to 0.6. This relatively soft threshold value ensures that also potency differences between remotely similar compounds are taken into account and thus enables a thorough assessment of putative activity cliffs, which is further emphasized by multiplication by pair-wise ligand similarity. 

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. 

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] ... 

Soft thresholding. In soft thresholding, all coefficients are shrunk towards zero by the threshold value. Mathematically, it can be written as. 

Usually, soft thresholding gives better visual quality of filtering than procedures based on minimizing the mean square error [25]. 

D.L. Donoho, Denoising by Soft Thresholding, Technical Report, Department of Statistics, Stanford University, (1992). 

D.L. Donoho, De-noising by Soft-Thresholding. IEEE Tran.sactions Information Theory M (1995), 613-627. 

Donoho, D.L. 1995. Denoising by soft thresholding. IEEE Trans. Information Theory 41(3) 613-627. Ephraim, Y. 1992. Statistical model based speech enhancement systems. Proc. IEEE 80(10) 1526-1555. Ephraim, Y. and Malah, D. 1984. Speech enhancement using a minimum mean-square error short time spectral amphtude estimator. IEEE Trans. Acoustics, Speech, and Signal Processing 32(6) 1109-1121. Ephraim, Y. and Van Trees, H.L. 1995. A signal subspace approach for speech enhancement. IEEE Trans, on Speech and Audio Processing 3 4) 25l-266. 

For a comparison, we also show the residue of wavelet de-noising with Donoho s soft thresholding (Donoho, 1995) in Figure 4.14. As can be seen, the residue of wavelet de-noising is typically a Gaussian noise. 

A soft threshold time series 4, where ie 1,2,- , N, is given by Eq. 1, where t is a threshold value. Hence, the low-power components of jc, are de-noised (removed) ... 

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