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In the linear penalization method every wavelet coefficient is affected by a linear shrinkage particular associated to the resolution level of the coefficient. A mathematical expression for this type of approach using linear shrinkage is shown in Equation .
In Equation , parameter $s$ is the known smoothness index of the underlying signal $g$ , while parameter $\lambda $ is a smooting factor whose determination is critical for this type of analysis.
It must be said that linear thresholding is adequate only for spatially homogenous signal with important levels of regularity. When homegeneity and regularity conditions are not met nonlinear wavelet thresholding or shrinkage methods are usuallymore suitable.
donoho1995 and donoho1995b proposed a nonlinear strategy for thresholding. Under their approach, the thresholding can be done by implementing either a hard or a soft thresholding rule. Their mathematicalexpressions are shown in Equation and Equation respectively.
In both methods, the role of the parameter $\lambda $ as a threshold value is critical as the estimator leading to destruction,reduction, or increase in the value of a wavelet coefficient.
Several authors have discussed the properties and limitations of these two strategies; hard thresholding, due to its induced discontinuity, can be unstable and sensitive even to small changes in the data. On the other hand, soft thresholdingcan create unnecessary bias when the true coefficients are large. Although more sophisticated methods has been introduced to account for the drawbacks of the described nonlinear strategies, the discussion in this report is limited to the hardand soft approaches.
One apparent problem in applying wavelet thresholding methods is the way of selecting an appropriate value for the threshold, $\lambda $ . There are indeed several approaches for specifying the value of the parameter in question. In a general sense, these strategies can be classified in two groups: global thresholds and level-dependent thresholds. Global threshold implies the selection of one $\lambda $ value, applied to all the wavelet coefficients. Level-dependent thresholds implies that a (possibly) different threshold value $lambd{a}_{j}$ is applied for each resolution level. All the alternatives require an estimate of the noise level $\sigma $ . The standard deviation of the data values is clearly not a good estimator, unless the underlying response function $g$ is reasonably flat. donoho1995 considered estimating $\sigma $ in the wavelet domain by using the expression in Equation .
donoho1995 obtained an optimal threshold value ${\lambda}^{M}$ by minimizing the risk involved in estimating a function. The porposed minimax threshold depends of the available data and also takes into account the noise level contaminating the signal (Equation ).
Where, ${\lambda}_{n}^{*}$ is equal to the value of $\lambda $ satisfying Equation
In Equation , ${R}_{\lambda}\left(d\right)$ is calculated following Equation .
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