Signal denoising using wavelet-based methods  (Page 6/9)

while $R_{oracle}\left(d\right)$ is an operators called oracle used to account for the risk associated to the modification of the value of a given wavelet coeficient. Two of these oracles were introduced by donoho1995: diagonal linear projection (DLP), and diagonal linear shrinker (DLS). The Equations and show the expressions for the two oracles.

antoniadis2001 provided values of the minimax threshold for both the hard and soft nonlinear thresholding rules. For the soft rule, 1.669 and 2.226 for $n$ equal to 128 and 1024; for the hard rule, 2.913 and 3.497 again for $n$ equal to 128 and 1024.

The universal threshold

donoho1995 proposed this threshold as an alternative to the minimax thresholds, applied to all the wavelet coefficients. The universal threshold is defined in Equation .

This threshold is easy to remember and its implementation in software is simpler and the optimization problem implicit in the minimax method are avoided. Also, the universal threshold ensures, with high probability, that every samplein the wavelet transform in which the underlying function is exactly zero will be estimated as zero, although the convergance rate (depending in the size of the sample) is slow.

The translation invariant method

It has been noted that wavelet thresholding with either minimax thresholds or the universal threshold presents some inconvenient features. In particular, in the vicinity of discontinuities, these wavelet thresholds can exhibitpseudo-Gibbs phenomena. While these phenomena are less pronounced than in the case of Fourier analysis and also are present in a local scale, this situation represents a challenge for the thresholding methods.

coifman1995 proposed the use of the translation invariant wavelet thresholding scheme. The idea is to correct mis-alignments between features in the studied signal and features in the basis usedfor the decomposition. When the signal contains an important number of discontinuities, the method applies a range of shifts to it, and average the results obtained after such transformations.

If a empirical contaminated signal $y\left[i\right],(i=1,...,n)$ is provided, the tranlation invariant wavelet thresholding estimator is calculated as (Equation ):

where ${\delta }_{\lambda }$ is either the hard of soft thresholding rule, $W$ is the size $n$ orthogonal matrix associated to the DWT, and $S_k$ is the shift kernel defined as:

In Equation , $I$ is the identity matrix and $O$ stands for a zero matrix with dimensions as indicated in the expression.

The sureshrink threshold

donoho1995 proposed a procedure to select a threshold value ${\lambda }_j$ for every resolution level $j$ . The method uses Stein’s unbiased risk criterion citep( ) to get an unbiased estimate of the $l^2$ -risk.

In mathematical terms, given a set $X_1,...,X_s$ of variables distributed as $N({\mu }_i,1)$ with $i=1,...,s$ , the problem consists in estimate the vector of means with minimum $l^2-$ risk. It turns out an estimator of $\mu$ , can be describes as $\widehat{\mu }\left(X\right)=X+g\left(X\right)$ , with $g$ a function from ${\mathbb{R}}^s$ to ${\mathbb{R}}^s$ is weakly differentiable. With this information, the risk of the estimation can be described as (Equation ):