Signal denoising using wavelet-based methods  (Page 7/9)

with,

If an operator SURE is defined as (Equation ):

where the operator $\#\left\{A\right\}$ returns the cardinality of the set A, it is found that SURE is an unbiased estimate of the $l^2-$ risk, i.e,

Now, the threshold $\lambda$ is found by minimizing SURE over the set $X$ of give data. Extending this principle to the entire set of resolution levels, an expression for ${\lambda }_j^s$ is found (Equation ):

where ${\lambda }^U$ is the universal threshold, and $\widehat{\sigma }$ is the estimator of the noise level (Equation ).

Classical methods: block thresholding

Thresholding approaches resorting to term-by-term modification on the wavelets coefficients attempt to balance variance and bias contribution to the mean squared error in the estimation of the underlying signal $g$ . However, it has been proven that such balance is not optimal. Term-by-term thresholding end sup removing to many terms leadingto estimation prone to bias and with a slower convergance rate due to the number of operations involved.

A useful resource to improve the quality of the aforementioned balanced is by using information of the set of data associated to a particular wavelet coefficient. In order to do so, a block strategy for thresholded is proposed.The main idea consists in isolating a block of wavelet coefficients and based upon the information collected about the entire set make a decision about decreasing or even entirely discard the group. This procedure will allow fastermanipulation of the information and accelerated convergence rates.

Overlapping block thresholding estimator

cai2001 considered an overlapping block thresholding estimator by modifying the nonoverlapping block thresholding estimator citep( ). The effect is that the treatment of empirical wavelet coefficients in the middle of each block depends on the data in the whole block. At each resolution level, this method packswavelet coefficients ${\widehat{d}}_{jk}$ into nonoverlapping blocks ( $j_b$ ) of length $L_0$ . Following this, the blocks are extended in each direction an amount $L_1=max(1;\left[\frac{L_0}{2}\right])$ , generating overlapping blocks ( $j_B$ ) of augmented length $L=L_0+2L_1$ .

If $S_{\left(j_B\right)}^2$ is the $L^2-$ energy of the empirical signal in the augmented block $j_B$ , the wavelet coefficients in the blocks $j_b$ will be estimated simultaneously using the expression in Equation

Once the estimated wavelet coefficients ${\breve{d}}_{jk}^{\left(j_b\right)}$ have been calculated, an estimation of the underlying signal $g$ can be obtained through using the new wavelet coefficients and the unmodified scaling coefficients ${\widehat{c}}_{j_{0}k}$ in the IDWT. The results from this method ( NeigBlock ) presented in this document used a value of $L_0=\left[log\left(\frac{n}{2}\right)\right]$ and a value of $\lambda =4.50524$ as suggested by cai2001.

Bayesian approach to wavelet shrinkage and thresholding

From Equations and it can be established that the empirical scaling and wavelet coefficients, conditional on their respective underlying coefficients, are independently distributed, i.e: