Signal denoising using wavelet-based methods

• Root Mean Squared Error : The mean squared error defined as (Equation )
  $$\frac{1}{N}\sum _{i=1}^N{\left(\left(f,\left(x_i\right),-,f_{dn},(,x_i\right)\right)}^2$$
  is computed for each realization and averaged over the 100 samples. Then, its square root is taken.
• Maximum Deviation : The average over the 100 samples of ${\displaystyle \underset{1<i<N}{max}\left|f,\left(x_i\right),-,f_{dn},\left(x_i\right)\right|}$

Computational efficiency has not been chosen as one of the criteria, since it is greatly depended on the individual programming skills of the individual. Therefore, in order to avoid a non-uniform programming approach which couldpossibly result in misleading conclusions, time efficiency has not been considered.

The test functions $f\left(x\right)$ and the sample sizes $N$ have been chosen as the factors of the comparison studies. To this aim, two samples, one of moderate moderate size ( $N=128$ ) and another of larger size ( $N=1024$ ) have been considered.

As far as the test functions are concerned, two smooth signals (Figures and ) and two discontinuous ones (Figures and ) were taken into account. In , the function consists of the sum of two sinusoids, whereas in , a time shifted sine is illustrated. Since the signals are smooth, linear methods are expected to be comparable to the nonlinear ones. On the other hand, nonlinear wavelet estimators areexpected to perform better for the functions in ( , ). These highly discontinuous signals have been used as examples in donoho1993

Results

The following plots, (Figures - ), illustrate the denoising performance for the 10 methods used. Each integer corresponds to a particular method as follows

• VisuShrink-Hard: Universal threshold with hard thresholding rule
• VisuShrink-Soft: Universal threshold with soft thresholding rule
• SureShrink: SureShrink threshold
• Translation-Invariant-Hard: Translation invariant threshold with hard thresholding rule
• Translation-Invariant-Soft: Translation invariant threshold with soft thresholding rule
• Minimax-Hard: Minimax threshold with hard thresholding rule
• Minimax-Soft: Minimax threshold with soft thresholding rule
• NeighBlock: Overlapping block thresholding (with $L_0=[logn/2]$ , $\lambda =4.50524$ )
• Linear Penalization: Term-by-term thresholding using linear shrinking
• Deterministic/Stochastic: Bayesian thresholding method for shrinkage estimates

Conclusions

A general comment can be made related to the Root Mean Squared Error (RMSE). As expected, the bigger the sample size the lower the value of the RMSE. It is readily seen that this is true for the same test function and denoising procedure.

Focusing on the smooth Wave function, the bayesian method performs well. However, the linear penalization method and the Translation-Invariant-Hard method are very competitive. The performance of the penalization method should not besurprising since the linear estimators are expected to achieve good results in smooth functions such as the Wave signal. Similar remarks can be made about the Time-Shifted Sine signal, a function that shares with the Wave signal the smoothnesfeature.

As far as the Bumps function and the Blocks function are concerned, the Bayesian method outperform the classical ones in terms of RMSE. This leads to the conclusion that using Bayesian methods for such type of functions is preferable ifcomputational efficiency is not an issue. In fact, it is well established that non-Bayesian methods uniformly outperform Bayesian methods in terms of CPU time.

Finally, as a general remark, larger values of MaxDeviation occur for functions with many spikes and discontinuities.

Acknowledgments

The authors wish to thank Professor C. Sidney Burrus for his help and guidance through the development of this work.