Wavelet transforms with application in signal denoising.
Wavelet transforms have gained wide acceptance as a valuable tool for common signal processing tasks. The most important difference between the wavelet transforms and transforms such as a Fourier transform is that the wavelets are localized in both the frequency and the time. This makes it possible to better localize properties of the analyzed signal. The result is a well known ability of the wavelet transforms to pack the main signal information into a very small number of large wavelet coefficients. Another advantage of wavelet transforms is that there is an indefinite number of basis functions. An appropriate wavelet can then be chosen for a specific signal which makes the transform adjustable and adaptable. Because of these excellent properties, wavelet transforms have been used with great success in many different applications, such as signal denoising and compression or feature detection. In this paper the focus will be on the application in signal denoising, where wavelets are used extensively.
2. WAVELET TRANSFORMS IN SIGNAL PROCESSING
For efficient application of the wavelet theory in signal processing it was necessary to simplify the calculations and allow the adaptivity to be introduced into the transform. In (Mallat, 1988; Mallat, 1989a; Mallat 1989b; Mallat 1989c) the theory was successfully connected with the concept of the multiresolution analysis. He also showed that the wavelet analysis can be performed using iterated filter banks of certain properties. Calculating the discrete wavelet transform (DWT) became as easy as performing a simple signal filtering. In a typical scenario, the analysis filter bank consists of a low-pass and a high-pass channel. Output from the low-pass channel represents the coarse approximation of the signal, while the output from the high-pass channel contains the fine signal details. Both, the approximation and the details, are decimated by a factor of 2 and low-pass channel output is iteratively filtered with the same filter bank until the desired level of decomposition is reached. This is illustrated in Fig. 1. Details coefficients at each decomposition level are also called the wavelet coefficients. To reconstruct the original signal, synthesis filter bank is used, with its inputs upsampled by a factor of 2, as shown in Fig. 2.
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3. DENOISING WITH WAVELET SHRINKAGE
The above procedure covers the basics of the wavelet analysis of the given signal but it does not provide a tool to remove a noise from the noisy signal. For denoising purposes, the wavelet shrinkage technique is used. It is based on the fact that the magnitude of the wavelet coefficients is important. Coefficients of higher magnitude are more likely to represent the main signal properties, while the coefficients of smaller magnitudes are more likely to be caused by the noise, as shown in Fig. 3 and Fig. 4. To remove the noise, a thresholding is applied to wavelet coefficients before signal reconstruction. A work was done in this area by Donoho and Johnstone (Donoho & Johnstone, 1994; Donoho & Johnstone, 1995; Donoho et al., 1995; Donoho & Johnstone, 1998). They successfully used the wavelet shrinkage technique for signal denoising and done significant research of the topic.
Although it has been showed that the wavelet transforms perform well in signal denoising they still exhibit unwanted visual artifacts, such as a Gibbs phenomena around signal discontinuities. Since the traditional wavelet transform is not translation invariant the size and the impact of the artifacts might heavily depend on the position of the discontinuity within the signal. To reduce the artifacts regardless of the discontinuity position, the translation-invariant denoising by means of the undecimated wavelet transform was proposed (Coifman and Donoho, 1996). In this transform the decimation step is not used, which results in an overcomplete representation of the signal. This was proved to both, significantly lower the overall root mean square error (RMSE) of the denoised signal, and also improve its visual appearance--subjective quality. The undecimated wavelet transform is now a common approach to a signal denoising problem. An example of denoising is shown in the Fig. 5. Additive white Gaussian noise with standard deviation equal to 5% of maximum signal magnitude is added to "Blocks" signal and Haar wavelet is used to construct a 4-level decomposition tree, as in Fig. 1. and 2.
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Since many real-world signals are susceptible to noise, signal denoising is a very important topic in a wide range of applications. Efficient denoising algorithms can improve otherwise unusable noisy signals to an acceptable level. The advantage of the wavelet transform when compared to traditional transforms such as a Fourier transform is its ability to localize the signal properties in both the frequency and the time. Because of this, wavelet transform packs the main signal properties in a small number of higher magnitude coefficients, while less important properties and noise get constrained to smaller coefficients, which can easily be observed in a provided denoising example. Denoising is performed by thresholding the wavelet coefficients at each decomposition level before the reconstruction. This way only main signal properties are reconstructed during the synthesis while the noise is attenuated.
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Wavelet transforms proved to perform very well in signal denoising. Further improvements are achieved with introducing adaptivity into wavelet transform, where research is commonly focused on choosing different wavelets for a different class of signals.
Coifman R. R. and Donoho, D. L. (1996) Translation-invariant denoising, Lecture Notes in Statistics: Wavelets and Statistics, New York: Springer-Verlag, pp. 125-150
Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81(3):425-455
Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage, Journal of the American Statistical Association, 90(432):1200-1224
Donoho, D. L. et al. (1995). Wavelet shrinkage: Asymptopia?, J. R. Statist. Soc. B., 57(2):301-337
Donoho, D. L. and Johnstone, I. M. (1998). Minimax estimation via wavelet shrinkage, Annals of Statistics, 26(3):879-921
Mallat, S. G. (1988). Multiresolution representations and wavelets, University of Pennsylvania, Philadelphia, PA, USA
Mallat, S. G. (1989a). Multiresolution approximation and wavelet orthogonal bases of l2(R), Transactions of the American Mathematical Society, Vol. 315, No. 1, pp. 69-87
Mallat, S. G. (1989b). Multifrequency channel decompositions of images and wavelet models, IEEE Transactions on Acoustics, Speech, and Signal Processing, Volume 37, Issue 12, Page(s):2091 - 2110
Mallat, S. G. (1989c). A theory for multiresolution signal decomposition: The wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11(7):674-693
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|Publication:||Annals of DAAAM & Proceedings|
|Date:||Jan 1, 2008|
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