Advantages and Disadvantages of Wavelet Denoising_Analysis and Comparison of Wavelet Denoising Methods

Wavelet analysis is a new mathematical theory and method developed in the past ten years and has been successfully applied in many fields. As a new time-frequency analysis method, wavelet analysis is known as "mathematical microscope" because of its multi-resolution analysis and ability to focus on any detail of the signal for multi-resolution time-frequency domain analysis.

This paper mainly introduces four commonly used wavelet denoising methods, such as wavelet decomposition and reconstruction method, nonlinear wavelet transform threshold method, translation invariant wavelet method and wavelet transform modulus maximum method. They are used to denoise the simulation examples respectively, and compare the application, denoising performance, calculation speed and influencing factors of these methods. Finally, the selection of wavelet denoising methods is summarized.

Essentially equivalent to a bandpass filter with multiple channels, it is mainly suitable for the case of deterministic noise when the bands of useful signals and noise are separated from each other. In this case, the method can basically remove noise, and the denoising effect is good. However, in the case where the frequency bands of useful signals and noise overlap each other (for example, the signal is mixed with white noise), the effect is not satisfactory.

advantage:

The scope of application is not very extensive. It is very effective when the frequency range of the noise is known in a specific case and the frequency bands of the signal and noise are separated from each other. For white noise, which is widely used in practical applications, the denoising effect is poor.

Fig.1 Denoising by wavelet decomposition and reconstruction

It is mainly used when there is white noise mixed in the signal. The advantage of denoising by the threshold method is that the noise is almost completely suppressed, and the characteristic spikes reflecting the original signal are well preserved. Denoising by soft threshold method can minimize the maximum mean square error of the estimated signal, that is, the estimated signal after denoising is an approximate optimal estimate of the original signal; and the estimated signal is at least as smooth as the original signal without additional oscillation. .

The method also has wide adaptability and is thus the most widely used one of many wavelet denoising methods. The calculation method of the threshold method is fast, which is O(N), where N is the signal length.

In some cases, such as at the discontinuous point of the signal, a pseudo-Gibbs phenomenon occurs after denoising. When denoising by this method, the choice of threshold has a significant influence on the denoising effect.

Figure 2 Soft threshold denoising

It is mainly applied to the case where the signal is mixed with white noise and contains several discontinuities, which is an improvement based on the threshold method.

The pseudo- Gibbs phenomenon generated at the discontinuous point of the signal can be effectively removed from the threshold method denoising, showing a better visual effect than the threshold method. From the point of view of L2 norm error, denoising by this method can obtain a smaller root mean square error than the threshold method, and the signal-to-noise ratio is also improved to some extent;

The calculation speed is not as fast as the threshold method. When the signal length is N, the calculation speed is O(NlogN).

Figure 3 Translation Invariant Wavelet Method for Denoising

It is mainly used when there is white noise mixed in the signal and the signal contains more singular points.

The method effectively preserves the singular point information of the signal while denoising, and the denoised signal has no unnecessary oscillation, which is a very good estimation of the original signal and has better picture quality.

When using the modulus maxima for reconstruction, the alternate projection method is used. To ensure the accuracy of the reconstructed signal and improve the signal-to-noise ratio, it is usually performed dozens of iterations. The speed of each iteration is O(NlogN). Therefore, the calculation speed is very slow, usually ten times slower than the previous methods.

Figure 4 Wavelet transform modulus maxima denoising

(1) For the denoising processing of deterministic noise in which the frequency bands of signal and noise are separated from each other, the wavelet decomposition and reconstruction denoising method with simple selection method and fast calculation speed are most suitable.

(2) For the denoising processing of Gaussian white noise, the threshold method, the translation invariant method and the modulus maximum method can be selected. Which method to choose should be based on the characteristics of the actual signal and the advantages and disadvantages of these methods. 1 threshold method because it has the original signal

Approximate optimal estimation, fast calculation speed and wide adaptability are the most widely used methods of wavelet denoising. In general, this method can be used to denoise. 2 The translation invariant method is applicable to the case where the signal contains several discontinuities. Usually the denoising performance is better than the threshold method, but at the expense of computational speed. 3 wavelet transform modulus maximum method When the signal contains more singular points, the denoising performance is quite good, but its biggest disadvantage is that the calculation speed is too slow. In the application, the relationship between the denoising effect and the calculation speed should be weighed.

(3) Wavelet denoising method combined with other methods may achieve better results

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