The basic principle of wavelet threshold denoising_How to choose the wavelet denoising threshold

The basic idea of ​​wavelet threshold denoising is to first set a critical threshold λ. If the wavelet coefficient is smaller than λ, the coefficient is mainly caused by noise, and the coefficient is removed. If the wavelet coefficient is larger than λ, the coefficient is mainly caused by the signal. This part of the coefficient is retained, and then the wavelet coefficients are inversely transformed by the processed wavelet coefficients to obtain a denoised signal. Specific steps are as follows:

(1) wavelet transform the noisy signal f(t) to obtain a set of wavelet decomposition coefficients Wj,k;

(2) By performing threshold processing on the wavelet decomposition coefficients Wj, k, the estimated wavelet coefficients Wj, k are obtained, so that Wj, k-uj, k are as small as possible;

(3) Wavelet reconstruction is performed using the estimated wavelet coefficients Wj, k to obtain an estimated signal f(t), which is a denoised signal.

A very simple method is proposed to estimate the wavelet coefficient Wkj. After several times of wavelet decomposition for f(k), there are spatially distributed non-uniform signals s(k) on the scale of the wavelet coefficients Wkj, which have larger values ​​at certain specific positions, which correspond to the original signal s(k) The odd position and important information, while most other positions of Wkj are smaller; for white noise n(k), its corresponding wavelet coefficient Wkj, the distribution on each scale is uniform, and scale Increase Wkj, the magnitude of the coefficient decreases. Therefore, the usual denoising method is to find a suitable number  as a threshold (threshold), and to set the wavelet function Wkj below λ (mainly caused by the signal n(k)) to zero, and for higher than  The wavelet function Wkj, (mainly caused by the signal s(k)), is retained or shrunk to obtain the estimated wavelet coefficient Wkj, which can be understood as basically caused by the signal s(k), and then reconstructed by Wkj. The original signal can be reconstructed.

(1) Perform s-layer orthogonal redundant wavelet transform on the noisy image g(i,j) to obtain a set of wavelet decomposition coefficients Wg(i,j)(s,j), where j=1,2,...s , s represents the number of layers of wavelet decomposition.

The wavelet threshold denoising method has good mathematical theory support, and it is simple and very effective. Therefore, it has achieved great success and attracted many scholars to further research and improve it. These studies focus on two areas: the study of threshold selection and the study of threshold functions.

The determination of the threshold is very important in the denoising process. The thresholds currently used can be divided into two categories: global threshold and local adaptive threshold. The global threshold is to select the same threshold for all wavelet coefficients of each layer or wavelet coefficients of different directions in the same layer. The local threshold is selected according to different directions of different layers.

Global threshold is calculated as follows

among them:

For the noise standard deviation, M, N are the scale of the image signal.

The threshold is derived from the joint distribution of multidimensional independent normal variables under the Gaussian model;

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