# An Improved Image Filtering Algorithm for Mixed Noise
## 2 The Filtering Algorithm for Mixed Noise with Salt and Pepper + Gaussian Noise
### The Improved Wavelet Threshold Denoising Algorithm
#### 2.2.2 The Improved Wavelet Threshold Function  
Aiming at the discontinuity and constant deviation of the traditional threshold function, this paper proposes a new wavelet threshold function. When the new threshold function is used to process the wavelet coefficients, the wavelet coefficients after denoising are closer to the wavelet coefficients of the original image, so the distortion of the restored image is smaller. The threshold function expression is:  
\[\widetilde{W}_{j,k}=\left\{\begin{array}{cc}sign(W_{j,k})\left(\left|W_  
Therefore, \(\lim\limits_{W_{jk}\rightarrow-\infty}\widetilde{W}_{j,k}=\lim\limits_{W_{jk} \rightarrow-\lambda^{+}}\widetilde{W}_{j,k}\), thus it can be seen that the improved threshold function is continuous at \(-\lambda\). Therefore, it is concluded that the improved threshold function is continuous at \(\pm\lambda\). * The asymptotic property analysis of the function When \(W_{j,k}\rightarrow+\infty\) : \[ \begin{split}\lim\limits_{W_{jk}\rightarrow+\infty}\frac{ \widetilde{W}_{j,k}}{W_{j,k}}&=\lim\limits_{W_{jk}\rightarrow+ \infty}\frac{sign\big{(}W_{jk}\big{)}\bigg{(}\big{|}W_{jk}\big{|}-\frac{2(1-p) \lambda}{1+e^{\alpha(|W_{jk}|-\lambda)^{n}}}\bigg{)}}{W_{jk}}\\ &=1-\lim\limits_{W_{jk}\rightarrow+\infty}\frac{\frac{2(1-p) \lambda}{1+e^{\alpha(|W_{jk}|-\lambda)^{n}}}}{W_{jk}}=1\end{split} \] (11) when \(W_{jk}\rightarrow-\infty\) : \[ \begin{split}\lim\limits_{W_{jk}\rightarrow-\infty}\frac{ \widetilde{W}_{j,k}}{W_{jk}}&=\lim\limits_{W_{jk}\rightarrow- \infty}\frac{sign\big{(}W_{jk}\big{)}\bigg{(}\big{|}W_{jk}\big{|}-\frac{2(1-p) \lambda}{1+e^{\alpha(|W_{jk}|-\lambda)^{n}}}\bigg{)}}{W_{jk}}\\ &=-\lim\limits_{W_{jk}\rightarrow-\infty}\frac{\bigg{(}\big{|}W _{jk}\big{|}-\frac{2(1-p)\lambda}{1+e^{\alpha(|W_{jk}|-\lambda)^{n}}}\bigg{)}} {W_{jk}}=1-\lim\limits_{W_{jk}\rightarrow-\infty}\frac{\frac{2(1-p)\lambda}{1+ e^{\alpha(|W_{jk}|-\lambda)^{n}}}}{W_{jk}}=1\end{split} \] (12) when \(W_{jk}\rightarrow\infty\) : \[ \begin{split}\lim\limits_{W_{jk}\rightarrow\infty}(\widetilde{W}_{j,k}-W_{j,k})\\ &=\lim\limits_{W_{jk}\rightarrow\infty}\bigg{[}sign\big{(}W_{j,k}\big{)}\bigg{(}\big{|}W_{jk}\big{|}-\frac{2(1-p)\lambda}{1+e^{\alpha(|W_{jk }|-\lambda)^{n}}}\bigg{)}-W_{j,k}\bigg{]}\\ &=\lim\limits_{W_{jk}\rightarrow\infty}\Big{[}sign\big{(}W_{j,k}\big{)}\Big{|}W_{j,k}\Big{|}-W_{j,k}\Big{]}=0\end{split} \] (13) Thus, the asymptote of the improved threshold function is \(\widetilde{W}_{j,k}=W_{j,k}\). * The deviation analysis of the function \[ \begin{split}\lim\limits_{W_{jk}\rightarrow+\infty}(\widetilde{W }_{j,k}-W_{j,k})\\ &=\lim\limits_{W_{jk}\rightarrow+\infty}\bigg{[}sign\big{(}W_{j,k}\big{)}\bigg{(}\big{|}W_{j,k}\big{|}-\frac{2(1-p)\lambda}{1+e^{\alpha(|W_{jk }|-\lambda)^{n}}}\bigg{)}-W_{j,k}\bigg{]}\\ &=\lim\limits_{W_{jk}\rightarrow+\infty}\bigg{[}\bigg{(}\big{|}W _{j,k}\big{|}-\frac{2(1-p)\lambda}{1+e^{\alpha(|W_{jk}|-\lambda)^{n}}}\bigg{)} -W_{j,k}\bigg{]}\\ &=\lim\limits_{W_{jk}\rightarrow+\infty}\Big{(}W_{jk}-W_{j,k} \Big{)}=0\end{split} \] (14)\[ \begin{split}\lim_{W_{j,k}\rightarrow-\infty}(\widetilde{W}_{j,k}-W_{j,k})\\ =\lim_{W_{j,k}\rightarrow-\infty}\biggl{[}sign\Big{(}W_{j,k} \Big{)}\biggl{(}\Big{|}W_{j,k}\Big{|}-\frac{2(1-\rho)\lambda}{1+\varepsilon^{ (|W_{j}|k|^{-\lambda)^{n}}}}\biggr{)}-W_{j,k}\biggr{]}\\ =\lim_{W_{j,k}\rightarrow-\infty}\biggl{[}-\biggl{(}\Big{|}W_{j, k}\Big{|}-\frac{2(1-\rho)\lambda}{1+\varepsilon^{(|W_{j}|k|^{-\lambda)^{n}}}} \biggr{)}-W_{j,k}\biggr{]}\\ =\lim_{W_{j,k}\rightarrow-\infty}\biggl{(}-\Big{|}W_{j,k}\Big{|} -W_{j,k}\biggr{)}=\lim_{W_{j,k}\rightarrow-\infty}\Bigl{(}W_{j,k}-W_{j,k} \Bigr{)}=0\end{split}  \tag{15}\]  
Therefore, \(\lim_{W_{j,k}\rightarrow+\infty}(\widetilde{W}_{j,k}-W_{j,k})=\lim_{W_{j,k} \rightarrow-\infty}(\widetilde{W}_{j,k}-W_{j,k})\)  
The deviation between \(\widetilde{W}_{j,k}\) and \(W_{j,k}\) will gradually decrease when \(\widetilde{W}_{j,k}\rightarrow\infty\), which indicates that the improved threshold function overcomes the deviation of the traditional threshold function.