## 5 Experiment and Application
### Experiment with Speckle Noise  
Speckle noise, which is known as multiplicative noise, added to original images forms noisy images. It can be described as: \(g(x,\,y)=f(x,\,y)\cdot n(x,\,y)\), in which \(n(x,y)\) is the multiplicative noise. The original images were polluted by Speckle noise using Matlab function IMNOISE in the experiment. Different noise variances were set to pollute the original image, and different methods were utilized to compare the denoising performances. Denoising results are listed in Tables 5-7. Figure 16: Denoising results of Peppers with Gaussian noise using IFOA-DTCWT-BF. (**A**) Peppers of noise \(\sigma\) = 0.01; (**B**) Peppers of noise \(\sigma\) = 0.05; (**C**) Peppers of noise \(\sigma\) = 0.1; (**D**) Peppers of noise \(\sigma\) = 0.2; (**E**) Denoised image of \(\sigma\) = 0.01; (**F**) Denoised image of \(\sigma\) = 0.05; (**G**) Denoised image of \(\sigma\) = 0.1; (**H**) Denoised image of \(\sigma\) = 0.2. Figure 15: Denoising results of Columbia with Gaussian noise using IFOA-DTCWT-BF. (**A**) Columbia of noise \(\sigma\) = 0.01; (**B**) Columbia of noise \(\sigma\) = 0.05; (**C**) Columbia of noise \(\sigma\) = 0.1; (**D**) Columbia of noise \(\sigma\) = 0.2; (**E**) Denoised image of \(\sigma\) = 0.01; (**F**) Denoised image of \(\sigma\) = 0.05; (**G**) Denoised image of \(\sigma\) = 0.1; (**H**) Denoised image of \(\sigma\) = 0.2. As shown in Tables 5-7, when the noise variance of Speckle noise was 0.01, the DTCWT, BM3D, Wiener and bilateral filter did good jobs. However, the proposed method performed much better than other methods when removing the Speckle noise of big noise variances. The results indicated that the proposed method was insensitive to the Speckle noise. In other words, it can be adopted in any condition. All denoised images are shown in Figures 17-19. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**MethodsParameters**} & \multicolumn{2}{c}{\(\sigma\) = 0.01} & \multicolumn{2}{c}{\(\sigma\) = 0.05} & \multicolumn{2}{c}{\(\sigma\) = 0.1} & \multicolumn{2}{c}{\(\sigma\) = 0.2} \\ \cline{2-9}  & **PSNR** & **r** & **PSNR** & **r** & **PSNR** & **r** & **PSNR** & **r** \\ \hline Noised & 28.57 & 0.9782 & 21.68 & 0.9046 & 18.75 & 0.8313 & 15.93 & 0.7250 \\ Median & 29.27 & 0.9812 & 26.95 & 0.9678 & 25.35 & 0.9531 & 23.37 & 0.9260 \\ Wiener & 32.09 & 0.9899 & 26.16 & 0.9615 & 23.34 & 0.9283 & 20.72 & 0.8742 \\ WDF & 28.05 & 0.9743 & 27.22 & 0.9688 & 26.26 & 0.9610 & 24.93 & 0.9468 \\ Bilateral & 31.20 & 0.9879 & 25.01 & 0.9495 & 21.09 & 0.8861 & 17.47 & 0.7777 \\ BM3D & 33.08 & 0.9920 & 29.70 & 0.9825 & 24.13 & 0.9397 & 18.99 & 0.8292 \\ DTCWT & 31.55 & 0.9888 & 25.10 & 0.9517 & 22.40 & 0.9142 & 19.95 & 0.8554 \\ IFOA-DTCWT-BF & 28.21 & 0.9759 & 27.81 & 0.9735 & 27.29 & 0.9701 & 26.35 & 0.9630 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the six methods (Columbia with Speckle noise). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**MethodsParameters**} & \multicolumn{2}{c}{\(\sigma\) = 0.01} & \multicolumn{2}{c}{\(\sigma\) = 0.05} & \multicolumn{2}{c}{\(\sigma\) = 0.1} & \multicolumn{2}{c}{\(\sigma\) = 0.2} \\ \cline{2-9}  & **PSNR** & **r** & **PSNR** & **r** & **PSNR** & **r** & **PSNR** & **r** \\ \hline Noised & 25.76 & 0.9715 & 18.89 & 0.8795 & 16.09 & 0.7965 & 13.42 & 0.6788 \\ Median & 30.14 & 0.9892 & 26.19 & 0.9678 & 23.81 & 0.9548 & 21.24 & 0.9212 \\ Wiener & 31.51 & 0.9921 & 24.74 & 0.9632 & 22.03 & 0.9325 & 19.43 & 0.8792 \\ WDF & 28.04 & 0.9822 & 26.52 & 0.9748 & 25.22 & 0.9659 & 23.18 & 0.9459 \\ Bilateral & 31.95 & 0.9928 & 22.01 & 0.9338 & 17.81 & 0.8464 & 14.34 & 0.7147 \\ BM3D & 33.12 & 0.9945 & 26.20 & 0.9734 & 19.68 & 0.8918 & 15.28 & 0.7518 \\ DTCWT & 30.73 & 0.9905 & 24.40 & 0.9604 & 22.23 & 0.9351 & 20.13 & 0.8948 \\ IFOA-DTCWT-BF & 28.66 & 0.9848 & 28.01 & 0.9823 & 27.14 & 0.9786 & 25.45 & 0.9703 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of the six methods (Peppers with Speckle noise).