## 2 Adaptive PCNN Combined with GWO and BEMD
### The Proposed BEMD-GWO-PCNN Algorithm  
Unlike most artificial neural networks, the PCNN does not require training at all. The PCNN parameters have a critical effect on denoising performance, which means that the PCNN limits itself considerably. We have engaged in numerous experiments to explore the parameter optimization problem. As we discovered previously, \(\alpha_{T}\), \(\beta_{T}\) and \(\upsilon_{T}\) are the key parameters for a guaranteed denoising effect. The GWO therefore comes in handy in meeting the exact parameter optimization demands. The GWO algorithm begins optimization by generating a set of random solutions to act as the first population. The optimized \(\alpha\), \(\beta\) and \(\delta\) solutions are saved. Through encircling the prey, hunting, attacking and searching for the prey, the optimized parameters go through exploration iterations and avoidance of local optima. Using a combination of the PCNN effect assessment with the optimization criteria of GWO, the fitness function [17] is given as shown below:  
\[h=10\cdot log\bigg{(}\frac{255^{2}}{\theta}\bigg{)} \tag{19}\]  
\[\theta=10\cdot log\bigg{[}\frac{\|S-Y\|^{2}}{M\cdot N}\bigg{]} \tag{20}\]  
where \(h\) is the fitness value acting as the judgement criterion; \(\theta\) is regarded as the mean square error value; \(S\) is the raw input image; \(Y\) is the noise-reduced image; \(\|S-Y\|\) is the \(p\)-norm of the matrix \(S-Y\), and \(p\) = 2. \(M\) and \(N\) are the dimensions of the original image. Each set of optimized parameters must be able to withstand the trials of the objective function. The set of parameters with the highest function fitness values is obtained after the screening process. BEMD can be applied to obtain a series of mode components. The valid components should be reserved as far as possible and heavy noise disturbances, such as salt and pepper noise, must be suppressed effectively. The BIMF, and residue obtained, all pass through the noise reduction process via the PCNN filter with optimized parameters. In this paper, the peak signal-to-noise ratio (PSNR), the mutual information (MI), the structural similarity (SSIM), and the standard deviation (STD) [21] are used as the evaluation indexes and are defined as follows:  
\[MSE=\frac{1}{m\cdot n}\sum_{l=0}^{m-1}\sum_{j=0}^{n-1}[I(i,j)-K(i,j)]^{2} \tag{21}\]  
\[PSNR=10\cdot log_{10}\bigg{[}\frac{MAX_{l}{}^{2}}{MSE}\bigg{]} \tag{22}\]  
where \(M\) and \(N\) are the dimensions of the original image; \(I(i,j)\) is the raw image; and \(K(i,j)\) is the noise-reduced image  
\[SSIM(x,y)=\frac{\big{(}2\mu_{x}\mu_{y}+c_{1}\big{)}\big{(}2\sigma_{xy}+c_{2} \big{)}}{\big{(}\mu_{x}{}^{2}+\mu_{y}{}^{2}+c_{1}\big{)}\big{(}\mu_{x}{}^{2}+ \mu_{y}{}  
assessment. To evaluate the optimization algorithms more comprehensively, Table 1 shows the best fitness values (BFV) through a range of different iterations from 20 to 500, along with the run time (T). In terms of run time, the GWO has a higher speed. The GACA iteration speed is relatively slow because of the larger influence of the combination factor. From the fitness values obtained, the ordering rule for these methods generally follows GWO > PSO > GACA. A table containing Algorithm 3 for the proposed hybrid image denoising algorithm, when the adaptive PCNN is combined with the GWO and BEMD, is shown below.