## 2 Adaptive PCNN Combined with GWO and BEMD
### Theories of the BEMD  
As a data driven method, the BEMD is developed based on envelope analysis and a sifting process, and can decompose complex signals into a number of components that have the local characteristics of the input source image [29]. The number of BIMFs is determined by the image data itself, where different images correspond to different numbers of BIMFs after decomposition. A BIMF must meet two constraints: (1) the local mean of the original bidimensional signals must be symmetrical and the mean is zero; and (2) the maximum of the BIMF is positive, while the minimum is negative [30]. The local BIMF frequencies range from high to low order when they are extracted layer-by-layer. The image signal energy is mainly concentrated in the low-frequency region; in contrast, the interfering noise is mostly located in the high-frequency region. The local BIMF frequencies include strong physical contrast information, including highlight edges of the source images, and the line and zone boundaries [31]. Each BIMF component contains different frequency coefficients, and the high--frequency coefficients are in a lower order. The residue function image contains the low-frequency information of the source image, for which the grey-level distribution and the gradient information are smooth. The BEMD method can then effectively extract the various details and edges of the source image. From the perspective of the primary function, the BEMD can generate different primary functions \(g_{i}\) adaptively, corresponding to different source image characteristics. BEMD differs from Fourier transforms and wavelet decomposition, which have functions \(g_{is}\) that are predetermined [32]. In this work, we have adapted BEMD to decompose a raw image into several BIMFs and residue images through an interpolation and envelope operation. The image BEMD decomposition result \(f(x_{1},y_{1})\) of \(u\times v\) neighboring pixels is as shown below:  
\[f(x_{1},y_{1})=\sum_{l=1}^{v}c_{l}(x_{1},y_{1})+r_{v}(x_{1},y_{1}) \tag{2}\]  
where \(c_{l}(x_{1},y_{1})\) represents the decomposed \(l\)-th components, and \(r_{v}(x_{1},y_{1})\) is the residue or trend term. The basic decomposition process steps of BEMD are as follows:  
1. Initialize the raw image and set the pending image \(f(x_{2},y_{2})\), where \(r_{l}(x_{2},y_{2})=f(x_{2},y_{2})\). 2. If \(z=1\), set \(h_{lz}(x_{2},y_{2})=r_{l}(x_{2},y_{2})\); otherwise, set \(h_{l+1}(x_{2},y_{2})=r_{l}\), and \(l=l+1\). 3. Calculate the maximum spectrum value \(p_{lz}\) and the minimum spectrum values \(q_{lz}\) corresponding to \(v_{lz}(x_{2},y_{2})\). 4. Interpolate the calculated maximum spectrum \(p_{lz}\) and the minimum spectrum values \(q_{lz}\) draw the upper envelope \(u_{max}(x_{2},y_{2})\) of \(h_{lz}(x_{2},y_{2})\), and draw the lower envelop \(u_{min}(x_{2}y_{2})\) of \(h_{lz}(x_{2},y_{2})\). 5. Calculate the mean envelope \(v_{lz}(x_{2},y_{2})\) of \(h_{lz}(x_{2},y_{2})\) using the following formula: \[v_{lz}(x_{2},y_{2})=\frac{u_{max}(x_{2},y_{2})+n_{min}(x_{2},y_{2})}{2}\] (3)
6. Set \(h_{l(z+1)}(x_{2},y_{2})\) to be the partial information of the source image, which can be calculated using (4): \[h_{l(z+1)}(x_{2},y_{2})=h_{lz}(x_{2},y_{2})-v_{lz}(x_{2},y_{2})\] (4)
7. Check whether the characteristics of \(h_{l(z+1)}(x_{2},y_{2})\) satisfy the BIMF properties; use the standard deviation (STD) between \(h_{z+1}\) and \(h_{z}\) as the stopping criterion from (5). It can be used to determine whether \(h_{l(z+1)}(x_{2},y_{2})\) meets the BIMF characteristics from two consecutive sifting results. Limitation of the size of the STD allows a criterion for stopping of the sifting process to be determined. From prior experience, the STD value can generally be set in the 0.2-0.3 range. \[SD=\frac{\sum_{x_{2}=1}^{u}\sum_{y_{2}=1}^{v}\left|h_{l(z+1)}(x_{2},y_{2})-h_ {lz}(x_{2},y_{2})\right|^{2}}{\sum_{x_{2}=1}^{u}\sum_{y_{2}=1}^{v}\left|h_{lz} (x_{2},y_{2})\right|}\] (5)
8.