# Using an Artificial _Physarum polycephalum_ Colony for Threshold Image Segmentation
## 3 Methodology
### Symbol Definitions  
In our proposed model, each _Physarum polycephalum_ hyphae is a feasible solution and is evaluated by a fitness function, which determines the searching and learning of hyphae and the optimization process of threshold image segmentation. The proposed model employs [PERSON]'s entropy [13] to calculate the fitness function and variance for _Physarum polycephalum_ colony to improve the segmentation performance. The symbol definition is shown in Table 1.  
In threshold image segmentation, it processes an original image \(O\) into a segmented image \(A\) with a grayscale level \(L\). Both images have the same image size \(H\) in the horizontal direction and the same image size \(V\) in the vertical direction. Hence, the total number of pixels on the original image is \(N=HV\). The threshold segmentation can be implemented by an artificial _Physarum polycephalum_ colony with a population size of \(S\), and instructed by a fitness function _fit_ based on Kapur's entropy of \(K\). The segmentation of image \(A\) can be  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & \multicolumn{1}{c}{**Definition**} \\ \hline \(O\) & An original image \\ \(A\) & A segmented image with additive noise \\ \(B\) & A benchmark image \\ \(H\) & The image size in the horizontal direction \\ \(V\) & The image size in the vertical direction \\ \(N\) & The total number of pixels on the original image \\ \(L\) & The grayscale level of an image \\ \(a_{ij}\) & A pixel with a pair of coordinates \((i,j)\) on a gray image \(A\) \\ \(\mu_{A}\) & The average gray level of a gray image \(A\) \\ \(\sigma_{A}\) & The standard deviation of a gray image \(A\) \\ \(A_{ij}\) & The intensity value on each pixel with a pair of coordinates \((i,j)\) \\ \(K\) & Kapur’s entropy \\ \(K_{i}\) & Kapur’s entropy on a segmentation area \\ \(w_{i}\) & The possibility for each segmentation cluster \\ \(p_{i}\) & The possibility of the occurrence of pixels with a gray value \(i\) \\ \(t\) & A set of thresholds \(t=\{t_{1},t_{2},t_{3},\ldots,t_{n}\}\) \\ \(M\) & The population size of _Physarum polycephalum_ \\ \(m\) & The number of _Physarum polycephalum_ \\ \(S\) & The population size of hyphae \\ \(k\) & The hyphae number \\ \(p_{s}\) & The possibility of social learning \\ \(p_{f}\) & The possibility of free learning \\ \(x^{k}(Ite)\) & A solution of the hyphae \(k\) at the iteration of _Ite_ \\ \(x^{k}_{self}(Ite)\) & The self-learning hyphae \(k\) at the iteration of _Ite_ \\ \(x^{k}_{social}(Ite)\) & The social-learning hyphae \(k\) at the iteration of _Ite_ \\ \(x^{n}_{free}(Ite)\) & The free-learning hyphae \(k\) at the iteration of _Ite_ \\ _fit_ & A fitness function \\ _Ite_ & An iterative computation counter \\ _Ite_ \(\_{max}\) & The maximization of the iterative computation counter \\ \(\epsilon_{th}\) & The error threshold of iterative computation \\ _MSE_ & The mean squared error \\ _PSNR_ & The peak signal-to-noise ratio \\ _CorCoe_ & The correlation coefficient between two images \\ _Jaccard_ & Jaccard’s index, also called Intersection-Over-Union, IoU \\ _Dice_ & The Sørensen Dice coefficient \\ \(SSIM\) & The structural similarity index \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbol definitions.  
compared with the benchmark image \(B\), where a pixel \(a_{ij}\) has a pair of coordinates \((i,j)\) on the gray image \(A\) and a pixel \(b_{ij}\) has a pair of coordinates \((i,j)\) on the benchmark image \(B\).